My daughter has this Articles of Faith page hanging on her cork board so she can always see it!
Thanks Renee! So glad you could use them for your girls!
How sweet are you Karin? You are very welcome.
Thank you, Thank you, Thank you! I've only just found your blog, but I've printed almost 20 pages in under 10 mins! All of your Articles of Faith printables are so beautiful! I can't wait to discover other beautiful things on your blog to share with my family as I teach them the gospel! Thank you for sharing your talents!
Amanda, Thank YOU for taking the time to comment. It makes my day!
This is great. Thanks so much for sharing!
Thank you! Just printed this for my kids!
I hope your kids enjoy them! You're welcome.
Thanks so much! This is so cute! Love it! Using it for my Activity Days girls!
I'm so happy to receive comments like this. I hope your Activity Days girls love them and are inspired to memorize the articles of faith!
You're very welcome and congratulations on your son's baptism. I bet it was a beautiful day for your family!


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Lakeland Community College
Athletic Event Jobs
Equity in Athletics 2020
Athletic and Fitness Center
Lakeland Foundation
Lakeland loses a pair of games to Patrick Henry by a total of two runs, 10-9 and 10-9
Patrick Henry Community College
Lakeland Community College (2-3) 1 0 2 0 4 0 2 9 10 1
Patrick Henry Community College (0-0) 5 0 0 0 2 1 2 10 12 0
Patrick Henry Community College (0-0) 0 1 2 0 7 0 X 10 15 2
2B: Cameron Malkamaki
3B: none
HR: Kallista Hribar; Justine Barnes 2
2B: Kayleigh Krause; Sydney Viar; Taylor Conklin
HR: Kayleigh Krause; Meagan Puckett
2B: Kallista Hribar; Elizabeth Montgomery; Justine Barnes; Cameron Malkamaki
3B: Kallista Hribar
HR: Elizabeth Montgomery
HR: Brandi Woods
MARTINSVILLE, Va. – Lakeland softball could not have been closer in both games of a doubleheader against Patrick Henry Community College but the Lakers were unable to keep late leads and lost but games by the score of 10-9 Saturday afternoon.
With the pair of losses, Lakeland falls to 2-4 overall after a 0-4 road trip this weekend.
Patrick Henry 10 – Lakeland 9
Lakeland held a 9-8 lead with three outs remaining but Patrick Henry was able to add two runs in the bottom of the seventh to earn the 10-9 win to open the doubleheader.
Cameron Malkamaki turned in a 2-for-4 game with a double and a single while bringing in three runs. Justine Barnes belted two solo home runs while Kallista Hribar added a two-run shot. Dana Turchon added a pair of singles and brought in a run as well.
Lakeland was first on the scoreboard in the opening inning when Barnes belted a solo home run with two outs to left for a 1-0 lead. Patrick Henry came back with five runs in the bottom of the first to take a 5-1 lead after an inning.
The Lakers used the long ball to get back into the game as Abby Takacs led off with a walk and then Hribar followed two batters later with a home run to right to get within 5-3 in the third.
Lakeland took the lead away in the fifth inning and it started with a single from Jamie Matso and a walk from Hribar and Barnes. Malkamaki cleared the bases with a double to center for a 6-5 lead. Turchon followed with an RBI single to left for a 7-5 advantage. Patrick Henry tied the game up in the bottom half with a pair of runs to knot the game at 7-7 and then took an 8-7 lead with a single run in the sixth.
Down to just two outs in the seventh, Barnes nailed a solo home run to center to tie the game at 8-8. The Lakers kept going in the inning and loaded the bases and Takacs brought in a run with a bases loaded walk for a 9-8 lead. Unfortunately, the lead did not hold as Patrick Henry scored two runs in the bottom of the seventh to steal away the win.
Lakeland did not lose the second game in the final inning but it was giving up seven runs in the fifth that did the Lakers in as Patrick Henry secured a 10-9 victory.
Hribar hit 3-for-4 in the game with a triple, double and a single while bringing in two runs. Hribar was just a home run shy of hitting for the cycle. Elizabeth Montgomery belted a home run and a double to bring in four runs. Barnes, Malkamaki and Cassidy Yarnell each had an RBI in the loss.
Takacs had a decent start as she went into the fourth inning and gave up one earned run in the no decision.
Lakeland again struck first in the opening inning when Hribar led off with a double and Malkamaki hit an RBI double to right for a 1-0 lead. Patrick Henry tied the game up in the second with a single run and then took a 3-1 lead in the third with two runs.
The Lakers cut into the lead in the fourth as Barnes started the inning with a walk and then was brought in with a single by Yarnell to center to get within 3-2.
Lakeland went after the lead in the fifth as Nicole Ulogar singled to begin the inning and then was brought in on a single by Hribar to tie the game at 3-3. Lakeland wasn't done as Montgomery brought in two with a double to left for a 5-3 lead. Barnes added a run with a double to left to make the score 6-3. Patrick Henry scored seven in the fifth to take a 10-6 lead.
The Lakers still showed some fight in the sixth as Ulogar singled with one out and was brought in with a triple to right to get within 10-7. Montgomery delivered a big blow with a home run to left to get within 10-9 but that would be as close as the Lakers would get as Patrick Henry held on for the win.
Lakeland will be on the road on Wednesday, March 10 with a short trip to face Notre Dame College for a doubleheader beginning at 4 p.m.
For more information about Lakeland athletics, visit www.athletics.lakelandcc.edu.
About Lakeland Community College
Founded in 1967, Lakeland Community College was the first college in Ohio created by a vote of the people. True to our mission, Lakeland provides quality learning opportunities to meet the social and economic needs of the community.
Offering day, evening, weekend and online classes, the college serves more than 10,800 credit and 2,800 noncredit students (2016). For more information about the College, visit www.lakelandcc.edu.
Lakeland Community College7700 Clocktower Dr | Kirtland, Ohio 44094440.525.7702

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Photos from the Canadian Atheist Bus Campaign.
The Freethought Association of Canada, who solicited donations to pay for the ads and who received approval from the TTC and CBS Outdoor for them last month, told Torontoist that the ads started appearing on buses about a week ago, but that the organization was only able to catch and photograph one of the busses this past weekend (above). And, sure, the United Church has already issued its response, and in Calgary the Islamic Supreme Council of Canada is planning to book bus ads of their own to counter the impending atheist ones. But creating a dialogue is entirely the point, and the Atheist Bus folk are obviously happy and relieved that their busses are finally out in the wild somewhere, writing on their website that "it's nice to finally see this in Canada isn't it?" That'll have to be the last word; God, unfortunately, could not be reached for comment.


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Mathematics Points the Way to a Perfect Head of Beer
Simple formula may lead to a host of improved materials—and the perfect pour
By JR Minkel on April 25, 2007
Researchers may have cracked the code for the perfect head on a glass of beer, and perhaps much more in the process. The key lies in a long sought equation for the growth and shrinkage of individual bubbles in foam and crystalline grains in metals, semiconductors and other materials.
The finding extends a formula that specifies how the area of two-dimensional shapes will change, discovered in part by famed mathematician John von Neumann in 1952. Researchers say the new math may help improve a wide range of industrial processes, from treating metals with heat to controlling the amount of foam in poured beer.
Metals, foams and multicellular organisms are all mosaics of microscopic spaces or domains that jostle with each other, grow or shrink, cave in or bulge out. The driving force behind this evolution is surface tension, the same property that lets a bug sit on water and draws liquid up a narrow straw.
According to the new equation, the change in volume of such a tension-driven domain is essentially the sum of the lengths of the domain's edges (imagine a honeycomb) minus six times the mean width of the domain, all multiplied by a constant that is particular to the material in question.
The key to the discovery was applying the pure math concept of mean width, which is trickier to measure than its cousins—surface area and volume, says materials scientist David Srolovitz of Yeshiva University in New York City, who, along with mathematician Robert MacPherson of the Institute for Advanced Study in Princeton, N.J., published the finding online today in Nature.
"It's exhilarating," Srolovitz says. "I've always found this problem very sexy." He says he does not know where it will be applied, but "the ideas are so general it's going to really change the way we think about geometric objects."
"It's very universal. It will touch everything" in materials design, says mathematician David Kinderlehrer of Carnegie Mellon University in Pittsburgh, who studies materials. He predicts it may lead to longer lasting, more efficient materials for everything from airplane wings to nuclear reactors to microprocessors.
For that to happen, researchers must learn to crunch numbers on groups of domains. That is no problem in two dimensions, but the 3-D case adds a new twist, Kinderlehrer says, because the domains have more edges that can shorten or lengthen affecting their neighbors.
"It's a very complicated type of evolution," he says. "It's going to be much harder to figure out how the network behaves."
JR Minkel was a news reporter for Scientific American.
Recent Articles by JR Minkel
7 Radical Energy Solutions
More Vitamin D Could Prevent Some Psychosis
Student Surveys Contradict Claims of Evolved Sex Differences
Two-Second Drafts
Brenda Goodman
The Science of Bubbly
Gérard Liger-Belair


What would you do if you had unlimited time and money? Me? I'd be constantly on the move. I'd visit all the places I've ever wanted to see, knitting all the way.
I'd also see a lot of bands play live. But hey, I can do that anyway. Next week I'm heading over to the Toad Tavern (formerly the Horny Toad until the local residents complained about the name) to see L'il Ed and the Blues Imperials.
Check out the hat. Doesn't that look like a fun one? Now, if only I had the time to make one, perhaps a good felting + needle felting project? Just kidding. I'm not a felting fan. Oh well.
I think I'll take along a sock project to work on before the performance. Anyone else going? If so, bring your knitting and I'll see you there!
Right there with you. Lots of travel and knitting. Reading too. I'd park myself by the beach or a Colorado stream and knit up a storm.


Michaela is a licensed Esthetician who graduated from The Salon Professional Academy in 2017. She is always continuing her education and learning new techniques. She has been doing makeup since 2012 and she specializes in Wedding & Event makeup.


