Tampere is squeezed between two amazing lakes!
After spending the week relaxing, skiing and trying to figure out the Tampereen bus system, we hopped on a shuttle Saturday morning to head to JamiJarvi, the site of the Jami42 ski marathon. Upon arrival we were thrilled to see the American flag flying high above the stadium next to the Finnish and Estonian flags. The Americans had arrived!
The most popular question to ask us at the race: why are you here?
Tyler bought a cool Jami42 hat. Well, he thinks it's cool at least.
There is a strong trend in classic marathon skiing (and classic skiing in general) for athletes to double-pole the entire course if possible. This allows skiers to race classic races on skate skis without kick wax and this makes the skis much faster. It is definitely weird to think about classic racing on skate skis, but if the conditions are right, double-poling a race can be much faster. We looked at the course profile, and after our kick wax debacle the week before we thought seriously about using skate skis and double-poling. After doing a little reconnaissance and seeing one other person with skate skis we decided it was a great idea and put our skate skis down on the starting line. We were a little nervous about our ski choice so to take our minds off of it we dutifully performed our tried and true race warm-up routine of walking around for a while with our poles and getting cold. With five minutes to go we approached the start line to put on our skis. As we clipped in our bindings a Norwegian man started doing one leg pushups, tricep presses, and other intimidating exercises on ground next to us. We aren't sure whether this was an attempt to warm-up or to intimidate us. Either way, when he asked to get in front of Jackson in the start line Jackson quickly obliged while Tyler looked forward hoping not to be noticed.
Our post race meals continue to impress and we happily ate homemade soups, bread, coffee and delicious crepes after the race. As it turns out our intimidating Norwegian competitor, Morthen, was quite friendly and we enjoyed a two hour conversation with him, his wife, and two young kids while we ate lunch. He has raced many world loppets and other ski races all over Europe. When he and his wife had their second child a few years ago he brought his skis to the hospital so that he could be sure to make it to the Finlandia Hiihto race later in the day. If you can't tell, we idolize him - and are very impressed (and sort of surprised) with how supportive his wife is!
At the end of the day we were waiting outside of the arena for our shuttle when Morthen and his family came outside to head back home. He was sorry that there was not room for him to give us a ride and he offered us each a headband and hat from his sponsor company. We then enjoyed a Finnish post race tradition of his outside their car as they packed up. A glass of whiskey consumed from a sippy cup has never tasted better!
On Sunday, Jackson went to watch World Cup skiing at the Lahti Ski Festival and Tyler enjoyed some restful down time. I arrived early enough to watch all of the pre-race preparation, which is a little more intense than it is in college skiing. Each skier warms up with one, or in the case of some of the Norwegian stars, several wax techs as they try to choose the fastest pair of skis for the day. The Lahti world cup was a pursuit race, meaning that halfway through the race, the skiers switched from classic to skate skis, so ski choice was even more rigorous than usual. Although a lot of fans came, there was more room to spectate than there was at the Holmenkollen Ski Festival and I had a chance to practice some action photography. I indulged myself, taking over 2000 pictures over the course of the day. The success of the American women over the last few years, and especially this year, has made being an American fan at these events really exciting. Despite the Finnish commentary, American skier Jessie Diggins's name could be heard over the loud speaker many times, and she received the loudest cheers of any non-Finnish athletes. The races themselves were exciting but, unfortunately for the host country, ended in Norwegian podium sweeps for both the men and women's races.
American Jessie Diggins, Number 7.
This is really a photo of what Tyler looked like 7 years ago and what he will look like 20 years from now.
In Finland, there are about 5.5 million people and about 2 million saunas. We've been lucky to enjoy several of them, including one in the house of our current hosts, Anssi and Emmi, who live in Helsinki. The steam, or löyly (which is Finnish word to describe sauna steam and is distinct from the word höyry, which describes steam in all other contexts) can create an intense experience, but saunas are integral to experiencing Finnish culture and probably good preparation for races, at least mentally. We travel to Lahti, Finland in a couple of days, which is where our next marathon, the Finlandia-Hiihto, is held.
You guys continue to impress and entertain! Thank you for the laughs- much needed right now. Love you!
This post definitely had some memorable quotes! The starting line scenario is too entertaining -- I can picture it so clearly. Miss you guys!!
Finally catching up on reading, and it's a good thing no-one is in my office right now since I guffawed when you described the Start Line with Morthen.


Students in grades 3-5 who have Coach Haller will begin a 3 vs. 3 speed volleyball tournament on Friday. Speed volleyball is when mini games are played. Each game is 4 minutes. After 4 minutes a winner is determined and teams then rotate to play someone different. It's a lot of fun!
Students in grades 3-5 who have Coach Bruce and Coach Kiefer will continue with basketball. Students in the primary grades will take a break from basketball and volleyball and play a cooperative game to be determined.


4 FRONT LOWER CONTROL ARM BUSHING AND 2 BALL JOINT CAN FIT. NISSAN 350Z 03-09 2WD INFINITI G35 03-07 2WD. All items are follow international standard to produce. Item are made of carbon steel with heat treatment, and ball stud are made of alloy steel.
All raw materials are OE standard. All items we sell are all high quality item.
We have been doing chassis parts for more than 20 years. We do our research carefully and experience before we sell items. It is rare chance of having any mistake. Which is quick and more reliable to pay.
We deliver to worldwide no matter where you are. The item "4 FRONT LOWER CONTROL ARM BUSHING 2 BALL JOINT FOR INFINITI G35 03-07 2WD 350Z" is in sale since Sunday, February 14, 2016. This item is in the category "eBay Motors\Parts & Accessories\Car & Truck Parts\Suspension & Steering\Control Arms & Parts".
The seller is "autochassisparts" and is located in Taichung. This item can be shipped worldwide.


text:
Internet improvements needed but costly: report
The report, Northern Connectivity: Ensuring Quality Communications, was completed by a pan-territorial group led by Yukon's Economic Development Department.
Without better Internet connectivity, the North will suffer.
That's according to a pan-territorial report released last week.
"In the Canadian Arctic, accessible, reliable and affordable communication services are seen as a foundation for northerners to meet many of the socio-economic challenges they face, enabled by networks capable of handling 21st century applications," the report says.
"There is much documented evidence showing the telecommunications infrastructure serving Yukon, Northwest Territories and Nunavut has not kept pace with services available in many large Canadian cities."
The report predicts that if things are not changed, things could get worse.
"Without a dedicated strategy to address the communications infrastructure deficit, the residents of the North are in danger of being left behind even as their territories lead Canada in
GDP growth through increased development – development that benefits Canada, but may not benefit northerners without strategic investments in the communications infrastructure."
The report recommends that transfer speeds be improved to a minimum average target of 9 Mbps for downloads and 1.5 Mbps for uploads.
"This target should be achieved by 2019 in order to meet projected consumer, business and government needs, while recognizing the constraints posed by the backbone infrastructure."
According to the Yukon's Economic Development Department, the only communities in Whitehorse that meet those standards currently are Whitehorse and Carcross.
But the cost of doing that is not cheap.
The report looks at costs as well as what kind of government subsidy "may be required to provide an operator with a reasonable business case for deploying and providing affordable service."
If you combine the three territories, the costs for a network upgrade range from $622.68 million to $2.2 billion, depending on which option is chosen.
When it comes to who will pay for these upgrades, the report notes that financial resources inside the territories would not likely be enough to pay for the recommended network upgrades. "Some outside funding of an initial capital investment would be necessary," it says.
In countries like Australia, the U.S. and the U.K., where broadband expansion has been successful, that money usually comes from the federal government.
Improvements to reliability, availability and service quality are also recommended.
Improvements to the Internet would spur economic growth, create jobs and produce consumer savings, the report says.
"Without improved broadband connectivity in the North, the three territories may lose potential growth and their ranking relative to other economies worldwide that are experiencing these positive impacts," the report says.
"As a result, the North might lose its competitive advantage in the global market, experience losses in jobs (migration of workers) and a decrease in quality of life."
Better broadband connectivity would make rural communities more attractive to businesses.
"As reports from the Canadian Chamber of Commerce and the National Aboriginal Economic Development Board conclude, current infrastructure deficits are serious impediments to economic activity, adding to the costs of doing business and increasing the logistical challenges to development in the area," the report says.
It also notes that better Internet means better education opportunities.
It even goes as far as suggesting that better Internet would mean better healthcare.
"Average per capita healthcare expenditure (2012 estimates) are much higher in the territories: Yukon ($8,916), Northwest Territories ($9,853) and Nunavut ($13,250) than the Canadian average $5,948. Similarly, the portion of GDP spent on healthcare is higher in the territories," it says.
"Broadband connectivity – when combined with e-health technology – can play a large role in bringing these costs down for the territories and closer to the national average."
First responders in emergencies also need fast, reliable communications to be effective, the report says.
No official for NorthwesTel was available for an interview in time for today's deadline. In a written statement, company spokesperson Eric Clements said: " The report raises many valid points about connectivity in the North.
"NorthwesTel has taken steps to address a large majority of them. In 2013, Northwestel's $233 million modernization plan proposed a comprehensive strategy to address many concerns around speed and transport. Last year, we began implementing that plan and have proven our commitment to providing broadband speed that is faster than the minimum target set by the CRTC and NCIS report.
Over the next several years, NorthwesTel will continue to roll out high-speed packages across the North, which will further enable programs such as tele-health and e-learning."
The entire 156-page report is available online at www.northernconnectivity.ca.
First Nations demand royalty sharing deal
Convention planned for Liberal leadership

redpajama_set_name: RedPajamaCommonCrawl


Cute, charming and comfortable home in immaculate Cypress Lake! New, 2 year old roof! New kitchen cabinets & quartz counter tops! New washer & dryer! New wall to wall carpeting! Newer Carrier A/C w/programmable remote! Stainless steel appliances! Dine-in kitchen/family room, with separate living room & dining area. Vaulted ceilings for that large open space feeling. Master en-suite bathroom with separate dual sink vanity & large walk in closet. Covered, screened-in patio for entertaining, with double sliding glass doors leading into kitchen/family room. 2 separate fenced-in yard areas for relaxing and entertaining in private. Corner lot! 2 car garage. Monitored alarm system. Gated community. Low HOA dues. Community pool, tennis courts & clubhouse. The list just goes on and on!