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Museums in the district
Guides and Catalogues
Church and Convent
of San Bonaventura
in Bosco ai Frati
According to sources, the first foundation of a chapel dates back to the 7th century. The original complex, whose first patrons were the Ubaldini family, was independent until was it donated in 1212 to the Franciscans on the occasion of the saint's passage throughj Borgo San Lorenzo. It was included in the parish of San Michele al Bosco and was later named San Francesco al Lago or al Bosco.
Saint Bonaventure of Bagnoregio, general of the Franciscan Order, was there some years later and received in 1273 the cardinal investiture by Pope Gregory X there. His exemplum virtutis is famous: the friar, who was caught washing the dishes by the papal legates sent to deliver his cardinal's insignias, did not stop working and asked them to leave the paraments on a small cornel tree nearby.
The 1349 plague led the Franciscans to abandon the convent in a period when the Ubaldini's power was slowly dying while that of the Medici family, originally from the Mugello, was increasingly asserting itself. In fact, they had extensive properties in the area, including the large fortress-villas of Trebbio and Cafaggiolo.
Around the 1420s, Cosimo the Elder purchased the land where the religious complex still stands today with the surrounding woods. Having obtained its patronage, he began modernization works. The project was entrusted to Michelozzo, together with works for Trebbio and Cafaggiolo, as attested by Vasari's biography dedicated to the famous architect.
In this simple and essential architectural complex, with a penchant for the classical canon that confirms the choices of the Medici patrons, there was a decorative device that visitors today must recreate using their imagination as it was altered by the later renovations.
The main altar was decorated with the wonderful altarpiece by Fra Angelico, known exactly as "the Bosco ai Frati altarpiece", depicting the Madonna and Child among saints, that was first transferred to the Uffizi Gallery by Pietro Leopoldo and is now found in the San Marco Museum in Florence.
On one of the altars was also found the famous triptych with the Resurrection of Lazarus by Nicolas Froment, dated 1461 and housed today in the Uffizi Gallery. Also the Donatellian-school Crucifix dates back to the same years. It is the only surviving work from the subsequent expropriations and was rediscovered by Alessandro Parronchi in the church's crypt.
The Crucifix is found in the hall which was the former chapterhouse and now houses the collection of sacred art that includes some 16th-and 17th-century paintings as well as sacred furnishings and vestments.
The loggia overlooking the garden, where the tiny chapel of Saint Bonaventure opens, closes the splendid and striking tour.
I tre capolavori di Donatello, in Primo Rinascimento in Santa Croce. Beccherucci L., Firenze, Editore: Città di Vita Edizioni, 1962.
Guida del Mugello e della Val di Sieve. Niccolai F., Borgo San Lorenzo, Multigrafiche Editrice, 1914.
Il Crocifisso del Bosco ai Frati di fronte ai modelli di Donatello e di Brunelleschi, in Mugello culla del Rinascimento. Cagliotti F., Firenze, Editore: Polistampa, 2008, note: catalogo della mostra.
Giuliano Ughi della Cavallina, Relazione del convento del Bosco di Mugello. Romby G.C., San Piero a Sieve, Editore: Comune di San Piero a Sieve, 1984.
Il Crocifisso del Bosco, in Scritti di storia dell'arte in onore di Mario Salmi, a cura di Martinelli V. Parronchi A., Roma, Editore: De Luca, 1962.
Date: ca. 1520-1530
Technique and Materials: oil on a wooden panel
Attributed to: Antonio del Ceraiolo (active between 1520 and 1538)
The panel painting is found on the left wall of the nave. Against a background with an arcade opening on a distant landscape, the angel and the Virgin are in affectionate conversation while, above to the left, the benedictory Father appears. The devout charm of the poses and a luminosity of color are to appreciated.
Allegory of the Cord of Saint Francis
Date: 1586 (signed and dated)
Technique and Materials: oil on canvas
Artist: Jacopo Ligozzi (Verona 1547 – Florence 1627)
This painting follows an iconography that was very dear to the Franciscans who wanted to disseminate images of the saint's life for didactic and moralizing purposes. Francis appears in the act of handing the cord of his robe to the Church, which is personified by the female figure on the throne. There is a crowd of figures at her feet, among whom is the pope, awaiting the holy cord.
The Madonna Giving her Child to Saint Francis
Date: end of the 16th century
Production: 16th-century Italian production
Found on the left wall of the nave, the strongly devotional painting is characterized by a simple and clear subject. The rich and bright chromatic range and the delicate brush strokes in the painting of the clothes and faces place it near the style of those artists moving between Tuscany and Rome towards the end of the century, like Ciampelli, Commodi or Giovanni de' Vecchi.
Madonna and Child with Saints
Date: post-1450-1452
Technique and Materials: tempera on a wooden panel
Size: 174 cm. x 173 cm.
Artist: Guido di Piero, later called Fra Giovanni da Fiesole, known as Fra Angelico (Vicchio di Mugello, documented in Florence from 1417 – Rome 1455)
Provenance: Convent of San Bonaventura in Bosco ai Frati
Collection: Florence, San Marco Museum
The painting, housed in the San Marco Museum, was commissioned by Cosimo the Elder for the Church of Bosco ai Frati. The Madonna with Child sits on a bench, a reference to Franciscan simplicity and Mary's humilitas, among the saints Francis, Louis of Toulouse, and Anthony as well as Peter, Cosmas and Damian, the Medici family protectors. The Child clutches a pomegranate, symbol of the Church. the seeds are the fruits as all the faithful are tied to the brothers in a single family.
Saint Catherine of Alexandria
Date: ca.1620-1630
Attributed to: Pietro Confortini (first half of the 17th century)
The artist Pietro Confortini, the brother of the more famous Jacopo, left many of his works in the Mugello where he was active in the 1620s. This painting, found in the room of the Crucifix, depicts the saint with the usual wheel of her martyrdom.
Technique and Materials: oil on a curved canvas
Artist: Ludovico Cardi, known as Cingoli
Collection: Convent of San Bonaventura in Bosco ai frati
On the left wall, as you enter the church, this curved painting centers on the figure of Christ in the foreground, supported by Saint John and the Virgin. Behind them, Joseph of Arimathea holds the oils for the burial rite while the Magdalene is seen from behind at Jesus's feet.
Wooden Crucifix
Date: ca.1460
Technique and Materials: pear wood
Attributed to: Donatello (?)
Collection: Sacred Art Collection of the Convent of San Bonaventura in Bosco ai Frati
It is a work of great fascination because of the quality of the intaglio and above all the profound expressivity of the face, captured at the tragic moment of death. Its naturalism almost seems highlighted by the damage of time that has partly altered the features. Some scholars have suggested that this crucifix ( and not the one in Santa Croce) is the one mentioned by Vasari in a famous anecdote recorded in his Lives where Donatello and Brunelleschi challenged each other around the theme of Man.
Resurrection of Lazarus
Artist: Nicolas Froment (Usèz ca.1435– Avignone ca.1438)
Collection: Florence, Uffizi Gallery
The painting's typology follows a northern painting tradition: a larger-sized central panel, depicting the Resurrection of Lazarus is flanked by two painted doors. On the left is Jesus in the house of Martha and on the right is the Washing of Feet. The triptych is currently found in the Uffizi Gallery.
Saint Helena, Empress
The painting in the room of the Crucifix, depicts the saint in half-figure. Sumptuously clothed, she holds the cross. The artist Pietro Confortini, the brother of the more famous Jacopo, left many of his works in the Mugello where he was active in the 1620s.
Church and Convent of San Bonaventura in Bosco ai Frati
at Bosco ai Frati
San Piero a Sieve
The complex and the collection of sacred art can be visited upon appointment with one of the friars.
PICCOLI GRANDI MUSEI - Discovering the Land
Via Bufalini, 6 - 50122 Firenze - c/o Ente Cassa di Risparmio di Firenze
redazione@piccoligrandimusei.it | Tel. 055 5384243
A project promoted by
© 2016 FONDAZIONE CASSA DI RISPARMIO DI FIRENZE - CF 00524310489
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Modest Music Fest brings psych-rock, R&B and folk to downtown Moscow | Music News | Spokane | The Pacific Northwest Inlander
Walters Photographers
The R&B sounds of Ural Thomas and the Pain are a highlight of this year's Modest Music Fest lineup.
It's fitting that the Moscow restaurant Humble Burger would give a name like Modest Music Fest to its signature annual event. It really tells you everything you need to know. The two-day festival, now in its third year, takes over downtown Moscow this weekend, mixing local artists with a diverse roster of out-of-town acts. Of course it's no Coachella, but that's the point, isn't it?
There's plenty to hear at Modest Music Fest, with more than 40 local and regional artists playing in venues that include not only Humble Burger but also One World Cafe and the Kenworthy Theater, among others.
Of course, there are too many good artists to list here, but we've got a rundown of some of the weekend's headliners that we're most excited to see. Go to humbleburger.com for the full festival lineup.
Ural Thomas & the Pain
If you dig old school R&B, Ural Thomas is the real deal. He's been involved with music since the '50s. He's toured with Otis Redding. He's played with the likes of Stevie Wonder and James Brown. And now his work has been given new life by a sprawling backing band known as the Pain. The longtime Portland resident has become an inextricable part of the city's scene, and songs like "Smoldering Fire" and "Pain (Is the Name of Your Game)" take you right back to the heyday of Motown. Sunday at 10:30 pm, Kenworthy Theater.
When he broke out of Portland's music scene as Horse Feathers, singer-songwriter Justin Ringle was producing the kind of hushed, acoustic folk you'd expect from… well, a Portland singer-songwriter. But with his last two albums, Ringle has been experimenting with styles and sounds. He's a little louder, too. Last year's eclectic LP Appreciation opens with the glossy, Fleetwood Mac-esque Laurel Canyon pop of "Without Applause," breezes by on the ethereal, experimental lilt of "Born in Love," and thumbs through the classic country-rock record bin on "Altamont" and "The Hex." Saturday at 11:30 pm, Mikey's Gyros.
Bryan John Applebee
Listening to Seattle singer-songwriter Bryan John Appleby's 2015 LP The Narrow Valley, it's hard not to think back to Sufjan Stevens' once-lofty goal of producing an album inspired by the histories and cultures of each of the states. The sounds and themes of the record channel the topography and ambience of the small California town where Appleby grew up: Awash in strings, brass and woodwinds, the album seems to be arranged into movements, bookended by an intro and outro and even breaking for an intermission in the middle. Sunday at 6:30 pm, Mikey's Gyros.
Sama Dams
This project formed by husband-and-wife duo Sam and Lisa Adams is known for its heartrending lyrics, lush vocals and unpredictable arrangements. The songs tend to have an experimental bent, coffee shop ballads that twist around time signatures and morph between melodies. The band's latest single, titled "Principle of Loving," is a nice synthesis of their house style: It draws you in with its warm soundscapes, flirts with synth-pop throughout, and sticks in your head. Saturday at 10:30 pm, Humble Burger; Sunday at 9:30 pm, Kenworthy Theater.
The Shivas
If music could physically shimmer, the Shivas certainly would. The Portland group's sound recalls 1960s psychedelia, freak-folk and even '70s bubblegum pop, with the occasional detour into pastoral sketches reminiscent of the Kinks or Love. But just when you think you've got 'em pigeonholed, they throw out a song like "You Make Me Wanna Die," a gem of garage-rock noise filtered through classic doo-wop, or the fuzzed-out rock 'n' roll of "Gloria." It's exciting to be whipsawed like that. Sunday at 7:15 pm, Humble Burger (And if you can't make it to Moscow, the Shivas are also playing Lucky You Lounge in Spokane on Saturday, Sept. 28). ♦
Modest Music Fest feat. Ural Thomas and the Pain, Horse Feathers and more • Sat-Sun, Sept. 28-29 • $30 • Downtown Moscow • humbleburger.com/
modest-music-fest
Tags: Bars Calendar Coeur d'Alene community newspaper drinkspotter happy hour Inland Northwest News Politics Restaurants Spokane
Barbara Sobel Of Sobel Promotions Named 'Queen Mother of Music Promotions' by Just Circuit Magazine
Faculty performs female pieces, 'Autumn Music'


Minute by minute updates from the Copenhagen climate summit provide a perspective that is both distorted and unhelpful. Watching the negotiations at COP-15 is like watching the Dow Jones at the beginning of a recession. Daily gyrations mean nothing. You have to take the long view.
Certainly momentum matters. President Obama, and other leaders in Copenhagen, has built some momentum for compromises on a climate change agreement. Recent commitments from the U.S., China, India and Brazil have produced both excitement and expectation.
The close of negotiations today is where the irresistible force of hope and expectation run smack into the immovable object of political reality and bureaucratic inertia. There was never going to be a binding political document satisfactory to 193 countries coming out of Copenhagen. That fantasy was abandoned long ago. However, what has happened over the last two weeks has been a productive use of time.
Negotiators have come to some understanding about the size of target reductions for green house gas (GHG) emissions. Calculations about the base year – whether its 1990 or 2005 – are becoming simply a matter of mathematics not politics. Most importantly, there is a clearer picture of how much money may be available for an adaptation fund for developing countries to help wean them off carbon in the same manner as developed countries. This is more than one could have expected as little as three months ago.
What ultimately matters at the COP-15 is continuing progress not a treaty. Given the present commitments from large carbon emitters and the attention that world leaders have focused on climate change, there is evidence that countries are far closer to concluding a successor agreement to the Kyoto Protocols than they have ever been.