text:
Summer training enrichment for African immigrant women
http://mshale.com/2012/09/13/summer-training-enrichment-african-immigrant-women/
African women at a summer training session offered by the Minnesota African Women's Association (MAWA). Photo: Ken Forbin/Mshale
The Minnesota African Women's Association (MAWA) this past summer, engaged women from different parts of Africa across the Twin cities, in a series of training programs and activities that made the summer very enriching.
On Saturday August 18th 20012, about fifteen women from different African countries attended a two- hour training program at the MAWA office conference room in Brooklyn Park, Minnesota.
The training, which is part of the "Community Power Project", was funded by the Solid Waste Management Coordination Board. The focus was health-related: Smart choices: Creating Non Toxic Environments for Children.
At the training session, female community leaders from different African communities were trained on how to effectively educate women groups in African communities to make smart choices when using cleaning products that contain harmful chemicals at home. Participants at the training, were from various health and human service professions, and came from African countries, like Cameroon, Liberia, Nigeria and Togo.
Mostly made up of nurses, dental hygienists and social workers from across the Twin cities, the participants learned and engaged each other in information sharing on how best they can each transfer the knowledge and skills gained, to their respective communities in a culturally competent manner.
According to one of the participants, Mrs. Maureen Tambi, the most exciting and educational part of the training was the hands-on Activity session where the women learned how to make non toxic cleaning products by themselves. She said the training provided both useful information on how to avoid toxic chemicals at home and a skill set on how to make homemade cleaning supplies that are cheap and environmentally friendly.
Julia Earl, the trainer and presenter of the Smart Choices session said that the training was both useful and timely because of the increase in children's exposure to toxic materials, especially from cleaning supplies used at home. She added that the prevalence of some illnesses like; autism and hyper activity in children have now been associated with exposure to toxic chemicals.
Commenting on the fact that the group consisted mostly of health professionals, Julia said it was an added advantage because the women grasped the knowledge faster and engaged each other in really resourceful conversation.
But the Smart choice training session was only one of the several other programs MAWA is offering this summer. On August 11th 2012, MAWA sponsored a Shoulder-to-Shoulder training for parents of Amakolo participants and other African refugee and asylee community leaders. "Shoulder to Shoulder" is an approach that teaches parents some innovative ways of dealing with their teenage children.
Last year, the training session was offered only to MAWA staff that in turn carried the message to the various African communities. This year however, thanks to more funding, MAWA extended the training to involve leaders of community based organizations. Participants came from Somalia, Sudan, Liberia and Cameroon and in the course of the training; these parents learnt that the best way to deal with their teenage children, is through Positive Parenting.
The Positive Parenting approach is where one the parents are warm, supportive and encouraging while being firm, consistent and clear with limits and boundaries. The training is particularly important to African mothers because due to cultural differences, child up bringing in America has now become a little more challenging.
A third workshop offered by MAWA was a health-related one for young people. MAWA is offering the nationally acclaimed curriculum, BART: Becoming A Responsible Teen, to 180 African teenagers. This training targets teenagers from African immigrant families and its first session begins August 28th through the 31st. MAWA's objective for this program is to respond to a growing need.
Explaining the structure of the program to Mshale, Nambangi of MAWA noted that the BART program is gender based. There will be groups of 8 to 15 teenagers for girls and another for boys. Also commenting on the relevance of the program, she said that with the rise in teen pregnancy among African teenage girls, such a program is not only necessary but urgent. Statistics from the Minnesota HIV/AIDS Surveillance System on the prevalence of HIV/AIDS among African teens also reiterates the urgency for such a program that responds to building HIV/AIDS awareness as well as teen pregnancy prevention education.
A total of 144 girls and 36 boys are anticipated to benefit from the program. While three African Community groups have already indicated their interest in having their teenagers go through this training, MAWA is willing to extend the program beyond its own AGILE participants to include more teenagers from two more African community groups. Leaders of teen initiatives in African communities are encouraged to put together groups of 12-15 girls and boys and register with MAWA.


text:
var express = require('express');
var path = require('path');
var httpProxy = require('http-proxy');

var proxy = httpProxy.createProxyServer();
var app = express();

var isProduction = process.env.NODE_ENV === 'production';
var port = isProduction ? process.env.PORT : 3000;
var publicPath = path.resolve(__dirname, 'public');

app.use(express.static(publicPath));

if (!isProduction) {
    var bundle = require('./server/bundle.js');
    bundle();
    app.all('/build/*', function (req, res) {
        proxy.web(req, res, {
            target: 'http://localhost:8080'
        });
    });
}

app.get('*.png', function (req, res) {
    res.sendFile(publicPath + req.path);
});

proxy.on('error', function (e) {
    console.log('Could not connect to proxy, please try again...');
});

app.listen(port, function () {
    console.log('Server running on port ' + port);
});

redpajama_set_name: RedPajamaGithub


Do you have an idea (or no idea) of what your personal version of optimal wellness looks and/or feels like? What is the wall or blockage between you and your vision of optimal wellness? Do you want to take control of your life?
Wellness and Nutrition Coaching services are designed for exploring and discovering while promoting long-term lifestyle and wellness changes. Based on your needs this may include directed reading, nutritional education and support, prepared meals, Reiki, meditation, QiGong, and / or referral for body work / personal training. Each program is designed for your unique needs.


text:
Q: pip install error under virtual env I am trying to install packages from inside the virtualenv
(Ubuntu 14.04LTS)
pip install scipy

I end up with the following error:
Collecting scipy
Exception:
Traceback (most recent call last):
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/basecommand.py", line 215, in main
    status = self.run(options, args)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/commands/install.py", line 324, in run
    requirement_set.prepare_files(finder)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/req/req_set.py", line 380, in prepare_files
    ignore_dependencies=self.ignore_dependencies))
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/req/req_set.py", line 554, in _prepare_file
    require_hashes
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/req/req_install.py", line 278, in populate_link
    self.link = finder.find_requirement(self, upgrade)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/index.py", line 465, in find_requirement
    all_candidates = self.find_all_candidates(req.name)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/index.py", line 423, in find_all_candidates
    for page in self._get_pages(url_locations, project_name):
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/index.py", line 568, in _get_pages
    page = self._get_page(location)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/index.py", line 683, in _get_page
    return HTMLPage.get_page(link, session=self.session)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/index.py", line 792, in get_page
    "Cache-Control": "max-age=600",
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/sessions.py", line 488, in get
    return self.request('GET', url, **kwargs)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/download.py", line 386, in request
    return super(PipSession, self).request(method, url, *args, **kwargs)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/sessions.py", line 475, in request
    resp = self.send(prep, **send_kwargs)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/sessions.py", line 596, in send
    r = adapter.send(request, **kwargs)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/cachecontrol/adapter.py", line 47, in send
    resp = super(CacheControlAdapter, self).send(request, **kw)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/adapters.py", line 390, in send
    conn = self.get_connection(request.url, proxies)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/adapters.py", line 290, in get_connection
    proxy_manager = self.proxy_manager_for(proxy)
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/adapters.py", line 184, in proxy_manager_for
    **proxy_kwargs
  File "/home/rduser/venvs/python2713/lib/python2.7/site-packages/pip/_vendor/requests/packages/urllib3/contrib/socks.py", line 154, in __init__
    "Unable to determine SOCKS version from %s" % proxy_url
ValueError: Unable to determine SOCKS version from socks://10.0.0.28:8080/

How do I get past this.?
P.S: I am using the latest version of pip

redpajama_set_name: RedPajamaStackExchange


Jane A. Kelley, LCSW is the founder and director of the Kelley Clinic, LLC. She has been practicing in the Milwaukee area since the mid-1980's. She received her Master's Degree in clinical social work from the University of Wisconsin-Milwaukee in 1988.
Jane's therapeutic approach is grounded in the body/mind/emotions/spirit connection, in her desire to help clients on their path to wholeness. She carries specialties in the area of addiction, i.e. AODA, relationship dependency, and compulsive overeating. In addition, she is trained in grief and loss; depression; anxiety; infertility and childbirth concerns; issues related to spirituality and sexual orientation.
In 2010, Jane completed a 5-year Analyst Training Program at the C. G. Jung Institute of Chicago and became a Jungian Psychoanalyst.
1991-present – Seminars given in 1-day; 2-day; weekend; weeklong formats.


text:
Preview Cart Checkout
WCF News
Initial Construction on National Willa Cather Center Begins
On October 9, 2014, phase one of construction began on the historic Moon Block building in downtown Red Cloud, Nebraska, which will become the National Willa Cather Center. "Beginning construction is a milestone in recognizing Willa Cather's enduring legacy. We are grateful to more than 350 individuals and foundations nationwide who have made this possible," said Honorary National Chair, Ken Burns.
The initial phase of construction will prevent further deterioration to the historic building and prepare it for reconfiguration and finishing work. Executive Director Ashley Olson stated, "After many years of careful planning, it's rewarding to see preliminary work proceed while we continue the campaign to raise the balance of funds necessary to begin the final phase of construction next spring."
When complete in 2016, the National Willa Cather Center's archive and public museum will safely house and provide access to countless original documents and objects related to the life and work of nationally-renowned author, Willa Cather. Other components include classroom and multimedia space, enlarged backstage facilities for the Red Cloud Opera House, exhibition space, an expanded bookstore, loft apartments for visitors, and street-level retail spaces.
The Willa Cather Foundation continues their capital campaign to raise the remaining $787,000 needed to bring the project to completion. The entire project is estimated to cost $6.7 million. To date, $5.9 million has been raised in private gifts and grants, including support from the Peter Kiewit Foundation, Robert B. Daugherty Foundation, and a community development block grant from the Nebraska Department of Economic Development.
Founded in 1955, The Willa Cather Foundation is a 501(c)(3) not-for-profit organization dedicated to preserving and promoting understanding and appreciation of the life, time, settings, and work of Pulitzer Prize-winning author Willa Cather. The Foundation currently owns or manages eleven properties connected with Willa Cather, including a 612-acre prairie, her childhood home, and the Red Cloud Opera House. Conferences, seminars, as well as arts and humanities events are held by the Foundation year-round in Red Cloud and internationally.
Donors may send contributions to the Willa Cather Foundation, 413 North Webster Street, Red Cloud, NE 68970, or give online. For more information call the Foundation toll-free at 866-731-7304.
Currently in the Gallery: Jorn Olsen Photography
Documentary filmmaker and lifelong Willa Cather enthusiast Ken Burns is the honorary national chair for the Center's public fundraising effort. According to Burns, who announced the creation of the Center in an online video, "The Center will have something for everyone. As an archive and museum, it will become the permanent home to some of Willa Cather's most personal treasures, items made famous in her stories, and artwork inspired by her vision of the prairie. For scholars, it will be a chance to explore, firsthand, some of her most insightful original works and papers."
In addition to Burns, former First Lady Laura Bush has noted, "Willa Cather's literature gives voice to the values that define America's frontier heritage. By reading her words and promoting her contributions to western culture, we commemorate the pioneering spirit that distinguishes us a nation."
Click here to learn more about the creation of the new National Willa Cather Center. For more information or to get involved, contact Ashley Olson, Executive Director, aolson@willacather.org, 866-731-7304. Please donate by Spring 2015 and share the news on social media! #fundnwcc
Related Website Links:
Member Profile: Meet Jerry and Susan Anderson
Both avid Willa Cather enthusiasts, the Andersons have traveled to various settings around North America that appear in Willa Cather's novels.
Executive Director Ashley Olson to be Honored as "Fundraiser of the Year" by Nebraska AFP
Ashley Olson to receive "Fundraiser of the Year" at Association of Fundraising Professionals Nebraska Chapter December 17 reception in Omaha, Nebraska.
Thanksgiving Traditions: Reflections from our Archives
Thanksgiving seems to have been especially meaningful to Willa Cather. Here's a peek into the Archives & Collections at the National Willa Cather Center of some related writing and correspondence.
Whey Jennings & The Unwanted
Julia Noyes Art Exhibit
Valentine's Day Weekend Package
Book Title Short Sleeve T-Shirt
There was nothing but land: not a country at all, but the material out of which countries are made.
– My Àntonia
National Willa Cather Center
Red Cloud Opera House
Cather Prairie
Willa Cather's Red Cloud
DVDs & Audio Books
My Ántonia Products
Willa Cather Review
Serena's Album
Conferences & CFP
Opera House Sponsor
Conference Sponsor
The National Willa Cather Center•413 North Webster|Red Cloud, NE 68970|1.866.731.7304
© All Rights Reserved The Willa Cather Foundation. This institution is an equal opportunity provider & employer.
The National Willa Cather Center is the registered tradename of the Willa Cather Foundation, a not-for-profit, tax-exempt charitable organization (tax identifcation number 47-0485401) under Section 501(c)(3) of the Internal Revenue Code.
Website Design and Development by UNANIMOUS


text:
Home » Carter takes creative mantle at Fremantle
Carter takes creative mantle at Fremantle
Guider, Elizabeth
Daily Variety;7/6/2006, Vol. 292 Issue 2, p2
The article reports on the promotion of Gary Carter as one of the key creative executives responsible for FremantleMedia's global programming ideas for both fiction and nonfiction. He will also be responsible for the company's drama and entertainment programming and lead the company on the inhouse program development. He will also remain as chief creative officer of FMX, and will mediate with producers to fashion brands and concepts. He will be reporting to CEO Tony Cohen.
Fremantle getting creative with Carter. Guider, Elizabeth; Clarke, Steve // Daily Variety;1/18/2006, Vol. 290 Issue 13, p7
The article reports that FremantleMedia Ltd. has promoted Gary Carter to the new chief creative officer position. The move was a step toward filling the vacancy left by the departure of the company's CEO. Carter will be responsible for creating new brands and concepts for new programs including...
THE REALITY FACTORY. Sloan, Paul // Time International (South Pacific Edition);8/9/2004, Issue 31, p46
The article reports on the growing popularity of reality television shows. FremantleMedia Ltd. has been instrumental behind several reality shows. FremantleMedia recently launched the reality show titled "Malaysian Idol." The show has been brilliantly produced. According to Tony Cohen, CEO of...
FremantleMedia hikes Mackay to regional CEO. Guider, Elizabeth // Daily Variety;7/30/2003, Vol. 280 Issue 18, p5
Reports on the promotion of Catherine Mackay to regional chief executive officer (CEO) in the U.S., Australasia and Asia for international producer FremantleMedia Ltd. Job background of Mackay at the company; Responsibilities as regional CEO; Career highlights.
Fremantle shakeup's exec factor. // Daily Variety;4/20/2012, Vol. 315 Issue 14, p1
The article announces that Cecile Frot-Coutaz was appointed chief executive officer (CEO) at FremantleMedia Ltd.
Fremantle vet logging off. Guider, Elizabeth // Daily Variety;1/16/2006, Vol. 290 Issue 11, p10
The article reports on the resignation of Catherine Mackay as chief executive officer of Enterprises and New Platforms division and as a board member of FremantleMedia Ltd. in the U.S. Mackay had been responsible for the growth of FremantleMedia in terms of strategy and profits. Mackay had been...
New chief at Spiegel Catalog. // Home Textiles Today;12/18/2000, Vol. 22 Issue 16, p1
Reports the promotion of Melissa Payner as president and chief executive officer of Spiegel Catalog in Illinois.
People Moves. // License! Europe;Apr/May2006, p12
Career developments involving licensing officers in Great Britain are discussed. David Berry has joined Diageo as director of licensing. Celador International has promoted Tina-Louise Phipps as head of licensing and marketing. Gary Carter has been appointed senior executive vice president of...
Minor League Flyers Win Championship, Business Hurdles Prove More Challenging. d'Hippolito, Joseph // Orange County Business Journal;10/6/2008, Vol. 31 Issue 40, p1
The article reports on the success of the baseball team Orange County Flyers in winning the championship game against the Calgary Vipers in California. Harris Tulchin, managing partner of Orange County Flyers, describes the game as fun and inexpensive family entertainment. Gary Carter, the...
BRIEFLY. // Modern Healthcare;9/17/2007, Vol. 37 Issue 37, p4
The article offers news briefs related to U.S. healthcare industry. Gary Carter, New Jersey Hospital Association president and chief executive officer, announced his plans to retire in June 2008. Minnesota Governor Tim Pawlenty reveals that the medical records for about 60 percent of the state's...


text:
Protestors hit parliament over Newman's industrial reforms
Queensland Council of Unions 'Rally for Right' protest outside Queensland Parliament on Tuesday, November 19, 2013. Picture: Adam Davies
by adavies
Adam Davies Senior Journalist
Adam was born in New South Wales and was educated at the prestigious Scots College in Sydney. He has worked both in Australia and United Kingdom for some of the largest newspapers in the two respective countries. He joined APN as a senior journalist at The Chronicle in Toowoomba in 2010, before moving to APN'S Brisbane Newsdesk in 2013 where he covered politics and court. Adam won a 2015 Queensland Clarion Award - the state's premier journalism awards - and was named 2011 APN Daily Reporter of...
MORE than 500 disgruntled workers took the opportunity in the final week of State Parliament to express their anger over a host of changes which they claim threatens their livelihoods.
The protest outside State Parliament today organised by the Queensland Council of Unions saw teachers, lawyers, nurses, emergency services personnel and union members united in their disapproval surrounding a host of industrial reforms introduced this year by the State Government.
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text:
\section{Concluding Remarks}\label{sec:conclusion}

To evaluate the performance of our main results, we randomly generate the system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$ by $500$ times according to the following conditions:
\begin{itemize}
\item $n=m=5$, $W_i=\{i\}$ for any user $i\in [5]$, and ${\bf A}$ is generated based on a randomly chosen graph $\mathcal{G}$ from the $9608$ nonisomorphic 5-vertex directed graphs \cite{arbabjolfaei2018fundamentals} such that $A_i=\{ j\in [5]:(j,i)\in \mathcal{G} \}$.
\item For any $i\in [5]$, $\mathcal{X}_i=\{0,1\}$ and $X_i\sim {\rm Bern}(p_i)$, where $p_i$ is uniformly randomly chosen from range $(0,1)$;
\item For any $i\in [5]$, $d_i={\tilde{d}}_i\cdot H_{\infty}({X}_{G_i})$, where ${\tilde{d}}_i$ is uniformly randomly chosen from range $(0,1)$;
\item $P\subseteq [5]$ is randomly generated assuring that $|P|\le 2$.
\end{itemize}
For each system, we compute the lower bounds $\mathcal{L}_{\rm max}^{\rm Thm. 1}$ and $\mathcal{L}_{\rm max}^{\rm Thm. 2}$ by Theorems \ref{thm:lower:bound:confusion} and \ref{thm:lower:bound:pm}, respectively.
An interesting observation is that we have $\mathcal{L}_{\rm max}^{\rm Thm. 2}>\mathcal{L}_{\rm max}^{\rm Thm. 1}$ for only $2$ among the $500$ tested systems, while $\mathcal{L}_{\rm max}^{\rm Thm. 1}>\mathcal{L}_{\rm max}^{\rm Thm. 2}$ for all the $498$ remaining systems.

We also compute the MaxL according to the privacy mechanism $P_{Y|{X}_{[n]}}$ given by Algorithm \ref{alg:agglomerative}, denoted as $\mathcal{L}_{\rm max}^{\rm Alg. 1}$.
Then we compute the ratio $R = \mathcal{L}_{\rm max}^{\rm Alg. 1}/\max( \mathcal{L}_{\rm max}^{\rm Thm. 1},\mathcal{L}_{\rm max}^{\rm Thm. 2} )$.
A lower ratio $R$ (close to $1$) means that our converse and achievable results perform well and are quite close to the optimal $\mathcal{L}_{\rm max}^*$, while a higher ratio indicates bad performance.
We summarize the values of $R$ from \textcolor{black}{$500$} tests in Table \ref{tab:test}, from which we can see that the proposed techniques achieves a satisfactory level of performance for the majority of tested problems. \textcolor{black}{Notably, we have $\mathcal{L}_{\rm max}^{\rm Thm. 1}=\mathcal{L}_{\rm max}^{\rm Alg. 1}$ and thus $R=1$ for $162$ among the $500$ tests. }

\newcommand{\opertypesmallone}[1]{\begin{minipage}{14mm}\centering\vspace{1mm} #1\vspace{1mm}\end{minipage}}
\newcommand{\opertypesmall}[1]{\begin{minipage}{9mm}\centering\vspace{1mm} #1\vspace{1mm}\end{minipage}}
\newcommand{\opertypesmalltwo}[1]{\begin{minipage}{15mm}\centering\vspace{1mm} #1\vspace{1mm}\end{minipage}}
\newcommand{\opertypemed}[1]{\begin{minipage}{70mm}\centering\vspace{1mm} #1\vspace{1mm}\end{minipage}}

\begin{table}[thb]
\begin{center}
\caption{Performance of the Lower Bounds versus Algorithm \ref{alg:agglomerative}.}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\opertypesmallone{Ratio, $R$}&\opertypesmall{$=1$}&\opertypesmall{$<1.05$}&\opertypesmall{$<1.1$}&\opertypesmall{$<1.2$}&\opertypesmall{$\ge 1.2$}\\
\hline
\opertypesmallone{Number of Systems} &\opertypesmall{$162$}&\opertypesmall{$401$}& \opertypesmall{$429$} &\opertypesmall{$460$} &\opertypesmall{$40$}\\
\hline
\end{tabular}\label{tab:test}
\end{center}
\end{table}

Future directions include improving the privacy mechanism and the converse results for the multi-terminal guessing problem \eqref{eq:model:tradeoff}, as well as studying the multi-terminal privacy-utility tradeoff using different privacy and utility measures.