Whenever I refer to the term minimalism I often receive backlash from people. The criticism is probably because of lack of awareness and knowledge of the concept of minimalism. Even though the concept has gained some popularity in recent past, it is surrounded with misconceptions. The most important reason why people detest the idea is because they lack awareness and the word itself makes them feel threatened.
In simple words Minimalism is living a hassle-free life. It doesn't mean denying yourself or living like a pauper. It just means you spend mindfully; it can be money or your time and energy. It forces you to have clarity, introspect and make an informed decision instead of doing what is expected or doing what everybody else is doing. Possessing only the essentials is just a small part of being minimalist. In fact letting go materialistic possessions which have no purpose or relevance in your operational life relives you from sense of anxiety and pressure. In return you have more time, energy, better finances and space to utilise which serves your purpose than otherwise.
So why be a minimalist? How is it going to simplify and enrich your life?
Owning too much materialistic stuff demands time and attention in order to ensure it maintenance. That expensive home decor, your silk rug, carpeted floor, fancy dinner set, leather jacket and many such possessions drain your energy instead of letting you enjoy and focus on what is important to you. Why have that expensive dinner set for your guests where as cheap plastic one for you to have dinner in? Would you want to spend your weekend vacuuming your floor carpet or instead would want to do something that will nourish your soul?
Decluttering reduces your options and thus saves you from confusion. Mental de-clutter helps you focus on important tasks at hand than get bothered by complete irrelevant things thus helping you make an informed decision. You can also save some time by uninstalling apps which serves you no value but drain your minutes. Having too many choices can surprisingly be exhausting. The point is not to deny yourself from anything, it is to have fewer relevant choices than flooding with irrelevant ones.
When you consume as per your need you generate less waste. Indirectly this contributes towards preserving and protecting the earth. Your little indirect contribution helps reduce the burden of mother nature.
As a minimalist you mindfully invest your time, money and energy you get to invest more of it where you truly want to. Which means instead of splashing out cash at a sale you mind put aside of a trip you always wanted to or buy something meaningful or start a business you had once drawn the blueprint of. Minimalism lifestyle will also channelize your time. You will be surprised to see how many things you can accomplish and learn with the time you have, once which you spent on the internet scrolling mindlessly.
When you opt for minimal lifestyle you automatically let go the sense of wanting to possess better than your colleague or relatives. The want is replaced with need and it is truly satisfying to live life without the burden of wanting please people.
When you decide to downsize you can sell the stuff at various websites and make some money. Your precious seating items may increase the inflow of your reserves.
Minimalism is very flexible. You do not have to follow a person and instead do what suits you. Find out what you want, figure out a suitable way and get going.


Distractions while driving can occur everyday and seem to be everywhere; this can include text messages, phone calls, a quick meal on the road, all of which interrupts a driver's attention.
Obtaining additional coverage for accidents caused by uninsured or underinsured motorist has become crucial. Excess Uninsured/Underinsured (UM) Motorist coverage is available as part of a Personal Umbrella Policy. Umbrella Liability provides protection from civil suits as a result of your actions.
The Excess UM coverage provides coverage for you and members of your household, if the other party is at fault and does not have enough liability coverage to pay for injuries and damages. UM coverage also provides coverage for you as a pedestrian or bicyclist.


Saffron not only stands out due to its rich color and aroma, but due to its medicinal qualities that have been practiced for centuries. This expensive and comes from the flower of Crocus sativus, a lavender-colored flowers that bloom each season between October and November. Saffron is used all around the world not only for its unique taste when cooking, but as a medicine since it contains high vitamins and minerals. Vitamin A, Vitamin C, folic acid, riboflavin, niacin, and manganese, copper and iron are all found in saffron and all aid in maintaining a strong immune system and cardiovascular system.
If you ever need a boost, there are plenty of ways to consume saffron to ensure you intake all of its medicinal properties. First, you should purchase the best quality saffron in threads instead of powder so that all of its medicinal properties are intact. You should also keep it in a cool and dry place to make sure that they remain intact.
You can add a pinch of saffron threads to either warm or cold water and add to your tea or recipe. To get the best out of this super food, try out our Saffron Golden Milk recipe and our Vitamin C Tea Blend!


compile failed - Unable to load module 'ScriptGeneratorPlugin' - 0 instances of that module name found.
some1 knows what's going on here?
Hi @j3lly_m3lly , have you tried closing the editor and building from Visual Studio?
btw is it possible that it may b caused by some ue4 plugins that r incompatible 2gether 4 example?
mayb my vs needs c# components?


I had a 5 week old kitten we had her for about 2 weeks but we were skeptical about keeping her due to work schedules. We called this what we thought was an animal agency in our neighborhood well a lady came and picked her up saying she was going to take care of her until she can put her up for adoption. We now found out she is trading our kitten. I texted her numerous times telling her I want the kitten back if she's just going to give her to someone else who would take her.
Generally when a person surrenders his/her animal, such person has no further rights to that animal.
PLEASE NOTE: Responses to legal inquiries are not meant to replace seeking legal advice from an attorney in your state. The materials in this website and any responses to questions are for informational purposes only and are not intended, nor should they be construed, as legal advice. This website, the information contained herein, and any responses to questions directed to this column are not intended to create and do not establish an attorney-client relationship. You should not rely or act upon any information provided on this website or in any response to your inquiry without seeking the advice of an attorney in your state regarding the facts of your specific situation.
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text:
Wilhelm Friedrich Eisenlohr (Pforzheim, 1 de janeiro de 1799 – Karlsruhe, 10 de Julho de 1872) foi um físico alemão.

Família
Filho do jurista Wilhelm Eisenlohr e de Caroline Henriette nascida Sachs.

Vida
Eisenlohr estudou matemática e física na Universidade de Heidelberg. Em 1819 foi nomeado Professor für Mathematik und Physik no liceu em Mannheim. Em 1840 assumiu o posto de professor de física no Instituto Politécnico de Karlsruhe. Conhecido por suas investigações sobre óptica. Publicou os resultados de suas pesquisas em seu Lehrbuch der Physik.

Eisenlohr fundou a primeira escola profissional de Baden em Mannheim e em 1847 uma escola de relojoeiro na Floresta Negra.

Foi membro da Gesellschaft Deutscher Naturforscher und Ärzte. Em 1869 foi membro honorário da Gesellschaft für Naturkunde in Württemberg.

Sepultado no Alter Friedhof Karlsruhe.

Publicações
Die Wellenlänge der brechbarsten Lichtstrahlen. In: Poggendorffs Annalen der Physik und Chemie, 1856.
Lehrbuch der Physik. Krais & Hoffmann, Stuttgart 8., verb. u. verm. Aufl. 1860 Digitalisierte Ausgabe der Universitäts- und Landesbibliothek Düsseldorf

Bibliografia
Nekrologe. In: Unsere Zeit, 1872, p. 707 digitalisat

Professores do Instituto de Tecnologia de Karlsruhe
Físicos da Alemanha do século XIX
Naturais de Pforzheim

redpajama_set_name: RedPajamaWikipedia


I thought I was going to make this grand statement about the swearing-in of our new President, Barack Obama. Something akin to Peter Beinart's recognition that we liberals celebrate "the promise of a future that redeems the past".
But as I watched through the events today, I noticed at times I got choked up a bit. Not at who our President is, or the occasion of now having our first African-American POTUS. But at the great sense of hope and optimism that his taking office represents.
You can see it in the crowd of 2 million people who braved the cold to watch history unfold on the steps of the Capitol, or the many who traveled from overseas to witness this event. You can see it in the smile on Ted Kennedy's face as he made the effort to witness his family's political torch pass to another young, inspirational leader. You can see it in the realization of just how fast history has moved – or, as Obama said, "the ground has shifted underneath us".
It is an awe-inspiring moment. One that is very hard not to be moved by.
I have been watching politics in America for sixteen years, since I was 11 years old. I have been waiting that entire time to find my civic faith vindicated, to find a candidate, an elected official, a President whose values, ambitions, and message perfectly fit the times we are in and where the country is going (not where the punditry is pointing and distracting us toward).
I have not always been impressed by President Obama. The long campaign against Hillary and against McCain-Palin was exhausting. And there is no doubt that this President is human, is one of us, can make mistakes, and most certainly will. But his heart, his agenda, and his character are all in the right place. He knows how to speak to and inspire the better parts of our collective, national consciousness, to help convince us all to "dust ourselves off" and begin again.
On this day, there can be no doubt. I am immensely proud of our country.
We have finally got it right this time.
Can the Dems Reach 60?


text:
Q: How to find the MAC address of the switch I am connected to? On a linux server, I am trying to find the MAC addresses that my server is aware of. Specifically the MAC address of the switch port I am connected to.
Is this possible?
Update:
The Below answers assisted me. My switch is a managed switch, however because the IP addresses on my host machines were set to static, the switch did not pick up the MAC Addresses for some reason. I changed the hosts to DCHP, which allowed my switch to properly pick up the MAC addresses on the connected ports after they sent DHCP requests. This allowed me to use the switches MAC-Address table to figure out which hosts were connected to which ports and I put the hosts back to static IPs.

A: It depends what kind of switch you are connected to.
A dumb switch (commonly called an unmanaged switch, which is effectively a network bridge) won't have a MAC address
A managed switch (sometimes called a smart switch) will have a MAC address
If the switch is a managed switch, and you know its IP address, run the following command in your terminal. It is then just match the IP with the MAC address.
arp -a

If you are dealing with a managed switch with a console port, connect with a console cable and enter this command (assuming it is a Cisco switch):
show mac address-table

http://www.techexams.net/forums/ccna-ccent/45578-mac-address-switch.html
https://learningnetwork.cisco.com/thread/9625

A: If you're lucky, you're running a smart switch and it'll actually have a MAC address.
In that case you can probably use LLDP. Install lldpd on Linux, then run:
michael:~$ lldpctl
-------------------------------------------------------------------------------
LLDP neighbors:
-------------------------------------------------------------------------------
Interface:    eth0, via: LLDP, RID: 1, Time: 5 days, 22:33:31
  Chassis:
    ChassisID:    mac 28:a6:8e:03:57:9b
    SysName:      core-pri
    SysDescr:     Netgear Gigabit Smart Switch
    MgmtIP:       10.0.0.38
    Capability:   Bridge, on
  Port:
    PortID:       local g22
    PortDescr:    Not received
-------------------------------------------------------------------------------

A: Switches don't have MAC addresses generally; they work at a level below them (although they do know what the MAC addresses of the devices connected to them are).

redpajama_set_name: RedPajamaStackExchange


Face low disk space problem on Windows 10 system partition so you want to extend the partition with existing unallocated space? Is it always possible to add existing unallocated space to expand system partition in Windows 10? What if there is no existing unallocated space on the same hard disk on your computer? Probably, you still have many doubts when you attempt to increase the size of system partition on Windows 10. This page will share tips you need to pay attention to and step-by-step guide to extend system partition C drive with unallocated space on Windows 10.
How to Add Unallocated Space to Windows 10 System Partition?
Step1. Right click system partition (C drive) and select Extend Volume option.
Step2. Enter the amount of space you want to add to system partition.
Step3. Follow the Extend Volume Wizard to add the unallocated space to system partition successfully.
It is often very easy to add the unallocated space to C drive with Windows 10 Disk Management as long as the unallocated volume is showing right behind the C partition.
Why Failed to Extend C Drive with Unallocated Space on Windows 10?
1. The unallocated space is not adjacent to system partition. Usually, there are several partitions on your computer. If there is other partition (s) show between the unallocated space and system partition, you will fail to add the unallocated space to extend system partition on Windows 10.
2. The unallocated space is on the left of C drive, in which situation you will also fail to add it to C partition. Windows 10 cannot extend system partition with unallocated space on its left even if the unallocated volume is contiguous to C drive.
How to Increase System Partition without Unallocated Space in Windows 10?
Method1: You can delete D drive which shows behind C drive under Disk Management. The deleted partition will show as unallocated space behind C drive, so you can add it to increase the size of C drive on your PC.
Method2: You can right click D drive and select Shrink Volume option to squeeze some unallocated space from D partition. This method does not delete any partition or data, but the unallocated space will not show close to C drive.
If you meet any problem to expand system partition with unallocated space in Windows 10, you can use Windows 10 partition software to resolve the problem. IM-Magic Partition Resizer Free is a free partition magic tool, with which you can squeeze unallocated space from other partition and add unallocated space to any partition without needing to delete any partition. Therefore, you can use this partition magic to expand system partition whether there is existing unallocated space or not.