\section{Introduction}\label{sec:intro}
\vspace{-0.125mm}

In this paper, we consider an information-theoretic multi-terminal guessing framework with side information. Our model is inspired by that of the index coding problem \cite{Birk--Kol2006,bar2011index}, but with a significant twist to place emphasis on privacy.
Index coding is a communication problem where a sender aims to efficiently broadcast to multiple users through a noiseless channel. See \cite{arbabjolfaei2018fundamentals} and references therein.
\textcolor{black}{In particular, the security and privacy aspects of index coding have been investigated in \cite{dau2012security,ong2016secure,mojahedian2017perfectly,liu:vellambi:kim:sadeghi:itw18,narayanan2018private,liu2020secure,karmoose2019privacy}.}
Instead of \textcolor{black}{maximizing the communication rate}, in our framework the sender's goal is to balance the privacy and utility performance in the broadcast, both measured based on the success rate of correctly estimating a certain parameter of interest about the sources \textcolor{black}{through} a single guess.
We consider multiple independent sources available at a data curator and assume that there are multiple legitimate users, as well as one adversary in the system. Each party (a user or the adversary) knows some sources a priori as side information. The data curator broadcasts \textcolor{black}{(discloses)} a distorted function of the sources to the users, which is also overheard by the adversary.
\textcolor{black}{Such framework has applications in various real-world scenarios, such as field data broadcasting from a paddock aggregator in the presence of a malicious agent in precision agriculture. }

For the adversary, we adopt the maximal leakage introduced in \cite{issa2019operational} as the privacy metric. \textcolor{black}{It measures} the worst-case information leakage in terms of the gain of the \textcolor{black}{adversary in maximum a posteriori estimation} of any target function of the unknown sources \emph{after and before} observing the disclosed data. For legitimate users, we define our utility metric \textcolor{black}{such that it also reflects the improvement in users' guessing  ability}.
Quite often in practice different sources are of different levels of priority to a user.
We capture this by dividing the unknown sources of each user into two subsets.
Some essential source are required to be perfectly reconstructed by the user. That is, the correct guessing probability of such sources after observing the disclosed data is non-negotiable and must be $1$.
The remaining sources are less critical, for which the user requires the guessing gain \textcolor{black}{about each such source} to be larger than a certain negotiable threshold.
The privacy-utility tradeoff is cast as a constrained optimization problem, where the objective is to minimize the privacy leakage to the adversary, conditioned that \textcolor{black}{the requirements on the utility} of the \textcolor{black}{unknown sources for} the legitimate users are satisfied.

The contributions of this paper are twofold. First, we derive two lower bounds on the privacy leakage given the source distribution and utility constraints (cf. Theorems \ref{thm:lower:bound:confusion}-\ref{thm:lower:bound:pm}) from different persepctives.
These lower bounds serve as converse results (fundamental performance limits) for the privacy leakage.
\textcolor{black}{Second, we propose a greedy algorithm as \textcolor{black}{the privacy mechanism} (cf. Algorithm \ref{alg:agglomerative}) inspired by the agglomerative clustering method used in the information bottleneck and privacy funnel problems \cite{slonim2000agglomerative,makhdoumi2014information,ding2019submodularity}.}
We leverage the connection between our data disclosure problem and the index coding problem when investigating both the converse results and the greedy algorithm.

{\it Notation:} For non-negative integers $a$ and $b$, $[a]$ denotes the set $\{1,2,\cdots,a\}$ and $[a:b]$ denotes the set $\{a,a+1,\cdots,b\}$.
For a set $S$, $|S|$ denotes its cardinality.
For any discrete random variable $Z$ with probability distribution $P_Z$, we denote its alphabet by $\mathcal{Z}$ with realizations $z\in \mathcal{Z}$. We denote an estimator of $Z$ by ${\hat{Z}}$, whose alphabet is also $\mathcal{Z}$.

\section{Lower Bounds on the Privacy Leakage}  \label{sec:lower:bound}

We derive two information-theoretic lower bounds on the privacy leakage. One is based on the utility constraint \eqref{eq:model:utility:wi} only, while the other is obtained based on both \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi}.

\subsection{Lower Bound Based on the Confusion Graph}  \label{sec:lower:bound:confusion}

The utility constraint \eqref{eq:model:utility:wi} indicates that for user $i$, any different realizations ${x}_{W_i}\neq {x}_{W_i}'\in \mathcal{X}_{W_i}$ must be distinguishable based on the released $y\in \mathcal{Y}$, as well as the user's side information ${x}_{A_i}\in \mathcal{X}_{A_i}$.
To describe such distinguishability,
we recall the notion of confusion graph for index coding \cite{alon2008broadcasting}.

\begin{definition}[Confusion graph\cite{alon2008broadcasting}]
Any two realizations of the $n$ sources ${x}_{[n]}^1,{x}_{[n]}^2 \in \mathcal{X}_{[n]}$ are \emph{confusable} if there exists some user $i\in [m]$ such that ${x}_{W_i}^1\neq {x}_{W_i}^2$ and ${x}_{A_i}^1={x}_{A_i}^2$. A confusion graph $\Gamma$ is an undirected uncapacitated graph with $|\mathcal{X}_{[n]}|$ vertices such that every vertex corresponds to a unique realization ${x}_{[n]}\in \mathcal{X}_{[n]}$ and an edge connects two vertices if and only if their corresponding realizations are confusable.
\end{definition}

Therefore, to ensure \eqref{eq:model:utility:wi}, a group of realizations of ${X}_{[n]}$ can be mapped to the same $y$ with nonzero probability only if they are pairwisely not confusable.
More rigorously, for any $S\subseteq [n]$, defin
\begin{alignat}{2}
\mathcal{Y}({x}_S)&\doteq \{ y\in \mathcal{Y}:P_{Y|{X}_S}(y|{x}_S)>0 \}, &&\qquad \forall {x}_S\in \mathcal{X}_S,   \nonumber   \\
\mathcal{X}_S(y)&\doteq \{ {x}_S\in \mathcal{X}_S:P_{Y|{X}_S}(y|{x}_S)>0 \}, &&\qquad \forall y\in \mathcal{Y}.  \nonumber
\end{alignat}
Then, we have the following lemma. We omit the proof as it can be simply done by contradiction.
\begin{lemma}    \label{lem:confusion}
Given a $P_{Y|{X}_{[n]}}$ satisfying \eqref{eq:model:utility:wi}, for any two confusable ${x}_{[n]}^1,{x}_{[n]}^2 \in \mathcal{X}_{[n]}$, we have $\mathcal{Y}({x}_{[n]}^1)\cap \mathcal{Y}({x}_{[n]}^2)=\emptyset. $
\end{lemma}

Given a set $S\subseteq [n]$ and a specific realization ${x}_S\in \mathcal{X}_S$, we define $\Gamma({x}_S)$ as the subgraph of $\Gamma$ induced by all the vertices ${x}_{[n]}$ such that ${x}_{[n]}=({x}_S,{x}_{S^c})$ for some ${x}_{S^c}\in \mathcal{X}_{S^c}$.
Notice that for any ${x}_S^1\neq {x}_S^2\in \mathcal{X}_S$ and ${x}_{S^c}^1\neq {x}_{S^c}^2\in \mathcal{X}_{S^c}$, $({x}_S^1,{x}_{S^c}^1)$ and $({x}_S^1,{x}_{S^c}^2)$ are confusable if and only if $({x}_S^2,{x}_{S^c}^1)$ and $({x}_S^2,{x}_{S^c}^2)$ are confusable. Hence, given $S\subseteq [n]$, the subgraphs $\Gamma({x}_S)$, ${x}_S\in \mathcal{X}_S$ are isomorphic to each other, and thus we simply denote any such subgraph by $\Gamma(S)$.

We present our main result of this subsection as follows.
\begin{theorem}   \label{thm:lower:bound:confusion}
For the problem \eqref{eq:model:tradeoff} with confusion graph $\Gamma$:
\begin{align}
\mathcal{L}_{\rm max}\ge \log \omega(\Gamma(P)),   \label{eq:lower:bound:confusion}
\end{align}
where $\omega(\cdot)$ is the clique number (size of the largest clique) \textcolor{black}{of a graph}.
\end{theorem}
\begin{IEEEproof}
Consider any ${x}_P\in \mathcal{X}_P$. There exists some realizations ${x}_Q^1,{x}_Q^2,\ldots,{x}_Q^{\omega(\Gamma({x}_P))} \in \mathcal{X}_Q$ whose corresponding vertices in the subgraph $\Gamma({x}_P)$ form a clique,
which indicates that the realizations $({x}_P,{x}_Q^1),({x}_P,{x}_Q^2),\ldots,({x}_P,{x}_Q^{\omega(\Gamma({x}_P))})$ also form a clique in $\Gamma$.
That is, the realizations $({x}_P,{x}_Q^1),\ldots,({x}_P,{x}_Q^{\omega(\Gamma({x}_P))})$ are pairwisely confusable.
Then by Lemma \ref{lem:confusion}, for any $k\neq k'\in [\omega(\Gamma({x}_P))]$, we have
\begin{align}
\mathcal{Y}( ({x}_P,{x}_Q^k) )\cap \mathcal{Y}( ({x}_P,{x}_Q^{k'}) )=\emptyset.    \label{eq:thm:confusion:disjoint}
\end{align}

Therefore, we have
\begin{align}
&\sum_{y} \max_{{x}_Q} P_{Y|{X}_{[n]}}(y|{x}_P,{x}_Q)    \nonumber   \\
&\ge \sum_{k\in [\omega(\Gamma({x}_P))]} \sum_{y\in \mathcal{Y}({x}_P,{x}_Q^k)} \max_{\substack{{x}_Q}} P_{Y|{X}_{[n]}}(y|{x}_P,{x}_Q)   \nonumber   \\
&\ge \sum_{k\in [\omega(\Gamma({x}_P))]} \sum_{y\in \mathcal{Y}({x}_P,{x}_Q^k)} P_{Y|{X}_{[n]}}(y|{x}_P,{x}_Q^k)   \nonumber   \\
&=\sum_{k\in [\omega(\Gamma({x}_P))]} 1=\omega(\Gamma({x}_P))=\omega(\Gamma(P)),    \label{eq:thm:confusion:eachxp}
\end{align}
where the first inequality follows from \eqref{eq:thm:confusion:disjoint}.
Hence, we have
\begin{align}
\mathcal{L}_{\rm max
&=\log \sum_{{x}_P} P_{{X}_P}({x}_P) \sum_{y\in \mathcal{Y}} \max_{{x}_Q} P_{Y|{X}_{[n]}}(y|{x}_P,{x}_Q)      \nonumber  \\
&\ge \log \sum_{{x}_P} P_{{X}_P}({x}_P)\cdot \omega(\Gamma(P))=\log \omega(\Gamma(P)),   \nonumber
\end{align}
where the first equality is due to source independence,
and the inequality follows from \eqref{eq:thm:confusion:eachxp}.
\end{IEEEproof}

\subsection{Lower Bound Based on Guessing Gain and \textcolor{black}{Polymatroidal Functions}}

We introduce a key lemma that serves as the baseline in the lower bound to be developed in this subsection.
\begin{lemma}  \label{lem:lower:bound:core}
For the problem \eqref{eq:model:tradeoff},
we have
\begin{align}
&\mathcal{L}_{\rm max}\ge \max\{ I({X}_Q;Y|{X}_P), \max_{i\in [m]:G_i\subseteq Q}\Delta_i \},   \label{eq:lower:bound:core}
\end{align}
where $\Delta_i\doteq D_i+H({X}_{W_i\cap Q})+I({X}_{A_i\cap Q};Y|{X}_{P}).
\end{lemma}
\vspace{0.5mm}

The proof is presented in Appendix \ref{appendix:lem:core}.

For a given system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$, $H({X}_{W_i\cap Q})$ has a fixed value
and $D_i$ is lower bounded by $d_i$ according to \eqref{eq:model:utility:gi}.
Hence, the only terms in \eqref{eq:lower:bound:core} that still depend on $P_{Y|{X}_{[n]}}$ are the mutual information $I({X}_P;Y|{X}_Q)$ and $I({X}_{A_i\cap Q};Y|{X}_P)$.
We further bound these mutual information terms below.

We draw inspiration from the polymatroidal bound \cite{blasiak2011lexicographic,Arbabjolfaei--Bandemer--Kim--Sasoglu--Wang2013} for index coding.
The bound is based on the polymatroidal axioms, which capture Shannon-type inequalities on the entropy function and play a central role in computing converse results in network information theory \cite{yeung2008information}.

\iffalse
\subsection{Lower Bound Based on the Locally Acyclic Structure}

Inspired by the maximum acyclic induced subgraph (MAIS) bound \cite[Theorems 3 and 8]{bar2011index} (see also \cite[Section V-B]{liu2019tight}), we give the following definition.
\begin{definition}  \label{def:lower:bound:acyclic}
For the system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$,
a source set $S=\{ j_1,j_2,\ldots,j_{|S|} \}\subseteq [n]$ is called \emph{locally acyclic} with respect to some set $T\subseteq S^c$ if and only if for any $k\in [|S|]$, there exists some user $i_k\in [m]$ such that
\begin{align}
j_k&\in W_{i_k},   \label{eq:lower:bound:def:acyclic:wi}   \\
A_{i_k}&\subseteq T\cup \{j_1,j_2,\ldots,j_{k-1}\} \label{eq:lower:bound:def:acyclic:ai}.
\end{align}
\end{definition}

\begin{lemma}\label{lem:lower:bound:mais}
Consider the problem \eqref{eq:model:tradeoff}.
For any disjoint $V,Z\subseteq [n]$, let $\mathcal{M}(V,Z)$ denote the collection of $S\subseteq V$ such that $S$ is locally acyclic with repect to $Z\cup (V\setminus S)$. Then we have
\begin{align}
I({X}_V;Y|{X}_Z)\ge \max_{S\in \mathcal{M}(V,Z)}H({X}_S). \nonumber  \quad \IEEEQED
\end{align}

\end{lemma}

\begin{IEEEproof}
By Definition \ref{def:lower:bound:acyclic}, for any $S\in \mathcal{M}(V,Z)$, there exists some $i_k\in [m]$ such that \eqref{eq:lower:bound:def:acyclic:wi} and \eqref{eq:lower:bound:def:acyclic:ai} hold. Hence,
\begin{align}
I({X}_V;Y|{X}_Z)&\ge I({X}_S;Y|{X}_Z,{X}_{V\setminus S})   \nonumber   \\
&=\sum_{k\in [|S|]}I(X_{j_k};Y|{X}_{Z\cup (V\setminus S)\cup \{ j_1,\ldots,j_{k-1} \}})     \nonumber    \\
&\ge H({X}_{S})-\sum_{k\in [|S|]}H(X_{j_k}|Y,{X}_{A_{i_k}})    \label{eq:lower:bound:lem:mais:proof:1}   \\
&=H({X}_{S}),  \label{eq:lower:bound:lem:mais:proof:2}
\end{align}
where \eqref{eq:lower:bound:lem:mais:proof:1} follows from source independence and \eqref{eq:lower:bound:def:acyclic:ai}, and \eqref{eq:lower:bound:lem:mais:proof:2} follows from \eqref{eq:lower:bound:def:acyclic:wi} and \eqref{eq:model:utility:wi}.
\end{IEEEproof}

Combinning Lemmas \ref{lem:lower:bound:core} and \ref{lem:lower:bound:mais} gives the following result.

\begin{theorem}\label{thm:lower:bound:mais}
For the problem \eqref{eq:model:tradeoff},
we have
\begin{align}
&\mathcal{L}_{\rm max}\ge \max\{ \max_{S\in \mathcal{M}(Q,P)}H({X}_S), \max_{i\in [m]:G_i\subseteq Q}\Gamma_i^{\rm acyc} \},   \nonumber
\end{align}
where $\Gamma_i^{\rm acyc}\doteq d_i+H({X}_{W_i\cap Q})+\max_{S\in \mathcal{M}(A_i\cap Q,P)}H({X}_S)$. \IEEEQED
\end{theorem}

\fi

\iffalse
Polymatroidal functions capture Shannon-type inequalities on the entropy function and play a central role in computing converse results in network information theory \cite{yeung2008information}.
The polymatroidal bound \cite{blasiak2011lexicographic,Arbabjolfaei--Bandemer--Kim--Sasoglu--Wang2013}  is strictly tighter than the MAIS bound, but is often much harder to compute, especially for large systems.
\fi

\begin{lemma}\label{lem:lower:bound:pm}
Consider the system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$. For any disjoint $V,Z\subseteq [n]$, we have
\begin{align}
I({X}_V;Y|{X}_Z)=g(Z^c)-g(Z^c\cap V^c),
\end{align}
for some polymatroidal rank function $g(S),S\subseteq [n]$ such that for any $i\in [n]$, $W\subseteq W_i$, $G\subseteq W^c\cap A_i^c$,
\begin{align}
&H({X}_W)=g(G\cup W)-g(G),     \label{eq:lower:bound:lem:pm:rate}
\end{align}
and
\begin{align}
&g(\emptyset)=0,      \label{eq:lower:bound:lem:pm:nonnegativity}  \\
&g(S')\ge g(S),    \qquad \text{if $S\subseteq S'$},          \label{eq:lower:bound:lem:pm:monotonicity}   \\
&g(S')+g(S)\ge g(S'\cup S)+g(S'\cap S).        \label{eq:lower:bound:lem:pm:submodularity}
\end{align
\end{lemma}

\begin{IEEEproof}
Define $g(S)\doteq H(Y|{X}_{S^c})-H(Y|{X}_{[n]}),\forall S\subseteq [n]$. We have
$I({X}_V;Y|{X}_Z)=g(Z^c)-g(Z^c\cap V^c).$
It remains to show that this $g(S)$ satisfies \eqref{eq:lower:bound:lem:pm:rate}-\eqref{eq:lower:bound:lem:pm:submodularity}.

For \eqref{eq:lower:bound:lem:pm:rate}, consider any $i\in [n]$, $W\subseteq W_i$, $G\subseteq W^c\cap A_i^c$. Set $A=[n]\setminus W\setminus G$, and one can verify that $A_i\subseteq A$. Hence,
\begin{align}
H({X}_W)&=H({X}_W|{X}_A)-H({X}_W|Y,{X}_A)    \label{eq:lower:bound:lem:pm:proof:1}  \\
&=H(Y|{X}_A)-H(Y|{X}_W,{X}_A)    \nonumber   \\
&=g(W,G)-g(G),    \nonumber
\end{align}
where \eqref{eq:lower:bound:lem:pm:proof:1} is due to source independence, \eqref{eq:model:utility:wi}, and $A_i\subseteq A$.

For \eqref{eq:lower:bound:lem:pm:nonnegativity}, we have $g(\emptyset)=H(Y|{X}_{[n]})-H(Y|{X}_{[n]})=0$.

For \eqref{eq:lower:bound:lem:pm:monotonicity}, for any $S\subseteq S'\subseteq [n]$, $S'^c\subseteq S^c$, and thus
\begin{align}
g(S')=H(Y|{X}_{S'^c})\ge H(Y|{X}_{S^c})=g(S).   \nonumber
\end{align}

For \eqref{eq:lower:bound:lem:pm:submodularity}, consider any $S,S'\subseteq [n]$. Set $S^c\cap S'^c=S_0$, $S^c\setminus S_0=S_1$, and $S'^c\setminus S_0=S_2$.
We have
\begin{align}
&g(S')+g(S)    \nonumber  \\
&=H(Y, \hspace{-0.375mm} {X}_{S_0\cup S_2} \hspace{-0.25mm} ) \hspace{-0.375mm} + \hspace{-0.375mm} H(Y, \hspace{-0.375mm} {X}_{S_0\cup S_1} \hspace{-0.25mm} ) \hspace{-0.375mm} - \hspace{-0.375mm} H( \hspace{-0.25mm} {X}_{S_0\cup S_2} \hspace{-0.25mm} ) \hspace{-0.375mm} - \hspace{-0.375mm} H( \hspace{-0.25mm} {X}_{S_0\cup S_1} \hspace{-0.25mm} )    \nonumber    \\
&\ge H(Y, \hspace{-0.25mm} {X}_{S_0} \hspace{-0.375mm} ) \hspace{-0.375mm} + \hspace{-0.375mm} H(Y, \hspace{-0.25mm} {X}_{S_0\cup S_1\cup S_2} \hspace{-0.375mm} ) \hspace{-0.375mm} - \hspace{-0.375mm} H({X}_{S_0} \hspace{-0.375mm} ) \hspace{-0.375mm} - \hspace{-0.375mm} H({X}_{S_0\cup S_1\cup S_2} \hspace{-0.375mm} )    \nonumber    \\
&=g(S\cup S')+g(S\cap S'),   \nonumber
\end{align}
where the inequality follows from the submodularity of the entropy function, as well as source independence.
\end{IEEEproof}

The above bound can be solved using either linear programming (LP) or Fourier-Motzkin elimination \cite[Appendix D]{elgamal_yhk}.
In the system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$, for any disjoint $V,Z\subseteq [n]$, let
\begin{align*}
\Lambda(V,Z)\doteq \textcolor{black}{\min_{\substack{\text{$g$ satisfying \eqref{eq:lower:bound:lem:pm:rate}-\eqref{eq:lower:bound:lem:pm:submodularity}}}}\{ g(Z^c)-g(Z^c\cap V^c) \}}.
\end{align*}
Combining Lemmas \ref{lem:lower:bound:core} and \ref{lem:lower:bound:pm} gives the following result.

\begin{theorem}\label{thm:lower:bound:pm}
For the problem \eqref{eq:model:tradeoff},
we have
\begin{align}
&\mathcal{L}_{\rm max}\ge \max\{ \Lambda(Q,P),  \nonumber  \\
&\max_{\substack{i\in [m]:G_i\subseteq Q}}\big( d_i+H({X}_{W_i\cap Q})+\Lambda(A_i\cap Q,P) \big) \}.   \nonumber
\end{align}
\end{theorem}

\begin{remark}
\textcolor{black}{In the index coding problem, we usually assume uniformly distributed independent sources and a deterministic mapping $P_{Y|{X}_{[n]}}$. In contrast, Theorems \ref{thm:lower:bound:confusion} and \ref{thm:lower:bound:pm} generally hold for any discrete independent source distribution and make no assumption on the privacy mechanism.}
\end{remark}

In general, Theorems \ref{thm:lower:bound:confusion} and \ref{thm:lower:bound:pm} can outperform each other.