text:
FSEM 100 The First Americans: American Indians in Fiction, Film and Fact
A guide of resources for Dr. Ballenger's students
Access New York Times Articles
Accessing Federal Documents
Stetson librarians are here to help you find sources and answer your questions about research. When you need help, call or text us at 386-747-9028 or libref@stetson.edu.
What is Peer Review?
This page explains the basics of peer review and shares clues for determining if an article is peer reviewed.
This survey is intended for you to fill out after a librarian visits your class.
stetson.libinsight.com/duponteval
This guide was updated in October 2017 for Dr. Ballenger's FSEM students. It features library & web resources that may be useful for research related to Native American history, culture, and contemporary experience.
Resources for Native American Topics
Find Primary Sources & Reference Content
American Indian Experience
Full-text digital resource exploring the histories and contemporary cultures of the indigenous peoples of the United States. Offers access to more than 150 volumes of scholarship and reference content, hundreds of primary documents, and thousands of images.
Find Scholarly Books & Peer-Reviewed Journal Articles
Use OneSearch to search many library databases simultaneously. OneSearch contains scholarly books and peer-reviewed articles, along with non-scholarly content like news,magazines, and videos.
Use JSTOR to find scholarly, peer-reviewed journal articles.
Gale Literature Resource Center
A literature reference database that includes biographical, bibliographical, and critical content. The LRC is built on the Gale Group three author databases: Contemporary Authors Online, Contemporary Literary Criticism Select and Dictionary of Literary Biography Online.
Find Streaming Videos
A streaming video platform with thousands of documentaries, educational films and news reels. Content from Films on Demand can be shown in the classroom setting and linked in your Blackboard course.
Full-text of the Daytona Beach New-Journal from 1996 to the present. Part of the Access World News database.
History of Volusia County
Use this resource to locate information about Volusia County history including historic Native American sites.
Florida Memory Project
Florida Memory is funded under the provisions of the Library Services and Technology Act, from the Institute of Museum and Library Services, administered by the Florida Department of State, Division of Library and Information Services
duPont-Ball Library Archives
Access to the digital archives including Stetson memorabilia.
Find Web Sites
Google Advanced gives you more control over the content you get from Google search results. Note the option to search for content on a specific site or with a specific domain (such as .gov or .edu).
DuckDuckGo is the internet search engine that doesn't track you. Great for avoiding bias based on your own previous searches.
Next: Access New York Times Articles >>
URL: https://guides.stetson.edu/AmericanIndiansinFictionFilmFact
Subjects: *Course Guides - Past


text:
Teenagers involved in recent rash of Evansville gun violence | Webb
At least four Evansville teens have been accused in killings this year.
Teenagers involved in recent rash of Evansville gun violence | Webb At least four Evansville teens have been accused in killings this year. Check out this story on courierpress.com: https://www.courierpress.com/story/opinion/columnists/jon-webb/2018/11/27/teenagers-involved-recent-rash-evansville-gun-violence/2129646002/
Jon Webb, Evansville Courier & Press Published 3:39 p.m. CT Nov. 27, 2018 | Updated 3:42 p.m. CT Nov. 27, 2018
Terry Terell Adams(Photo: Supplied/Vanderburgh County Sheriff's Office)
A horrible headline splashed across courierpress.com on Tuesday.
"Teens arrested in shooting that sent 15-year-old to the hospital," it said.
The story chronicled the arrests of Terry Adams Jr., 18, and Kyavion Brown, 16. The two were accused of shooting the 15-year-old in the face during a botched weed deal on the East Side on Nov. 21.
The story is terrible. Brown faces a slew of robbery charges, while Adams is staring at a possible attempted murder conviction. A Level 1 felony like that could bring at least 20 years in prison. Meanwhile, the victim will live the rest of his life with the physical and mental effects of a gun wound.
Kyavion Amir Brown (Photo: Supplied/Vanderburgh County Sheriff's Office)
But it's not even the worst story involving guns and young people to happen in Evansville recently. The city has seen a depressing parade of violence in the last few weeks – and some of it has involved kids who aren't even old enough to vote.
Take Nov. 9. That night, 65-year-old Odie Carrier was shot in his home on Wood Duck Drive. The U.S. Air Force veteran who once owned his own electronics company died at a hospital three days later.
Three teenagers were arrested in the shooting. Connor Johnson, 18, Xzereus Dlamini, 19, and 17-year-old Mackayla Scarbrough have all been charged with murder, according to court records.
From left to right: Xzereus Dlamini, Connor Johnson and Mackayla Scarbrough. Vanderburgh County officials are pursuing charges against all three in the fatal shooting of a 65-year-old Evansville man. (Photo: Supplied/Vanderburgh County Sheriff's Office)
That means at least four Evansville teens have been accused in killings this year.
This summer, the Vanderburgh County Prosecutor's Office filed a murder charge against Jahkei Mitchell. Police say Mitchell, 16, shot and killed 23-year-old Kevin Colon on July 18 outside Sam's Food Mart on Washington Avenue during an attempted robbery. A 15-year-old kid was initially arrested in connection to the shooting as well, but he hasn't been formally charged or publicly named.
Gun violence can skew even younger.
A 14-year-old boy was shot back in August in an apparent act of self-defense, police said.
And in a horrific accident from October, a 2-year-old girl somehow got hold of a firearm and shot herself at Savannah Garden Apartments. Amazingly, the bullet passed through her body without inflicting serious damage, and, wonderfully, she's expected to survive, police said.
It's impossible to pin down any one reason why this is happening. There's poverty, drugs, and the ever-hovering presence of guns in every facet of American life. Plus 1,000 others.
Violence of all forms flares up hard in Evansville and often withers away just as fast. At one point last year, we recorded 10 homicides in 78 days.
So there's hope this recent rash of teenager-involved shootings will fade. But even if it slows here, it will continue everywhere else.
According to Everytown for Gun Safety, an average of 47 children and teenagers are shot every day. And more than 3 million kids witness some type of gun violence every year.
They see it at home. In their schools. On their sidewalks. And some of them are the ones behind it.
It's difficult to offer any type of solution for such a persistent problem. And the only "opinion" I can muster for this supposed opinion column is one that's popped up more and more as the years limp on: it's all just so horribly sad.
More: 'This happens way too often': Evansville shooting victim's family reflects on his life
Contact columnist Jon Webb at jon.webb@courierpress.com
Read or Share this story: https://www.courierpress.com/story/opinion/columnists/jon-webb/2018/11/27/teenagers-involved-recent-rash-evansville-gun-violence/2129646002/
Think it couldn't happen to you? Don't be so sure
Dead bodies often found near Pigeon Creek
Viewpoint: Why I am joining forces with a Democratic senator
Letters: Do the politicians even care?
Howey: Ross Perot defined the path for Donald Trump
Letter: Thoughts on immigration


text:
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\begin{document}

\runningtitle{Privacy-Preserving Inverse Probability Weighting}

\twocolumn[

\aistatstitle{Privacy-Preserving Causal Inference \\
via Inverse Probability Weighting}

\aistatsauthor{Si Kai Lee\textsuperscript{\normalfont 1}$^*$ \And Luigi Gresele \textsuperscript{\normalfont 1,2} \And Mijung Park \textsuperscript{\normalfont 1, 3} \And Krikamol Muandet\textsuperscript{\normalfont 1}}
\aistatsaddress{\textsuperscript{1}Max Planck Institute for Intelligent Systems, T\"ubingen, Germany\\
                \textsuperscript{2}Max Planck Institute for Biological Cybernetics, T\"ubingen, Germany\\
                \textsuperscript{3}University of T\"ubingen, T\"ubingen, Germany}]

\begin{abstract}
The use of inverse probability weighting (IPW) methods to estimate the causal effect of treatments from observational studies is widespread in econometrics, medicine and social sciences.
Although these studies often involve sensitive information, thus far there has been no work on privacy-preserving IPW methods.
We address this by providing a novel framework for privacy-preserving IPW (PP-IPW) methods.
We include a theoretical analysis of the effects of our proposed privatisation procedure on the estimated average treatment effect,
and evaluate our PP-IPW framework on synthetic, semi-synthetic and real datasets.
The empirical results are consistent with our theoretical findings.
\end{abstract}

\section{INTRODUCTION}

The increasing ubiquity of machine learning in our daily lives has created a pressing need for trustworthy artificial intelligence (AI).
One key tenet of trustworthy AI, defined in the European Commission's Ethics Guidelines for Trustworthy AI \cite{EU19:Ethics}, is \emph{privacy}.
Preserving patient privacy is critical in medicine as it builds trust and fosters thoughtful decision making, which in turn helps improve patient care.
Although the privacy requirements in the medical field are especially high, such requirements are not unique.
In observational studies for social sciences, it is commonly assumed that sensitive information such as employment, education and criminal records that are used in analyses would be kept private.
The datasets compiled from these studies often contain personal information, hence conclusions drawn from statistical analyses on such datasets runs the risk of violating the privacy of those present in the datasets.
Such risk extends to \emph{causal inference} methods~\cite{Glass13:CIPH,Imbens15:CIS,Shiffrin16:CIBig,Bareinboim16:Fusion} as they fall under the umbrella of statistical estimation tools.

We illustrate the problem of causal inference with an example.
In medicine, it is crucial to have a well-rounded understanding of the efficacy of different medical treatments since a treatment can produce a broad range of responses across patients and multiple treatments are usually available.
The treatment effect can be modelled from observational data, by looking at how patients responded to different treatments in the past. However, two formidable obstacles need to be overcome in order to obtain a sound \emph{causal} effect estimation.
First, for each patient we only observe the outcome associated with the treatment that patient received, and no other~\cite{Holland86:FPCI}.
Second, the treatments that patients receive are not assigned at random, as doctors assign the treatment they expect to work best for each patient; as a result, treatment assignments and outcomes are subject to \emph{confounding}, which could result in a biased estimate of the outcome of each treatment.
An accurate estimate of the treatment effect should take confounding into account, possibly modelling how individual characteristics of the patient determine the assigned treatment.
However, this often requires collecting sensitive information from patients.

The \emph{propensity score} is arguably one of the most used quantities in causal inference for observational studies.
It forms the basis of popular techniques such as matching, stratification, and inverse probability weighting (IPW) \cite{austin2011introduction, Rosenbaum83:PS, rosenbaum1984reducing, rosenbaum1985constructing} which are extensively used in econometrics, medicine, and social sciences
\cite{Imbens15:CIS}.
Moreover, the IPW estimator based on propensity scores is the backbone of several counterfactual inference algorithms in the machine learning literature \citep{Dudik11:DoublyRobust,Bottou13:Counterfactual,Swaminathan15:CRM,Swaminathan17:Slate}.
Despite its widespread use, the propensity score---defined as the probability of assignment of a particular treatment given observed covariates---could depend on sensitive information about the patients, such as age, gender, race, and ethnicity.
Little concern has thus far been raised regarding the privacy issues related to the use of propensity scores in such methods.
Previously, propensity scores have been used by \cite{Rassen10:Pooling} as privatised substitutes for individual covariates.
However, as we show in Section \ref{sec:det-ate}, this method still violates privacy \cite{dwork2014algorithmic} since sensitive observational data is used for \textit{estimating} propensity scores.

Inverse probability weighting (IPW) methods, which employs estimated propensity scores, are frequently used to estimate the average treatment effect (ATE) \cite{ robins2000marginal, rosenbaum1987model}. Since estimating the ATE with IPW methods still require sensitive observational data, privacy is further violated.

To address this important but neglected issue, we develop a novel framework for privacy-preserving IPW methods. This framework consists of two steps: (1) we learn a privacy-preserving propensity score estimator
and (2) we output a privacy-preserving ATE estimator.
In addition, we investigate the effect of privatisation in both steps on the performance of the resulting causal analysis.

\vspace{-0.8em}
\paragraph{Related Work.}
To the best of our knowledge, this paper is the first work which formally investigate the privatisation of the propensity score function and the average treatment effect estimates with IPW methods.
There have only been few prior attempts to privatise causal inference techniques in different contexts.
For example, in \cite{KusnerSSW16}, the authors demonstrated how one could privatise statistical dependence scores such as \textit{Spearman's $\rho$} and \textit{Kendall's $\tau$} under the additive noise model.
The main focus of \cite{KusnerSSW16} is to obtain privatised scores that still can correctly identify the causal direction between two random variables.
In \cite{8166616}, the authors developed a differentially private constraint-based causal graph discovery method for categorical data.
None of the above papers considered propensity score-based causal inference methods which is our focus.
Since propensity score-based methods are fundamental to econometrics, medicine, social sciences etc., we expect this work to impact diverse fields.

\vspace{-0.8em}
\paragraph{Contributions.}
In this paper, we propose a privacy-preserving framework for IPW methods which comprises a propensity score estimator and an IPW-based average treatment effect estimator
We privatise the parameters of the logistic regression model used to estimate the propensity scores, as well as the output of inverse probability weighting.
This guarantees the privacy of the individuals in the training dataset used to learn the propensity score estimator and individuals in the estimation dataset used to estimate the ATE.
We analyse the \textit{effect of the noise added to enforce privacy} on the resulting estimated causal effect.
Our analysis provides guidelines on how many samples we need to guarantee a certain level of privacy while providing accurate causal inference.
We test our method on synthetic, semi-synthetic and real-world datasets to illustrate its effectiveness.

The rest of this paper is organised as follows. Section \ref{sec:background} provides a review of propensity scores in causal inference and differential privacy.
The privacy-preserving propensity scores, IPW estimator, as well as their theoretical guarantees are then presented in Section \ref{sec:det-ate}, followed by experimental results in Section \ref{sec:experiments}.
Finally, Section \ref{sec:conclusion} concludes the paper.

\section{BACKGROUND}
\label{sec:background}
In this section, we introduce relevant concepts from causal inference and differential privacy.
The key quantities are summarised in Table \ref{tab:notation}.

\begin{table}[t!]
    \centering
    \caption{Key Quantities}
    \label{tab:notation}
    \begin{tabular}{cl}
         \toprule
         {\small\textbf{Symbol}} & {\small\textbf{Description}} \\
         \midrule
         $\epsilon$ & Privacy loss \\
         $\delta$ & Failure probability \\
         $\mu_t$ & Population mean under treatment $t$ \\
         $\hat{\mu}_t$ &  Estimated $\mu_t$ \\
         $\hat{\mu}_t^{\epsilon}$ & Privatised $\hat{\mu}_t$ \\
         $\tau$ & Average Treatment Effect (ATE), $\mu_1 - \mu_0$\\
         $\hat{\tau}$ & ATE estimate, $\hat{\mu}_1 - \hat{\mu}_0$\\
         $\hat{\tau}_n$ & Partially privatised ATE estimate\\% $\hat{\mu}^{\epsilon}_1 - \hat{\mu}^{\epsilon}_0$ \\
         $\hat{\tau}_n^{\epsilon}$ & Fully privatised ATE estimate\\%Privatised $\hat{\tau}_n$ \\
         $\pi_{\ensuremath \mathbf{w}}$ & Propensity score function \\
         $\pi_{\hat{\ensuremath \mathbf{w}}}$ & Estimated $\pi_{\ensuremath \mathbf{w}}$ \\
         $\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}$ & Privatised $\pi_{\hat{\ensuremath \mathbf{w}}}$ \\
         \bottomrule
    \end{tabular}
\end{table}

\subsection{Propensity Scores in Causal Inference}
\label{sec:ppci}
The \emph{potential outcomes} framework is one of the most widely-used approaches in causal inference \cite{Neyman1923:Causal, Rubin74:Causal, Rubin05:PO}.
It provides the mathematical basis for estimating the outcome of an experiment which has not been performed given outcomes observed under other experimental settings.

Consider the setting where we want to estimate whether a given treatment has a positive, negative or null effect on different units/individuals.
We define $T$ as the treatment variable and $Y_t$ as the random variable representing the potential outcome associated with treatment $T=t$.
In medicine, $T$ could represent different cancer treatments and $Y_t$ an indicator for patient recovery after treatment $t$.
Throughout this paper, we focus on the binary treatment setting, \textit{i}.\textit{e}., $T\in\{0, 1\}$, and refer to the subset of the population with $T=1$ as the \textit{treatment} group, and the rest, with $T=0$, as the \textit{control} group.
The random variables $Y_1$ and $Y_0$ are the outcomes associated with the treatment and control group, respectively.

The question we want to answer is: \emph{what is the effect of administering a treatment to a unit compared to not doing so?}
Quantitatively, this can be characterised by the differences of the outcome when the treatment is administered and the outcome when the treatment is not administered, \textit{i}.\textit{e}., $Y_1 - Y_0$.
To estimate this, we would require both the outcomes of treatment and no treatment to be observed for every unit.
However, for each unit we can observe only either $Y_1$ or $Y_0$.
In practice, we substitute each unobserved quantity with an estimate of its expected outcome, $\mu_t := \mathbb{E}[Y_t]$, and evaluate the \textit{average treatment effect} (ATE) with $\tau = \mathbb{E}[Y_1] - \mathbb{E}[Y_0]$.
Given a dataset $\mathcal{D}=\{(t_1,y_1),\ldots,(t_N,y_N)\}$, we can approximate $\hat{\mu}_t$ with $n_t^{-1}\sum_{i=1}^N \mathbbm{1}(t_i=t)\cdot y_i$, where $\mathbbm{1}(\cdot)$ is an indicator that returns 1 when $t_i=t$ and 0 otherwise.

If $\mathcal{D}$ is collected from a randomised experiment, $\hat{\tau} := \hat{\mu}_1-\hat{\mu}_0$ is an unbiased estimate of $\tau$.
However, the problem with most observational studies is that $\hat{\tau}$ is generally biased because of potential \emph{confounding} variables $X$ that affect both $T$ and $Y_t$.
For example, $X$ could be the current stage of a patient's cancer, which could influence both the decision of the physician regarding the treatment and the outcome of the treatment.
To obtain an unbiased ATE estimate despite the confounding variables $X$ for dataset $\mathcal{D}=\{(\ensuremath \mathbf{x}_i,t_i,y_i)\}_{i=1}^N$, we expand each $\mathbb{E}[Y_t]$ as $\mathbb{E}[Y_t] = \mathbb{E}_X[\mathbb{E}_{t}[Y_t|X,T=t]]$ and compute the difference of the quantity above with $t=1$ and with $t=0$.
The validity of this estimate can be assessed given three technical requirements, which we assume throughout:
\begin{enumerate}[label={\bf (\roman*)}]
\item \textbf{Stable Unit Treatment Value Assumption (SUTVA):} The observed outcome of the $\mathrm{i}^{th}$ unit $Y(i)$ is unaffected by the assigned treatment to other units.
\item \textbf{Ignorability:} $T \,\rotatebox[origin=c]{90}{$\models$}\, (Y_0,Y_1)\,|\, X$.
\item \textbf{Positivity:} $0 < \mathbb{P}(T=1|X=\ensuremath \mathbf{x}) < 1$ for all $\ensuremath \mathbf{x}$.
\end{enumerate}
See \cite{Imbens15:CIS} for a thorough exposition on these assumptions.

\paragraph{Propensity Scores.}
The \emph{propensity score} is one of the most widely used quantities for causal analysis in observational studies \cite{austin2011introduction, Rosenbaum83:PS, rosenbaum1984reducing, rosenbaum1985constructing}.
In the binary treatment setting, the propensity score $\pi(\ensuremath \mathbf{x})$ is defined as the probability of a unit with covariate $\ensuremath \mathbf{x}$ receiving treatment $T = 1$,
$\pi(\ensuremath \mathbf{x}) := \mathbb{P}(T = 1| X = \ensuremath \mathbf{x})$.
It has been shown that, under the above set of assumptions, the propensity score $\pi(\ensuremath \mathbf{x})$ summarises all the relevant information in $X$ for causal inference so that $T \,\rotatebox[origin=c]{90}{$\models$}\, (Y_0,Y_1)\,| \pi(X)$ holds \cite{Rosenbaum83:PS}.
Unfortunately, in most observational studies, the true treatment assignment mechanism is not known.
Thus, a common practice is to fit a propensity score function on data $\mathcal{D}$ using standard statistical models $\pi_\ensuremath \mathbf{w}(\ensuremath \mathbf{x}) = f_{\ensuremath \mathbf{w}}(\ensuremath \mathbf{x})$, where $f_{\ensuremath \mathbf{w}}$ represents a model parameterised by a parameter vector $\ensuremath \mathbf{w}$.
In this work, we focus on the logistic regression model since it is the most frequently used model for fitting propensity scores \cite{cepeda2003comparison}.
The model is defined as
\begin{align} \label{eq:logistic}
\pi_{\ensuremath \mathbf{w}}(\ensuremath \mathbf{x}) = \frac{1}{1+ e^{-\ensuremath \mathbf{w}^\top \ensuremath \mathbf{x}}}
\end{align}
\noindent where $\ensuremath \mathbf{w} \in\mathbb{R}^d$.
Other popular techniques for propensity score estimation include classification and regression trees (CART), boosted CART, random forests, \etc.

\paragraph{Inverse Probability Weighting (IPW).}
\label{sec:IPW}
One popular propensity score-based method for estimating $\tau$ from observational data is IPW \cite{ robins2000marginal, rosenbaum1987model}. Using IPW, we can obtain an unbiased ATE estimate $\hat{\tau} = \hat{\mu}_1 - \hat{\mu}_0$ with $\hat{\mu}_1$ and $\hat{\mu}_0$ defined as
\begin{align}
\label{eq:iptw}
\hat{\mu}_1 := \frac{1}{N} \sum_{i=1}^N \frac{y_i t_i}{\pi(\ensuremath \mathbf{x}_i)}, \quad
\hat{\mu}_0 := \frac{1}{N} \sum_{i=1}^N \frac{y_i (1-t_i)}{1-\pi(\ensuremath \mathbf{x}_i)},
\end{align}
\noindent where $N$ is the number of units in $\mathcal{D}$.
In practice, we replace the propensity score function $\pi$ with its empirical estimate $\pi_{\hat{\ensuremath \mathbf{w}}}$.

\subsection{Differential Privacy}
The notion of differential privacy (DP) \cite{dwork2014algorithmic} provides a well-defined framework to describe the privacy properties of statistical estimation algorithms.
DP states that a privacy-preserving, randomised algorithm behaves \textit{similarly} on \textit{similar} datasets. Specifically, the algorithm's behaviour is quantified in terms of a probability ratio, which describes how the algorithm's output changes when different datasets are used as an input. Intuitively, the probability ratio does not change much (behaves similarly) if the input datasets differ by a single entry (similar datasets). The formal definition  is given below.
\begin{definition}\label{def:dp}
A randomised algorithm $\mathcal{A}$ with domain $\mathbb{N}^{|\mathcal{X}|}$, where $\mathcal{X}$ is the data universe, satisfies $(\epsilon, \delta)$-differential privacy, i.e., is $(\epsilon, \delta)$-DP, if for all $\mathcal{S} \subseteq$ Range($\mathcal{A}$) and for all neighbouring $\mathcal{D}, \mathcal{D}' \in \mathbb{N}^{|\mathcal{X}|}$ such that $\|\mathcal{D} - \mathcal{D}'\|_1 \leq 1$, i.e., there is only one entry difference in the two datasets $\mathcal{D}, \mathcal{D}'$,
\begin{equation*}
\mathbb{P}[\mathcal{A}(\mathcal{D}) \in \mathcal{S}] \leq \exp(\epsilon)\mathbb{P}[\mathcal{A}(\mathcal{D}') \in \mathcal{S}] + \delta
\end{equation*}
where the probability space is over the outputs of $\mathcal{A}$.
\end{definition}
Here, $\epsilon$ is defined as the \textit{privacy loss} controlling the level of privacy.
For $0<\delta<1$, $\delta$ defines the \textit{failure probability}, \textit{i}.\textit{e}., an algorithm is $\epsilon$-DP with probability at least $1-\delta$.