\section{System Model and Problem Formulation}\label{sec:model}

Assume that a data curator observes $n$ independent discrete random sources $X_1,X_2,\ldots,X_n$. We assume a general distribution $P_{X_i}$ for each source.
\textcolor{black}{Without loss of generality, we assume every source has full support.}
For brevity, when we say source $i$, we mean source $X_i$. For any $S\subseteq [n]$, set $S^c=[n]\setminus S$, ${X}_S=(X_i: i\in S)$, ${x}_S=(x_i: i\in S)$, and $\mathcal{X}_S=\prod_{i\in S}\mathcal{X}_i$.
The data curator broadcasts a distorted version of ${X}_{[n]}$, denoted by $Y$, generated according to the privacy mechanism $P_{Y|{X}_{[n]}}$, to a number of legitimate users, which is also overheard by an adversary.

We consider a guessing framework where both the legitimate users and the adversary attempt to estimate a certain parameter $V$ of interest about the sources in a single trial.

Both the privacy and utility are measured in terms of a guessing gain $r$ as defined below.
Consider any party (a user or the adversary) that wishes to guess $V$ with the side information $Z$.
Note that $V\perp \!\!\! \perp Z$ due to source independence.
The party aims to maximize the correct guessing probability of $V$ upon observing $Y$ (i.e., the party employs the maximum \emph{a posteriori} estimator).
For each $(y,z)\in \mathcal{Y}\times \mathcal{Z}$,
we define the ratio between such maximized guessing probability after and before observing a $y \in \mathcal{Y}$ given $z \in \mathcal{Z}$ as
\begin{align}
r(V\to y|z)&\doteq  \frac{\max\limits_{P_{{\hat{V}}|Y=y,Z=z}} {\mathbb E}\big[ P_{{\hat{V}}|Y=y,Z=z}(V|y,z) \big]}{\max\limits_{P_{{\hat{V}}|Z=z}}{\mathbb E}\big[ P_{{\hat{V}}|Z=z}(V|z) \big]}   \label{eq:model:ratio}  \\
&=\frac{\max\limits_{v}P_{V|Y,Z}(\textcolor{black}{v}|y,z)}{\max\limits_{v}P_{V}(v)}.  \label{eq:model:ratio:dddd}
\end{align}

{\bf \textcolor{black}{Privacy Metric}:} We assume the adversary has side information ${X}_{P}$ for some $P\subseteq [n]$, and is interested in a (possibly randomized) discrete function $U$ of the sources ${X}_{Q}$ it does not know, where $Q\doteq P^c$.
\textcolor{black}{Quite often in practice,} this function is chosen by the adversary and is unknown to the data curator. Therefore, we consider a worst-case privacy leakage measure, a conditional version of maximal leakage (MaxL) from \cite{issa2019operational}, as our privacy metric.
\begin{definition}[Maximal Leakage,\cite{issa2019operational}]    \label{def:ML}
Given a finite discrete joint distribution $P_{{X}_{[n]},Y}$, the maximal leakage from ${X}_Q$ to Y given ${X}_P$ is defined as
\begin{align}
&\mathcal{L}_{\rm max}(\hspace{-0.0625mm} {X}_Q \hspace{-0.375mm} \to \hspace{-0.375mm} Y|{X}_P \hspace{-0.0625mm})\doteq \hspace{-1mm} \sup_{U:U-{X}_Q-(Y,{X}_P)} \hspace{-1mm} \mathcal{L}(\hspace{-0.0625mm} U \hspace{-0.375mm}\to \hspace{-0.375mm} Y|{X}_P \hspace{-0.0625mm}),   \label{eq:model:def:ml}
\end{align}
where
\begin{align}
\mathcal{L}(U\to Y|{X}_P)\doteq \log {\mathbb E}_{P_{Y,{X}_P}}\big[ r(U\to Y|{X}_P) \big].  \label{eq:model:def:l}
\end{align}
\end{definition}

\begin{remark}
Note that the MaxL in Definition \ref{def:ML} assumes a different Markov chain model from \cite[Section III-E]{issa2019operational}: for our problem, the Markov chain model studied in \cite{issa2019operational} always \textcolor{black}{reduces} to $U-{X}_{[n]}-Y$ regardless of $Q$.
\end{remark}

For the rest of the paper, we refer to $\mathcal{L}_{\rm max}({X}_Q \to Y|{X}_P)$ as $\mathcal{L}_{\rm max}$ when there is no ambiguity.
A computable expression of the MaxL in Definition \ref{def:ML} is presented as
\begin{align}
\mathcal{L}_{\rm max}=\log \sum_{y,{x}_P}\max_{{x}_{Q}}P_{Y,{X}_P|{X}_Q}(y,{x}_P|{x}_Q),   \label{eq:model:ml:sibson}
\end{align}
which can be obtained following a similar approach to \cite{issa2019operational}.
We omit the derivation details due to limited space.

{\bf \textcolor{black}{Utility Constraints}:} There are $m$ legitimate users. User $i\in [m]$ knows some sources ${X}_{A_i}$ a priori as side information for some $A_i\subseteq [n]$, and is interested in all the remaining sources ${X}_{A_i^c}$,
which are divided into two groups of different levels of priority:
\begin{itemize}
\item Source ${X}_{W_i}$ for some $W_i\subseteq A_i^c$ are indispensable to the user, and thus must be correctly guessed by user $i$ with probability of $1$ (i.e., perfect decoding).
\item The rest of the sources, ${X}_{G_i}$, where $G_i\doteq A_i^c\setminus W_i$, are less essential yet still useful/interesting. Thus, user $i$ requires the guessing ability gain upon observing $Y$ to be larger than a certain threshold $d_i$.
\end{itemize}
These result in the following two kinds of utility constraints. For any $i\in [m]$, we have
\begin{alignat}{2}
&H({X}_{W_i}|Y,{X}_{A_i})=0,                     && \qquad  \forall i\in [m],   \label{eq:model:utility:wi}   \\
&D({X}_{G_i}\to Y|{X}_{A_i})\ge d_i,          && \qquad  \forall i\in [m],   \label{eq:model:utility:gi}
\end{alignat}
where $D({X}_{G_i}\to Y|{X}_{A_i})$ is defined as
\begin{align}
&D({X}_{G_i}\to Y|{X}_{A_i}) \hspace{-0.25mm} \doteq \hspace{-0.25mm} {\mathbb E}_{P_{Y,{X}_{A_i}}} \hspace{-0.5mm} \big[ \log r({X}_{G_i} \hspace{-0.5mm} \to \hspace{-0.5mm} Y|{X}_{A_i}) \big].      \label{eq:model:utility:leakage:def}
\end{align}

{Note that for constraint \eqref{eq:model:utility:gi}, each legitimate user $i$ is interested in obtaining the source ${X}_{G_i}$ \textcolor{black}{rather than} a function/feature of ${X}_{G_i}$.
We simplify the notation $D({X}_{G_i}\to Y|{X}_{A_i})$ to $D_i$ when there is no ambiguity.
}

\begin{remark}
Note the subtle difference between $D_i$ and $\mathcal{L}({X}_{G_i}\to Y|{X}_{A_i})=\log {\mathbb E}_{P_{Y,{X}_{A_i}}}\big[ r({X}_{G_i}\to Y|{X}_{A_i}) \big]$ as in \eqref{eq:model:def:l}.
The latter is lower bounded by the former due to Jensen's inequality.
From the data curator's viewpoint, requesting $D_i$ to be above a certain threshold is more stringent than requesting $\mathcal{L}({X}_{G_i}\to Y|{X}_{A_i})$ to be above that threshold.
We use $D_i$ rather than $\mathcal{L}({X}_{G_i}\to Y|{X}_{A_i})$ as our utility measure as it leads to a simple closed-form result characterizing $\mathcal{L}_{\rm max}$ in terms of $D_i$ (cf. Lemma \ref{lem:lower:bound:core}).
\end{remark}

\begin{remark}  \label{rem:model:utility}
The two types of utility constraints in \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi} can be unified to be represented in terms of the same function $D$: The perfect decoding constraint \eqref{eq:model:utility:wi} is equivalent to requiring that
\begin{align}
D({X}_{W_i}\to Y|{X}_{A_i})&={\mathbb E}_{P_{Y,{X}_{A_i}}}  \big[ \log \frac{1}{\max\limits_{{x}_{W_i}}P_{{X}_{W_i}}({x}_{W_i})} \big]  \nonumber   \\
&=H_{\infty}({X}_{W_i}),   \label{eq:model:utility:remark}
\end{align}
where $H_{\infty}({X}_{W_i})$ denotes the min-entropy (i.e., R\'enyi entropy of order $\infty$ \cite{renyi1961measures}) of ${X}_{W_i}$. \textcolor{black}{One} can show that for any $i\in [m]$, $D_i\le H_{\infty}({X}_{G_i})$. Consequently,
we always require that $0\le d_i\le H_{\infty}({X}_{G_i})$ for any $i\in [m]$.
\end{remark}

{\bf Privacy-Utility Tradeoff:}
We denote such system by the 5-tuple
$(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$, where ${\bf A} \doteq (A_i,i\in [m])$, ${\bf W} \doteq (W_i,i\in [m])$, and ${\bf d} \doteq (d_i,i\in [m])$. Note that $G_i$ is determined by $W_i $ and $A_i$, and $Q$ is determined by $P$.

To design the privacy mechanism $P_{Y|{X}_{[n]}}$, we need to consider the fundamental tradeoff between the privacy and utility.
Any data distortion that reduces the information leakage to the adversary can decrease the utility obtained by the users.
Such tradeoff is formulated by the following constrained optimization problem.
\begin{align}
\inf_{\substack{P_{Y|{X}_{[n]}}\in \\ \mathcal{P}_{Y|{X}_{[n]}}(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)}}\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P),    \label{eq:model:tradeoff}
\end{align}
where $\mathcal{P}_{Y|{X}_{[n]}}(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$ denotes the collection of randomized mappings $P_{Y|{X}_{[n]}}$ that satisfy \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi} for the problem $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$.

Due to the non-convexity of \eqref{eq:model:tradeoff}, instead of providing an explicit solution, we derive lower bounds on $\mathcal{L}_{\rm max}$ by taking inspiration from index coding.
These bounds serve as fundamental performance limits that cannot be surpassed \textcolor{black}{by any mechanism} because they are enforced by the system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$.
\textcolor{black}{In Section \ref{sec:privacy:mechanism}, we design an achievable mechanism based on the idea of agglomerative clustering.}

\section{Privacy Mechanism Design}   \label{sec:privacy:mechanism}

We develop a greedy algorithm to provide a solution for the problem \eqref{eq:model:tradeoff}.
The algorithm is based on the agglomerative clustering method, which has been used in the information bottleneck \cite{slonim2000agglomerative} and the privacy funnel problem \cite{makhdoumi2014information}.

\iffalse
Consider a given system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$. A naive approach is to establish a deterministic mapping between ${X}_{[n]}$ and $Y$ through $p(y|{x}_{[n]})\in \{0,1\}$.
\textcolor{black}{The deterministic mapping allows every user to perfectly reconstruct the sources, which is a direct method to hold the constraint \eqref{eq:model:utility:wi}}. Nevertheless, such mapping also leads to the largest possible MaxL as the adversary can also perfectly reconstruct ${X}_Q$ and subsequently any function $U$ it is after.
\textcolor{black}{(I still have a problem with this statement for that it only relates to \eqref{eq:model:utility:wi} not \eqref{eq:model:utility:gi}.)}

The main idea for the greedy algorithm is to start from the one-to-one mapping $\mathcal{Y} = \mathcal {X}_{[n]}$ and then iteratively merge some elements of $\mathcal{Y}$ to make the MaxL smaller (in other words, to ``blur" the revealed information), while still ensuring the utility for the users at an acceptable level, i.e., satisfying \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi}.
We recall the notion of confusion graph for index coding from \cite{alon2008broadcasting}.
\fi

Consider a given system $(P_{{X}_{[n]}},{\bf A},{\bf W},{\bf d},P)$. To design the privacy mechanism $P_{Y|{X}_{[n]}}$, we start from the one-to-one deterministic mapping with \textcolor{black}{$\mathcal{Y} = \mathcal {X}_{[n]}$ and $P_{Y|{X}_{[n]}}(y|{x}_{[n]}) = 1$ iff $y={x}_{[n]}$}\footnote{Note that such one-to-one mapping allows every user to perfectly reconstruct every source and thus satisfies \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi} for sure. Nevertheless, it also leads to the largest privacy leakage as the adversary can also perfectly reconstruct ${X}_Q$ and subsequently any function $U$ it is interested in.}
and then iteratively merge some elements of $\mathcal{Y}$ to make the privacy leakage smaller (in other words, we ``blur" the revealed information), while still ensuring the utility for the users at an acceptable level, i.e., satisfying \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi}. \textcolor{black}{In particular, to ensure \eqref{eq:model:utility:wi}, we again utilize the notion of confusion graph\cite{alon2008broadcasting} introduced in Section \ref{sec:lower:bound:confusion}.}

Based on the merging idea discussed above, we propose an agglomerative clustering algorithm in Algorithm \ref{alg:agglomerative}.
Let $Y^{y_1,y_2}$ be the resulting random variable from merging any $y_1,y_2\in \mathcal{Y}$.

Let $\Theta$ denote the collection of $\{y_1,y_2\}$ such that merging them does not violate \eqref{eq:model:utility:wi} and \eqref{eq:model:utility:gi} and strictly reduces the privacy leakage to the adversary:
\begin{equation} \label{eq:Theta}
\small
    \begin{aligned}
        \Theta \doteq  & \big\{ \{y_1,y_2\} \in \mathcal{Y} \times \mathcal{Y} \colon y_1 \neq y_2, \text{ any two } \\
                 & \  {x}_{[n]},{x}_{[n]}'\in \mathcal{X}_{[n]}(y_1)\cup \mathcal{X}_{[n]}(y_2) \text{ are not confusable,} \\
                 & \ D({X}_{G_i}\to Y^{y_1,y_2}|{X}_{A_i})\ge d_i, \forall i \in [m], \\
                 & \ \mathcal{L}_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)< \mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)\big\}.
    \end{aligned}
\end{equation}
The algorithm terminates if $\Theta$ becomes an empty set.

\iffalse
\textcolor{black}{Original:
Let $\Theta$ denote the collection of $\{y_1,y_2\} \in \mathcal{Y} \times \mathcal{Y}$ that can be merged.
That is, $\Theta$ is the set of $\{y_1,y_2\} \in \mathcal{Y} \times \mathcal{Y}$ such that $y_1\neq y_2$, all realizations ${x}_{[n]}\in \mathcal{X}_{[n]}(y_1)\cup \mathcal{X}_{[n]}(y_2)$ are pairwisely not confusable, $D({X}_{G_i}\to Y^{y_1,y_2}|{X}_{A_i})\ge d_i$ for any $i\in [m]$, and $\mathcal{L}_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)< $ \textcolor{black}{$\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)$} \textcolor{black}{what is this?}.}
\fi

\iffalse
\begin{algorithm}[bth]\label{alg:agglomerative}
    \SetKwInOut{Input}{Input}
    \SetKwInOut{Output}{Output}
    \SetKwInOut{Initialization}{Initialization}
    \Input{Problem $(P_{{X}_{[n]}},{\bf W},{\bf A},{\bf d},P)$, $\Theta$.}
    \Output{Privacy mechanism $P_{Y|{X}_{[n]}}$.}
    \Initialization{$\mathcal{Y}=\mathcal{X}_{[n]}$, $P_{Y|{X}_{[n]}}(y|{x}_{[n]})=1$ if and only if $y={x}_{[n]}$.}
{\bf while} $\Theta \neq \emptyset$ {\bf do} $\{y_1,y_2\}=\arg\min_{\{y_1,y_2\} \in \Theta}L_{\rm max}({X}_Q\to Y^{y,y'}|{X}_P)$.

\begin{itemize}
\item[--] {\bf merge} $\{y_1,y_2\}\to \bar{y}$.
\item[--] {\bf update} $\mathcal{Y}=\mathcal{Y}\setminus \{y_1,y_2\}\cup \{{\bar y}\}$, and $P_{Y|{X}_{[n]}}({\bar y}|{x}_{[n]})=P_{Y|{X}_{[n]}}(y_1|{x}_{[n]})+P_{Y|{X}_{[n]}}(y_2|{x}_{[n]})$ for any ${x}_{[n]}\in \mathcal{X}_{[n]}$ while keeping the rest of $P_{Y|{X}_{[n]}}$ unchanged, as well as $\Theta$ \textcolor{black}{how do you update $\Theta$?}. \textcolor{red}{better writing?}
\end{itemize}

{\bf return} $P_{Y|{X}_{[n]}}$
\caption{Greedy algorithm: multi-terminal privacy-utility tradeoff \eqref{eq:model:tradeoff}}
\end{algorithm}
\fi

        \begin{algorithm} \label{alg:agglomerative}
	       \small
	       \SetAlgoLined
	       \SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
	       \SetKwFor{For}{for}{do}{endfor}
            \SetKwRepeat{Repeat}{repeat}{until}
            \SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{endif}
	       \BlankLine
           \Input{The system $(P_{{X}_{[n]}},{\bf W},{\bf A},{\bf d},P)$.}
	       \Output{Privacy-preserving mechanism $P_{Y|{X}_{[n]}}$.}
	       \BlankLine
            Initialization: $\mathcal{Y} \leftarrow \mathcal{X}_{[n]}$, $P_{Y|{X}_{[n]}}(y|{x}_{[n]}) \leftarrow 1$ iff $y={x}_{[n]}$ and obtain $\Theta$ based on $\mathcal{Y}$ by \eqref{eq:Theta}\label{step:initialization}\;
            \Repeat{$\Theta = \emptyset$}{
                $\{y_1^*,y_2^*\} \leftarrow \arg\min_{\{y_1,y_2\} \in \Theta} \mathcal{L}_{\rm max}({X}_Q\to Y^{y,y'}|{X}_P)$\label{step:Min}\;
                Merge $y_1^*$ and $y_2^*$ into $\bar{y}$: $ \bar{y} \leftarrow \{y_1^*,y_2^*\}$\;
                Obtain the new $Y$ by letting $\mathcal{Y} \leftarrow \mathcal{Y}\setminus \{y_1^*,y_2^*\}\cup \{{\bar y}\}$ and $P_{Y|{X}_{[n]}}({\bar y}|{x}_{[n]}) \leftarrow P_{Y|{X}_{[n]}}(y_1^*|{x}_{[n]})+P_{Y|{X}_{[n]}}(y_2^*|{x}_{[n]})$ for any ${x}_{[n]}\in \mathcal{X}_{[n]}$ while keeping the rest of $P_{Y|{X}_{[n]}}$ unchanged\;
                Obtain the new $\Theta$ by \eqref{eq:Theta} based on updated \textcolor{black}{$Y$}\label{step:Theta}\;
            }

            \Return $P_{Y|{X}_{[n]}}$\;
	   \caption{Agglomerative clustering algorithm for problem \eqref{eq:model:tradeoff}}
	   \end{algorithm}

\iffalse
Consider any hard mapping $P_{Y|{X}_{[n]}}$ such that each realization ${x}_{[n]}$ is mapped to only one $y\in \mathcal{Y}$ with probability of $1$. For any $y_1,y_2\in \mathcal{Y}$ merging to ${\bar y}$, we have
\begin{align}
&2^{\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)}-2^{\mathcal{L}_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)}   \nonumber   \\
&=\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}(y_1|{x}_P,{x}_Q)   \nonumber   \\
&\quad +\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}(y_2|{x}_P,{x}_Q)   \nonumber   \\
&\quad -\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}({\bar y}|{x}_P,{x}_Q)   \nonumber   \\%\label{eq:agglomerative:ML:reduction}   \\
&=P_{{X}_P}(\mathcal{X}_P(y_1))+P_{{X}_P}(\mathcal{X}_P(y_2))-P_{{X}_P}(\mathcal{X}_P({\bar y}))\ge 0,    \nonumbe
\end{align}
where the inequality follows from that $\mathcal{X}_P(y_1)\cup \mathcal{X}_P(y_2)=\mathcal{X}_P({\bar y})$ since $y_1,y_2$ merge to ${\bar y}$.
Hence we know that
\begin{align}
\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)\ge \mathcal{L}_{\rm max}({X}_Q\to Y^{y,y'}|{X}_P).
\end{align}
\fi

\begin{remark}
To compute Algorithm \ref{alg:agglomerative} more efficiently, notice that finding $\arg\min_{\{y_1,y_2\} \in \Theta}L_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)$ is equivalent to finding $\arg\max_{\{y_1,y_2\} \in \Theta}\big(2^{\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)}-2^{\mathcal{L}_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)\big)}$ in step~\ref{step:Min}, which can be computed as
\begin{align}
&2^{\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)}-2^{\mathcal{L}_{\rm max}({X}_Q\to Y^{y_1,y_2}|{X}_P)}   \nonumber   \\
&=\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}(y_1|{x}_P,{x}_Q)   \nonumber   \\
&\quad +\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}(y_2|{x}_P,{x}_Q)   \nonumber   \\
&\quad -\sum_{{x}_P}\max_{{x}_Q}P_{{X}_P}({x}_P)\cdot P_{Y|{X}_{[n]}}({\bar y}|{x}_P,{x}_Q)   \nonumber   \\
&=\sum_{{x}_P\in \mathcal{X}_P(y_1)}P_{{X}_P}({x}_P)\cdot 1+\sum_{{x}_P\in \mathcal{X}_P(y_2)}P_{{X}_P}({x}_P)\cdot 1   \nonumber   \\
&\quad -\sum_{{x}_P\in \mathcal{X}_P({\bar y})}P_{{X}_P}({x}_P)\cdot 1      \nonumber   \\
&=P_{{X}_P}(\mathcal{X}_P(y_1))+P_{{X}_P}(\mathcal{X}_P(y_2))-P_{{X}_P}(\mathcal{X}_P({\bar y})).    \nonumber
\end{align}
\end{remark}