\paragraph{Gaussian Mechanism.}
The Gaussian mechanism \cite{dwork2006our} is commonly used to privatise models (see, \textit{e}.\textit{g}., \cite{dwork2014algorithmic,SarwateC13} for other DP mechanisms).
The mechanism privatises a vector-valued function $f: \mathcal{D} \mapsto \mathbb{R}^p$ by adding Gaussian noise to it. The noise is calibrated based on the L2-{\it{sensitivity}} of $f$, which is defined by $S(f) = \max_{\mathcal{D}, \mathcal{D}', \|\mathcal{D}-\mathcal{D}'\|=1} \|f(\mathcal{D}) - f(\mathcal{D}') \|_2$.
Hence, the privatised function has the form $\tilde{f}(\mathcal{D}) = f(\mathcal{D}) + \mathcal{N}(0, \sigma^2 \mathbf{I}_d)$.
A choice of $\sigma \geq \epsilon^{-1}\sqrt{2\log(1.25/\delta)}S(f)$ produces the function $\tilde{f}(\mathcal{D}) $ that is $(\epsilon, \delta)$-DP.

\paragraph{Differentially Private Empirical Risk Minimisation (DP-ERM).}
Let $\ell:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_+$ be the loss function and vector $\ensuremath \mathbf{w}$ the model parameters of $\pi_{\ensuremath \mathbf{w}}$.
Under ERM framework, the optimal model parameters $\hat{\ensuremath \mathbf{w}}$ are obtained by minimising the empirical risk function $J(\ensuremath \mathbf{w}, \mathcal{D}) = m^{-1} \sum_{i=1}^{m} \ell(\pi_\ensuremath \mathbf{w}(\ensuremath \mathbf{x}_i), t_i) + \lambda \Omega(\ensuremath \mathbf{w})$, where $\Omega(\cdot)$ is the regulariser and $\lambda > 0$ the regularisation constant.
If logistic regression \eqref{eq:logistic} is used to model the propensity score function $\pi_{\ensuremath \mathbf{w}}$, the parameters $\ensuremath \mathbf{w}$ can be learned using ERM with a $L_2$-regulariser.
We assume throughout that $\mathcal{X}$ is contained in the $L_2$-unit ball\footnote{This is a typical assumption on the dataset in the differential privacy literature.}, \textit{i}.\textit{e}., $\|\ensuremath \mathbf{x}_i\|_2 \leq 1$ for all $\ensuremath \mathbf{x}_i \in \mathcal{X}$. Note that the dataset $\mathcal{D}_m = \{(\ensuremath \mathbf{x}_i,t_i,y_i)\}_{i=1}^m$ contains the observed treatment effects $y_i$, but that they are not used to fit $\pi_{\ensuremath \mathbf{w}}$.
In our case, the regularised cross-entropy loss $J(\ensuremath \mathbf{w}, \mathcal{D})$ is
\begin{align}\label{eq:loss}
- \frac{1}{m} \sum_{i=1}^m t_i \log p_i + (1 - t_i)\log (1-p_i) +  \frac{\lambda}{2} \|\ensuremath \mathbf{w}\|_2^2
\end{align}
\noindent where $p_i := p(\pi_\ensuremath \mathbf{w}(\ensuremath \mathbf{x}_i))$. This loss is equivalent to the logistic loss used in \cite{chaudhuri2011differentially}.
The L2-sensitivity of $\hat{\ensuremath \mathbf{w}}$ obtained by minimising \eqref{eq:loss} is
given by $S(\hat{\ensuremath \mathbf{w}}) = \max_{\mathcal{D}, \mathcal{D}', \|\mathcal{D}-\mathcal{D}'\|=1} \left\|\hat{\ensuremath \mathbf{w}}(\mathcal{D}) - \hat{\ensuremath \mathbf{w}}(\mathcal{D}')
  \right\|_2 \leq 2(m\lambda)^{-1}$ \cite{chaudhuri2011differentially}.

\section{Privacy-Preserving IPW}
\label{sec:det-ate}

We start by explaining why logistic-regression based estimation for the propensity score by minimising \eqref{eq:loss} is not private.
An intuitive explanation is: since an ERM solution can be written as a linear combination of training samples, the $\hat{\ensuremath \mathbf{w}}$ and $\hat{\ensuremath \mathbf{w}}'$ estimated from datasets $\mathcal{D}$ and $\mathcal{D}'$ differing by one single entry could be completely different, \textit{e}.\textit{g}., if the entry was an outlier.
Hence, the likelihood ratio of the two models $\hat{\ensuremath \mathbf{w}}$ and $\hat{\ensuremath \mathbf{w}}'$ estimated from two neighbouring datasets would be unbounded, which makes $\hat\ensuremath \mathbf{w}$ \textbf{\textit{not}} $\epsilon$-DP.
A more rigorous proof can be found in~\cite{chaudhuri2011differentially}.

Since existing propensity score estimation often relies on ERM,
standard methods for modelling the propensity score function would yield a model that violates differential privacy: given $\pi_{\hat{\ensuremath \mathbf{w}}}$ and all points in dataset $\mathcal{D}$ bar one, it is possible to infer the covariates $\ensuremath \mathbf{x}_i$ of omitted unit.
This is a serious problem since propensity score-based methods are frequently used to estimate the causal effect from observational studies containing sensitive data and highlights the need for privacy-preserving propensity score estimators and propensity score-based methods

Privatising the ATE estimated via IPW requires two layers of privatisation.

The first step of our proposed method is to divide the dataset $\mathcal{D}$ into $\mathcal{D}_m = \{\ensuremath \mathbf{x}_i, t_i, y_i\}_{i=1}^m$ and $\mathcal{D}_n = \{\ensuremath \mathbf{x}_i, t_i, y_i\}_{i=1}^n$, and use the first $m$ points to learn $\hat{\ensuremath \mathbf{w}}$ and the remaining $n$ points to estimate $\hat{\tau}$.
We separately privatise the estimated propensity score function with respect to the $m$ datapoints and the estimated ATE with respect to the $n$ datapoints to ensure that all $N$ datapoints (where $N=m+n$) in $\mathcal{D}$ are protected.

We describe each step in detail in the next two subsections.

\subsection{Privacy-Preserving Propensity Score}

To privatise the logistic regression model, we first compute a non-private version of the propensity score $\pi_{\hat{\ensuremath \mathbf{w}}}$ by minimising ~\eqref{eq:loss}. Next, we use the Gaussian mechanism to generate a privacy-preserving version of $\hat{\ensuremath \mathbf{w}}$, defined as $\hat{\ensuremath \mathbf{w}}_{\epsilon}$.

\begin{definition}
\label{def:pps}
Let $\hat{\ensuremath \mathbf{w}}$ be the solution
of \eqref{eq:loss}.
A privacy-preserving propensity score function is
\begin{align}\label{log_reg}
\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}) = \frac{1}{1+ \exp(-\hat{\ensuremath \mathbf{w}}_{\epsilon}^\top \ensuremath \mathbf{x})} = \frac{1}{1+ \exp(- \hat{\ensuremath \mathbf{w}}^\top \ensuremath \mathbf{x} - \ensuremath \mathbf{z}^\top \ensuremath \mathbf{x})},
\end{align}
where $\hat{\ensuremath \mathbf{w}}_{\epsilon} := \hat{\ensuremath \mathbf{w}} + \ensuremath \mathbf{z}$ with $\ensuremath \mathbf{z} \sim \mathcal{N}(\mathbf{0}, \sigma^2\mathbf{I}_d)$ and $\sigma = \epsilon^{-1}\sqrt{2\log(1.25/\delta)}S(\hat{\ensuremath \mathbf{w}})$ for $\epsilon \in (0,1)$ and any $\delta \in (0,1)$.
\end{definition}

Alongside Definition \ref{def:pps}, we define the counterparts of $\hat{\tau}$, $\hat{\mu}_1$ and $\hat{\mu}_0$ which use $\pi_{\hat{\ensuremath \mathbf{w}}}^\epsilon(\ensuremath \mathbf{x})$ in place of $\pi_{\hat{\ensuremath \mathbf{w}}}(\ensuremath \mathbf{x})$ over the $n$ points as $\hat{\tau}_n$, $\hat{\mu}_1^{\epsilon}$ and $\hat{\mu}_0^{\epsilon}$. This yields $$\mbox{\textbf{DP ATE  w.r.t. $\mathcal{D}_m$}}: \hat{\tau}_{n} := \hat{\mu}_1^{\epsilon} - \hat{\mu}_0^{\epsilon}$$ where
\begin{align}
\label{eq:formpropen}
\hat{\mu}_1^{\epsilon} := \frac{1}{n}\sum_{i=1}^{n_{1}} \frac{y_i}{\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}_i)}, \quad
\hat{\mu}_0^{\epsilon} := \frac{1}{n}\sum_{i=1}^{n_{0}} \frac{y_i}{1-\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}_i)},
\end{align}
with the first sum is over the $n_{1}$ points where $t_i = 1$ and the second sum is over the $n_0$ points where $t_i = 0$, and $n_0+n_1 = n$. The estimator $\hat{\tau}_n$ only safeguards the privacy of $m$ points within the dataset, those belonging to the split $\mathcal{D}_m$.
Privatisation of the points in $\mathcal{D}_n$ is discussed in the next subsection. Here, we focus on the effect of the noise added for protecting $\mathcal{D}_m$ on causal inference.

\begin{figure*}[t!]
    \centering
    \includegraphics[width=0.45\linewidth]{bias_plot_1}\hfill
    \includegraphics[width=0.45\linewidth]{bias_plot_2}

    \caption{Behaviour of $g(\epsilon,m,n,\lambda,\delta)$ w.r.t. the sample size $m$ and the privacy loss $\epsilon$. As we can see, the bias decreases as we expect when the sample size $m$ and the privacy loss $\epsilon$ increase.}
    \label{fig:g_sigma}
\end{figure*}

\paragraph{Characterising $\hat{\tau}_n$.} Both $\hat{\mu}_1^{\epsilon}$ and $\hat{\mu}_{0}^{\epsilon}$ are weighted sums of correlated log-normal random variables, where the magnitude and signs of the weights depend on the data. This can be seen from their formulae
\begin{align}
\label{eq:mu0-mu1}
\hat{\mu}_1^{\epsilon} &= \frac{1}{n}\sum_{i=1}^{n_{1}} y_i (1 + \exp(-\ensuremath \mathbf{w}^T \ensuremath \mathbf{x}_i)\exp(-\ensuremath \mathbf{z}^T \ensuremath \mathbf{x}_i)),\\
\hat{\mu}_0^{\epsilon} &= \frac{1}{n}\sum_{i=1}^{n_{0}} y_i (1 + \exp(\ensuremath \mathbf{w}^T \ensuremath \mathbf{x}_i)\exp(\ensuremath \mathbf{z}^T \ensuremath \mathbf{x}_i)),
\end{align}
and recalling that $\ensuremath \mathbf{z}$ is a Gaussian random variable.
Obtaining closed-form expressions for the above quantities with finite samples is highly nontrivial \cite{gulisashvili2016tail, lo2012sum}.

As a first step towards understanding the random variable $\hat{\tau}_n$, we show how its expected value can be rewritten in terms of the non-privatised ATE estimate $\hat{\tau}$ and variance of the added noise. See Appendix \ref{proof:exp-tau-hat} for proof.

\begin{lemma}\label{lem:exp-tau-hat}
    Let $\alpha_i := (-1)^{1-t_i}y_i\exp((-1)^{t_i}\hat{\ensuremath \mathbf{w}}^\top \ensuremath \mathbf{x}_i)$ be a constant for $i=1,\ldots,n$.
    Then, we have $$\mathbb{E}[\hat{\tau}_n] = \hat{\tau} + g(\epsilon,m,n,\lambda,\delta)$$
    where the function $g(\epsilon,m,n,\lambda,\delta)$ is given by
    \begin{equation} \label{eq:g-func}
    \frac{1}{n}\sum_{i=1}^n \alpha_i\left[\exp\left((-1)^{t_i} \frac{4\log(1.25/\delta)\|\ensuremath \mathbf{x}_i\|_2^2}{\epsilon^{2}m^{2}\lambda^2}\right) - 1\right].
    \end{equation}
\end{lemma}

Lemma \ref{lem:exp-tau-hat} allows us to interpret $\hat{\tau}_{n}$ as a biased estimate of $\hat{\tau}$, where the \emph{additive} bias term is a function of the privacy loss $\epsilon$, sample sizes $m$ and $n$, regularisation constant $\lambda$, and failure probability $\delta$.
Notice that the bias $g(\epsilon,m,n,\lambda,\delta)$ converges to zero as either the privacy budget $\epsilon$ or the total number of points $m+n$ goes to infinity as to be expected.
We complement our insights with numerical simulations (see Figure \ref{fig:g_sigma}) that describe the behaviour of this nontrivial bias term.

To study the probability of $\hat{\tau}_n$ having the opposite sign w.r.t. the non-private estimator, we assume that its support is bounded both from above and below. This allows us to employ standard concentration inequality results for variables with bounded supports.
We bound the support of the estimator by either \emph{deterministic}, or \emph{probabilistic} means. An in-depth discussion of how we do so is provided in Appendix \ref{sec:det-prob-bounds}.

The next theorem characterises the behaviour of $\hat{\tau}_n$.

\begin{theorem}\label{thm:main-thm-1}
Assume that $\hat{\tau} > 0$ and $\mathrm{sign}(\hat{\tau})=\mathrm{sign}(\tau)$. If $|\hat{\tau}_n| \leq \eta$ for some $\eta > 0$, we have
$$\mathbb{P}(\hat{\tau}_{n} \leq 0| \hat\tau>0) \leq \exp \left( -2\eta^{-2}(\hat{\tau} + g(\epsilon,m,n,\lambda,\delta))^2 \right).$$
\end{theorem}

\begin{proof}
Given that $|\hat{\tau}_{n}| \leq \eta$ (or the result in Lemma \ref{lem:prob-bdc} with probability at least $1-\gamma$), we can apply Lemma \ref{lem:exp-tau-hat} and Hoeffding's inequality for bounded variables to yield the result.
\end{proof}
Qualitatively, we expect this since the theorem implies that the larger the magnitude of the true estimated ATE is, the smaller the probability of drawing incorrect causal conclusions from the partially privatised estimator. Interestingly, the bound in Theorem \ref{thm:main-thm-1} provides a quantitative characterisation, showing that such probability decreases \emph{exponentially} as a function of $\hat{\tau}$.
The probability of drawing incorrect conclusions from $\hat{\tau}_{n}$ also depends exponentially on the bias $g(\epsilon,m,n,\lambda,\delta)$. We provide empirical results clarifying the dependency on this term in Section \ref{sec:experiments}.

\begin{algorithm*}[t]
    \caption{PP-IPW}
    \begin{algorithmic}[1]
    \REQUIRE Data $\mathcal{D} = \{(\mathbf{x}_i,t_i,y_i)\}_{i=1}^N$ and privacy loss ($\epsilon, \delta$)
    \STATE Split $\mathcal{D}$ into two random subsets $\mathcal{D}_m$ and $\mathcal{D}_n$ consisting of $m$ and $n$ data points.

    \\
    \textbf{A. Obtain DP propensity score function}
    %
    \STATE Minimise $J(\ensuremath \mathbf{w},\mathcal{D}_m)$ in \eqref{eq:loss} for the non-private estimate $\hat{\mathbf{w}}$.
    \STATE $\hat\ensuremath \mathbf{w}_{\epsilon} =  \hat\ensuremath \mathbf{w} + \mathbf{z} $ where $\ensuremath \mathbf{z} \sim \mathcal{N}(\mathbf{0}, \sigma_m^2\mathbf{I}_d)$ and $\sigma_m = \epsilon^{-1}\sqrt{2\log(1.25/\delta)}S(\hat{\ensuremath \mathbf{w}})$.
    \STATE Output DP propensity score function
    $\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}) = 1/(1+ \exp(- \hat{\ensuremath \mathbf{w}}_{\epsilon}^\top \ensuremath \mathbf{x}))$. \\
        %
    \textbf{B. Obtain DP ATE}
    %
    \STATE Compute $\hat{\tau}_{n} := \hat{\mu}_1^{\epsilon} - \hat{\mu}_0^{\epsilon}$, given $\mathcal{D}_n$, where
    $\hat{\mu}_1^{\epsilon} := n^{-1}\sum_{i=1}^{n_{1}} \frac{y_i}{\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}_i)}$ and
    $\hat{\mu}_0^{\epsilon} := n^{-1}\sum_{i=1}^{n_{0}} \frac{y_i}{1-\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}(\ensuremath \mathbf{x}_i)}$.
    \STATE Output DP ATE $\hat{\tau}_{n}^{\epsilon}= \hat{\tau}_{n} + e$, where $e \sim \mathcal{N}(0,\sigma_n^2)$ and $\sigma_n := \epsilon^{-1}\sqrt{2 \log (1.25 / \delta)} S(\hat{\tau}_{n})$.
    \ENSURE DP propensity score function $\pi_{\hat{\ensuremath \mathbf{w}}}^{\epsilon}$ (w.r.t. $\mathcal{D}_m$) and DP ATE  $\hat{\tau}_n^{\epsilon}$ (w.r.t $\mathcal{D}$).
    \end{algorithmic}\label{algo:PPIPW}
\end{algorithm*}

\subsection{Privacy-Preserving ATE}

We now proceed to privatising the $n$ points used to compute $\hat{\tau}_n$.
Given that $\mathbb{P}(T = 1 | x_i)$ is bounded above and below by $\omega_1$ and $\omega_2$ respectively for all $i \in n$, $\hat{\tau}_n$ is consequently bounded as well.
By further assuming that $|y_i| \leq C_y$ for all $i \in n$, we ensure that the L2-sensitivity of $\hat{\tau}_{n}$, $S(\hat{\tau}_{n})$, is bounded by $2 n^{-1} C_y \max[1/\omega_1, 1/(1 - \omega_2)]$. We show how this quantity is obtained in Appendix \ref{sec:l2-sens}.

We apply the Gaussian mechanism to $\hat{\tau}_{n}$ to obtain a privacy-preserving approximation $\hat{\tau}_{n}^{\epsilon}$ that is $(\epsilon, \delta)$-DP w.r.t to the $n$ points used to obtain the estimate.
$$\mbox{\textbf{DP ATE w.r.t both $\mathcal{D}_m$ and $\mathcal{D}_n$} : }\hat{\tau}_{n}^{\epsilon} = \hat{\tau}_{n} + e $$ where the noise is drawn from $e \sim \mathcal{N}(0,\sigma_n^2)$ and with the noise standard deviation $\sigma_n := \epsilon^{-1}\sqrt{2 \log (1.25 / \delta)} S(\hat{\tau}_{n})$.

\paragraph{Characterising $\hat{\tau}^\epsilon_n$.}
We now present our main result which bounds the probability that \emph{both} $\hat{\tau}_{n}^{\epsilon}$ and $\hat{\tau}_n$ yield incorrect causal conclusions. See Appendix \ref{proof:main-thm-2} for proof.

\begin{theorem} \label{thm:main-thm-2}
Assume that $\hat{\tau} > 0$ and $\mathrm{sign}(\hat{\tau})=\mathrm{sign}(\tau)$. If $|\hat{\tau}_{n}| \leq \eta$ for some $\eta > 0$, we have
\begin{align*}
\MoveEqLeft \mathbb{P}(\hat{\tau}_{n}^{\epsilon} \leq 0, \hat{\tau}_{n} \leq 0 | \hat\tau >0) \\
&\leq \frac{1}{2} \exp \left( \frac{-2(\hat{\tau} + g)^2}{\eta^2} \right)\left[1 + \mathrm{erf}\left(\frac{|\hat{\tau}_{n}|}{\sigma_n\sqrt{2}}\right)\right],
\end{align*}
\noindent where $g := g(\epsilon,m,n,\lambda,\delta)$.
\end{theorem}

The additional $1/2 [1 + \mathrm{erf}(|\hat{\tau}_{n}|/\sigma_n\sqrt{2})]$ term in Theorem \ref{thm:main-thm-2} represents the probability that the added noise from the Gaussian mechanism $e$ is greater than $|\hat{\tau}_{n}|$, computed via the Gaussian CDF. The bound in Theorem \ref{thm:main-thm-2} further accounts for $\hat{\tau}_{n}$ and $\sigma_n$: the probability that both $\hat{\tau}_{n}^{\epsilon}$ and $\hat{\tau}_{n}$ are negative increases exponentially with $|\hat{\tau}_{n}|$ and decreases exponentially with $\sigma_n$ respectively where the latter is a function of $\epsilon$, $\delta$ and $S(\hat{\tau}_{n})$.

This theorem provides a full characterisation of our proposed differentially private ATE estimation procedure. It summarises the effects of protecting all $m+n=N$ points, and show that the probability of drawing an incorrect causal conclusion decay exponentially w.r.t to the values of $\hat{\tau}$ and $\hat{\tau}_n$.

\paragraph{Remark.}
Notice that in Theorem \ref{thm:main-thm-2} we chose to bound the probability $\mathbb{P}(\hat{\tau}_{n}^{\epsilon} \leq 0, \hat{\tau}_{n} \leq 0 | \hat\tau>0)$. However, depending on the application, other quantities, such as $\mathbb{P}(\hat{\tau}_{n}^{\epsilon} \leq 0 | \hat\tau>0)$ or $\mathbb{P}(\hat{\tau}_{n}^{\epsilon} \leq 0 | \hat{\tau}_{n} \leq 0, \hat\tau>0)$, might be more interesting.
It is also possible to bound those quantities; the proofs follow from that of Theorem \ref{thm:main-thm-2} presented in Appendix \ref{proof:main-thm-2}, and we omit them for brevity.

Putting everything together, our algorithm is presented in \algoref{PPIPW}. For the sake of simplicity, we assigned the same privacy budget for privatising the propensity score and the average treatment effect but one could choose to have separate privacy levels for the two quantities. We also include in Appendix \ref{sec:att_and_atc} bounds for the average treatment effect for the treated (ATT) and the average treatment effect for the controls (ATC).
We leave the development of privatised estimators based on more sophisticated techniques as future work.

\section{EXPERIMENTS} \label{sec:experiments}
In this section, we demonstrate our theoretical findings with experiments on synthetic, semi-synthetic and real data.
We set the probability of failure $\delta = 10^{-6}$ and regularisation coefficient $\lambda = 0.1$ for all experiments.
The logistic regression model is implemented in PyTorch \cite{paszke2017automatic} and optimised via gradient descent using the entire dataset.
To ensure reproducibility, we set the random seed for both NumPy and PyTorch to 1.

\begin{figure*}[t!]
\centering
    \begin{subfigure}{0.45\linewidth}
\includegraphics[width=\linewidth]{sim_tau_0_1_epses_flip_prob.pdf}

    \vspace{-0.75cm}
    \caption{}
\label{fig:1a}
    \end{subfigure}\hfill
    \begin{subfigure}{0.45\linewidth}
\includegraphics[width=\linewidth]{sim_tau_2_epses_flip_prob.pdf}
    \vspace{-0.75cm}
    \caption{}

\label{fig:1b}
    \end{subfigure}
    \begin{subfigure}{0.45\linewidth}
    \includegraphics[width=\linewidth]{sim_tau_0_1_m.pdf}

    \vspace{-0.75cm}
    \caption{}
    \label{fig:1c}
    \end{subfigure}\hfill
    \begin{subfigure}{0.45\linewidth}
\includegraphics[width=\linewidth]{sim_tau_2_m.pdf}

    \vspace{-0.75cm}
    \caption{}
\label{fig:1d}
    \end{subfigure}
    \vspace{-0.25cm}
    \caption{Experimental results for synthetic data. (a) and (c) correspond to low-confidence with $\tau=0.1$. (b) and (d) correspond to the high-confidence with $\tau=2$. See main text for interpretation.}
    \label{fig:1}
\end{figure*}

\paragraph{Synthetic Data.}
For this set of experiments, we vary the $m$ points used to fit the logistic regression model, use $n = 1000$ points to estimate the ATE and average over 100 trials.

We generate $X \in \mathbb{R}^{50}$ by sampling $\ensuremath \mathbf{x}_i$s separately from $\mathcal{N}(\mathbf{0}, 9 \cdot\mathbf{I}_{50})$ for each trial and standardising each separate sampled set of $\ensuremath \mathbf{x}_i$s with the maximum L2-norm of the $\ensuremath \mathbf{x}_i$s in the set.
For each standardised $\ensuremath \mathbf{x}_i$, the treatment assignment $t_i$ and outcome $y_i$ are generated across trials in the following manner:
\begin{align*}
    t_i &\sim \text{Bernoulli}(s(\mathbf{a}^\top\ensuremath \mathbf{x}_i)), & \mathbf{a} & \sim \mathcal{N}(\mathbf{0},\mathbf{I}_{50}),\\
    y_i &= \mathbf{b}^\top\ensuremath \mathbf{x}_i + t_i \tau + \vartheta, & \mathbf{b} & \sim \mathcal{N}(\mathbf{0},\mathbf{I}_{50})
\end{align*}
\noindent where $\vartheta\sim \mathcal{N}(0,0.01)$, $\tau\in\{0.1,2\}$ is a non-zero bias, and $s(\cdot)$ is a sigmoid function.
We perturb the weights of the learned logistic regression model with a sample from $\mathcal{N}(\mathbf{0}, \sigma^2\mathbf{I}_{50})$ with $\sigma$ defined in \eqref{log_reg}.

Figures \ref{fig:1a} and \ref{fig:1b} shows that the probability that the signs of $\hat{\tau}$ disagreeing with both $\hat{\tau}_n$ and $\hat{\tau}_n^\epsilon$ decreases as $\epsilon$ and $\tau$ increases with $m$ set to 1000.
These results reflect the exponential dependence of $\mathbb{P}(\hat{\tau}_n \leq 0)$ and $\mathbb{P}(\hat{\tau}_n^\epsilon \leq 0, \hat{\tau}_n \leq 0)$ on $g(\epsilon,m,n,\lambda,\delta) = O(1/\epsilon^2)$ and $\hat{\tau}$ in Theorems \ref{thm:main-thm-1} and \ref{thm:main-thm-2}.

The convergence of $\hat{\tau}_n^\epsilon$ to $\hat{\tau}$ as $m$ increases from 500 to 2500 at intervals of 500 seen in Figures \ref{fig:1c} and \ref{fig:1d} reinforces the inverse relationship between $g(\epsilon,m,n,\lambda,\delta)$ and $m$ earlier demonstrated in Figure \ref{fig:g_sigma}.
The error bars represent the 95\% confidence interval of the mean of the various estimates.
The above plots also show that increasing $\tau$ yields exponentially larger mean ATE estimates which checks out with the form of $\mu_1^\epsilon$ and $\mu_0^\epsilon$ in \eqref{eq:mu0-mu1}, as the expectation of the log-normal random variables weighting $y_i$ monotonically increases with the variance of the Gaussian random variable.

\paragraph{Semi-synthetic Data.}
Next, we test our methods on the semi-synthetic binary-treatment Infant Health and Development Programme (IHDP) dataset that was introduced in \cite{hill2011bayesian}.
We use the train and test sets from \cite{shalit2017estimating} for fitting logistic regression and estimating the ATE respectively with the true ATE of the train/test splits being 4.
IHDP is a real-world dataset with 25 covariates describing 747 children and their mothers, de-randomised binary treatments and synthetic continuous outcomes that can be used to compute a ground truth ATE \cite{hill2011bayesian}.
We create balanced training and ATE estimation datasets where $m = 500$ and $n = 500$ by sampling with replacement 250 units with $T =1$ and $T=0$ and 100 units with $T =1$ and $T=0$ respectively from the above train and test sets.
As the IHDP dataset comes with 1000 different realisations of train and test data, we average over all realisations.
In Tables \ref{tab:ihdp}, we see that increasing $\epsilon$ generally increases the fidelity of the mean ATE estimate and reduces $\mathbb{P}(\textrm{sgn}(\hat{\tau_n}) \neq \textrm{sgn}(\hat{\tau}))$ and $\mathbb{P}(\textrm{sgn}(\hat{\tau_n^\epsilon}) \neq \textrm{sgn}(\hat{\tau}))$ respectively.
We did not include the average standard deviation of the estimates as estimating ATE with IPW is known to have high variance due to the unboundedness of the propensity score function $\pi_{\hat{\ensuremath \mathbf{w}}}$.
Lastly, another critical observation from Tables \ref{tab:ihdp}, especially for practitioners, is that too small an $\epsilon$ can lead to an unreliable estimate of $\hat{\tau}_{n}$ and $\hat{\tau}_n^{\epsilon}$, \textit{i}.\textit{e}., see the estimates for IHDP when $\epsilon=0.2$.

\paragraph{Real Data.}
To further verify our proposed method, we use the Lalonde observational studies benchmark \cite{lalonde1986evaluating} obtained from \cite{cbps}.
As we do not account for unbalanced datasets, we only use the original Lalonde dataset with 297 treated and 425 control individuals and subsample from it to create our training and ATE estimation datasets.
There are 9 covariates containing sensitive information such as age, education, and race, and the outcome is 1978 earnings.
As no train/test splits are provided, we sample without replacement 100 units of $T=1$ and $T=0$ to create the ATE estimation dataset of size 200 and sample with replacement 250 units with $T=1$ and $T=0$ from the remaining points to generate the training dataset of size 500.
We do the above 1000 times to obtain the same number of realisations as the IHDP dataset.
The results for Lalonde in Tables \ref{tab:lalonde} supplements those for IHDP: increasing $\epsilon$ improves the accuracy of the mean ATE estimate and decreases $\mathbb{P}(\textrm{sgn}(\hat{\tau_n}) \neq \textrm{sgn}(\hat{\tau}))$ and $\mathbb{P}(\textrm{sgn}(\hat{\tau_n^\epsilon}) \neq \textrm{sgn}(\hat{\tau}))$.

\begin{table}[t!]
    \centering
    \caption{Average $\hat{\tau}_{n}$, $\hat{\tau}_n^{\epsilon}$, $\rho(\hat{\tau},\hat{\tau}_n) := \mathbb{P}(\textrm{sgn}(\hat{\tau}_n) \neq \textrm{sgn}(\hat{\tau}))$, and $\rho(\hat{\tau},\hat{\tau}_n^{\epsilon}) := \mathbb{P}(\textrm{sgn}(\hat{\tau}_n^{\epsilon}) \neq \textrm{sgn}(\hat{\tau}))$ for various $\epsilon$ over 1000 runs on IHDP dataset.
    The average $\hat{\tau}$ is 4.80.}
    \label{tab:ihdp}
    \resizebox{\columnwidth}{!}{
    \begin{tabular}{cccccc}
        \toprule
        \multirow{2}{*}{\textbf{Estimate}} &
        \multicolumn{5}{c}{\textbf{Privacy loss} ($\epsilon$)} \\
        & $0.2$ & $0.4$
            & $0.6$ & $0.8$
            & $0.99$ \\

        \midrule\midrule
        $\hat{\tau}_n$ & -237487.30 & 32.34 & 8.37 & 5.22 & 5.31  \\
        $\hat{\tau}_n^{\epsilon}$  & -237477.13 & 32.20 & 8.50 & 5.09 & 5.40  \\
        \midrule
        $\rho(\hat{\tau},\hat{\tau}_n)$ & 0.515  & 0.425 & 0.344 & 0.287 & 0.229 \\
        $\rho(\hat{\tau},\hat{\tau}_n^{\epsilon})$ & 0.517  & 0.425 & 0.347 & 0.301 & 0.228 \\
        \bottomrule
    \end{tabular}}
\end{table}

\begin{table}[t!]
    \centering
    \caption{Average $\hat{\tau}_{n}$, $\hat{\tau}_n^{\epsilon}$, $\rho(\hat{\tau},\hat{\tau}_n) := \mathbb{P}(\textrm{sgn}(\hat{\tau}_n) \neq \textrm{sgn}(\hat{\tau}))$, and $\rho(\hat{\tau},\hat{\tau}_n^{\epsilon}) := \mathbb{P}(\textrm{sgn}(\hat{\tau}_n^{\epsilon}) \neq \textrm{sgn}(\hat{\tau}))$ for various $\epsilon$ over 1000 runs on Lalonde dataset. The average $\hat{\tau}$ is 902.11.}
    \label{tab:lalonde}
    \resizebox{\columnwidth}{!}{
    \begin{tabular}{cccccc}
        \toprule
        \multirow{2}{*}{\textbf{Estimate}} &
        \multicolumn{5}{c}{\textbf{Privacy loss} ($\epsilon$)} \\
        & $0.2$ & $0.4$
            & $0.6$ & $0.8$
            & $0.99$ \\

        \midrule\midrule
        $\hat{\tau}_n$ & 872.21 &  899.15 & 895.26 & 906.70 & 904.17  \\
        $\hat{\tau}_n^{\epsilon}$ & 876.83 &  897.75 & 893.37 & 903.95 & 900.25   \\
        \midrule
        $\rho(\hat{\tau},\hat{\tau}_n)$ & 0.143 & 0.072 & 0.049 & 0.027 & 0.035 \\
        $\rho(\hat{\tau},\hat{\tau}_n^{\epsilon})$ & 0.154 & 0.071 & 0.05 & 0.039 & 0.034 \\
        \bottomrule
    \end{tabular}}
\end{table}

\section{CONCLUSION}
\label{sec:conclusion}
We proposed a differentially private IPW method for average treatment effect under the inverse probability weighting framework.
A key element of our proposed method is the use of a newly defined private propensity score estimator. Unlike traditional propensity scores, ours can be deployed in causal analysis without running the risk of exposing the covariates of any unit used in estimating the propensity score function.
Furthermore, we demonstrate---both theoretically and empirically---that the ATE estimate resulting from an application of our method is consistent with its non-private counterpart with high probability.
In other words, the proposed propensity score function not only safeguards privacy, but also yields valid causal analyses with high probability.

We believe this work highlights long-neglected privacy concerns associated with the use of propensity scores in causal inference, but would also pave the way for subsequent developments at the intersection of differential privacy and causal inference.
Although the starting point of our work is a specific choice of a non-private method for ATE estimation, the analyses can be extended to more sophisticated estimators.
In particular, we note that IPW is mathematically equivalent to other estimation methods such as stratification and the backdoor correction (see, \textit{e}.\textit{g}., ~\citep{Hernan19:CI}).
Future extensions of our work will be dedicated to the development of privatised estimators based on alternative ATE estimation methods as well as other causal estimands such as conditional average treatment effect (CATE) \citep{shalit2017estimating}.
The effectiveness of private propensity scores in more complex methods and settings still remains an open question.

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