\section{Proof of Lemma \ref{lem:lower:bound:core}} \label{appendix:lem:core}

\iffalse
To show $\mathcal{L}_{\rm max}\ge I({X}_Q;Y|{X}_P)$, we have
\begin{align}
\mathcal{L}_{\rm max
&=\log\sum_{y,{x}_P}P_{Y,{X}_P}(y,{x}_P)  \max_{{x}_Q} \frac{  P_{Y,{X}_{P}|{X}_{Q}}(y,{x}_{P}|{x}_{Q}) }{ P_{Y,{X}_P}(y,{x}_P) }   \nonumber  \\
&\ge I({X}_Q;Y,{X}_P)=I({X}_Q;Y|{X}_P),   \nonumber
\end{align}
where the inequality follows from replacing maximum over ${x}_{Q}$ with expectation over $P_{{X}_{Q}|Y,{X}_P}$ and Jensen's inequality, and the last equality is due to source independence.
\fi

\textcolor{black}{Note that the right hand side of \eqref{eq:model:ml:sibson} is equal to the Sibson mutual information of order $\infty$ \cite{sibson1969information}, $I_{\infty}^{\rm S}$, and hence we have
\begin{align*}
\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)&=I_{\infty}^{\rm S}({X}_Q;Y,{X}_P)  \\
&\ge I({X}_Q;Y,{X}_P)=I({X}_Q;Y|{X}_P),
\end{align*}
where the inequality follows from \cite[Theorem 2]{verdu2015alpha}, and the last equality follows from source independence. }

It remains to show $\mathcal{L}_{\rm max}\ge \textcolor{black}{\Delta_i}$ for any user $i\in [m]$ with $G_i\subseteq Q$.
For brevity, we drop the subscript $i$ remembering that $W,A,G$ stand for $W_i,A_i,G_i$, respectively.
Set
\begin{align}
\begin{array}{cc}
W_{P}=P\cap W, & W_{Q}=Q\cap W, \\
A_{P}=P\cap A, & A_{Q}=Q\cap A.
\end{array}
\nonumber
\end{align}
Since $G\subseteq Q$, we have $P\cap G=\emptyset$, $Q\cap G=G$, and thus $P=W_P\cup A_P$, and $Q=W_Q\cup A_Q\cup G$.
We have
\begin{align}
&\mathcal{L}_{\rm max}({X}_Q\to Y|{X}_P)   \nonumber   \\%=\log\sum_{y,{x}_{P}}   \max_{{x}_{Q}} P_{Y,{X}_{P}|{X}_{Q}}(y,{x}_{P}|{x}_{Q})  \nonumber  \\
&\ge \sum_{y,{x}_P} P_{Y,{X}_P}(y,{x}_P)   \log \max_{{x}_Q} \frac{P_{Y,{X}_P|{X}_Q}(y,{x}_P|{x}_Q) }{P_{Y,{X}_P}(y,{x}_P)}   \label{eq:lower:bound:core:proof:jensen} \\
&=\sum_{y,{x}_P}P_{Y,{X}_P}(y,{x}_P)   \nonumber   \\
&\quad \log \big(   \max_{{x}_{W_Q},{x}_{A_Q}} (\frac{P_{Y,{X}_P|{X}_{W_Q},{X}_{A_Q}}(y,{x}_P|{x}_{W_Q},{x}_{A_Q})}{P_{Y,{X}_P}(y,{x}_P)} \nonumber  \\
&\quad  \cdot \max_{{x}_G} \frac{P_{Y,{X}_P|{X}_Q}(y,{x}_P|{x}_Q) }{P_{Y,{X}_P|{X}_{W_Q},{X}_{A_Q}}(y,{x}_P|{x}_{W_Q},{x}_{A_Q})})   \big)  \nonumber   \\
&=\sum_{y,{x}_P}P_{Y,{X}_P}(y,{x}_P)   \nonumber   \\
&\quad \log \big(   \max_{{x}_{W_Q},{x}_{A_Q}} (\frac{P_{Y,{X}_P|{X}_{W_Q},{X}_{A_Q}}(y,{x}_P|{x}_{W_Q},{x}_{A_Q})}{P_{Y,{X}_P}(y,{x}_P)} \nonumber  \\
&\quad  \cdot \max_{{x}_G} \frac{P_{Y,{X}_A|{X}_G}(y,{x}_A|{x}_G) }{P_{Y,{X}_A}(y,{x}_A)})   \big)   \label{eq:lower:bound:core:proof:markov}   \\
&\ge \sum_{y,{x}_P,{x}_{A_Q}}P_{Y,{X}_P,{X}_{A_Q}}(y,{x}_P,{x}_{A_Q})   \nonumber   \\
&\quad \big(   \log \max_{{x}_{W_Q}} \frac{P_{Y,{X}_P|{X}_{W_Q},{X}_{A_Q}}(y,{x}_P|{x}_{W_Q},{x}_{A_Q})}{P_{Y,{X}_P}(y,{x}_P)} \nonumber  \\
&\quad  +\log \max_{{x}_G} \frac{P_{Y,{X}_A|{X}_G}(y,{x}_A|{x}_G) }{P_{Y,{X}_A}(y,{x}_A)}   \big)   \label{eq:lower:bound:core:proof:max:ave:AQ}  \\
&=I(Y,{X}_P;{X}_{W_Q},{X}_{A_Q})   \nonumber   \\
&\quad +\sum_{y,{x}_{A}}\hspace{-0.25mm}P_{Y,{X}_{A}}(y,{x}_{A})\hspace{-0.25mm} \log \max_{{x}_G}\hspace{-0.25mm} \frac{P_{Y,{X}_A|{X}_G}(y,{x}_A|{x}_G) }{P_{Y,{X}_A}(y,{x}_A)}   \label{eq:lower:bound:core:proof:max:ave:WQ}    \\
&\ge I(Y,{X}_P;{X}_{W_Q},{X}_{A_Q})+D_i     \label{eq:lower:bound:core:proof:Dt:D}   \\
&=I(Y;{X}_{A_Q}|{X}_{P})+H({X}_{W_Q})+D_i=\textcolor{black}{\Delta_i},    \label{eq:lower:bound:core:proof:last}
\end{align}
where
\begin{itemize}
\item \eqref{eq:lower:bound:core:proof:jensen} follows from Jensen's inequality;
\item \eqref{eq:lower:bound:core:proof:markov} follows from that for any $y\in \mathcal{Y}$, ${x}_{W\cup A}\in \mathcal{X}_{W\cup A}$,
\begin{align*}
&\max_{{x}_G} \frac{P_{Y,{X}_P|{X}_Q}(y,{x}_P|{x}_Q) }{P_{Y,{X}_P|{X}_{W_Q},{X}_{A_Q}}(y,{x}_P|{x}_{W_Q},{x}_{A_Q})}   \\
&=\max_{{x}_G} \frac{P_{Y,{X}_{A\cup G}}(y,{x}_{A\cup G}) \hspace{-0.375mm}  \cdot \hspace{-0.375mm}   P_{{X}_W|Y,{X}_{A\cup G}}({x}_W|y,{x}_{A\cup G}) }{P_{{X}_G}({x}_G) \hspace{-0.375mm}  \cdot  \hspace{-0.375mm}  P_{Y,{X}_A}(y,{x}_A) \hspace{-0.375mm}  \cdot  \hspace{-0.375mm}  P_{{X}_W|Y,{X}_A}({x}_W|y,{x}_A)}   \\
&\stackrel{(a)}{=} \max_{{x}_G} \frac{P_{Y,{X}_A|{X}_G}(y,{x}_A|{x}_G) }{P_{Y,{X}_A}(y,{x}_A)},
\end{align*}
where (a) follows from the Markov chain ${X}_W - (Y,{X}_A) - {X}_G$ as a result of the utility constraint \eqref{eq:model:utility:wi};
\item \eqref{eq:lower:bound:core:proof:max:ave:AQ} follows from replacing maximum over ${x}_{A_Q}$ with expectation over $P_{{X}_{A_Q}|Y,{X}_P}$, and Jensen's inequality;
\item \eqref{eq:lower:bound:core:proof:max:ave:WQ} follows from the fact that ${X}_{W_Q}$ is a deterministic function of $(Y,{X}_P,{X}_{A_Q})$ according to \eqref{eq:model:utility:wi};
\item \eqref{eq:lower:bound:core:proof:Dt:D} follows from that
\begin{align*}
&\sum_{y,{x}_{A}} P_{Y,{X}_{A}}(y,{x}_{A}) \log \max_{{x}_G} \frac{P_{Y,{X}_A|{X}_G}(y,{x}_A|{x}_G) }{P_{Y,{X}_A}(y,{x}_A)}   \nonumber   \\
&\ge \sum_{y,{x}_{A}} P_{Y,{X}_{A}}(y,{x}_{A}) \log \frac{ \max\limits_{{x}_G}P_{Y,{X}_A,{X}_G}(y,{x}_A,{x}_G) }{\max\limits_{{x}_G}P_{{X}_G}({x}_G)\cdot P_{Y,{X}_A}(y,{x}_A)}   \nonumber   \\
&={\mathbb E}_{P_{Y,{X}_{A_i}}} \big[ \log r({X}_{G_i} \to Y|{X}_{A_i}) \big]=D_i;
\end{align*}
\item \eqref{eq:lower:bound:core:proof:last} follows from source independence, as well as \eqref{eq:model:utility:wi}.
\end{itemize}

This concludes the proof.

redpajama_set_name: RedPajamaArXiv


text:
Q: HTML - How do I assign a text value to another attribute's value? I am trying to assign a HTML text attribute's value to a hidden attribute's value.
The text code:
<input type="text" name="number" id="number" maxlength="4" onBlur="myno=this.value; concatno=myno.concat('0001')" />

I've used alert to try the output of the concatno value. For example, if user enter 1010, then the output will be 10100001.
Then my hidden code:
<input type="hidden" id="hide" name="hide" value=concatno>

I want my hidden value to be 1010001, but instead the value became "concatno". How should I assign the value in my hidden attribute?

A: The problem here is that you never updated your #hide element.
You need to use some javascript, for example:
document.getElementById('hide').value = concatno;

Working snippet:

<input type="text" name="number" id="number" maxlength="4" onkeyup="var myno = this.value; var concatno = myno.concat('0001'); document.getElementById('hide').value=concatno;" />
<input id="hide" name="hide" value=concatno disabled>

Note that even if the event is not the issue here, I suggest you to use another trigger, like onkeyup, so that the value is updated more often.
I've also changed your hidden element to disabled to make it visual.
Moreover, you should learn to avoid inline JavaScript.
Here is how I'll do it:

document.getElementById('number').addEventListener("keyup", function() {
  document.getElementById('hide').value = this.value.concat('0001');
});
<input type="text" name="number" id="number" maxlength="4" />
<input id="hide" name="hide" value=concatno disabled>

Documentation: getElementById
Hope it helps.

A: The issue is that you never actually update the value of your #hide element. You need to set its value inside of your event binding (just made the input visible for reference):

<input type="text" name="number" id="number" maxlength="4" onblur="var myno = this.value; var concatno=myno.concat('0001'); document.getElementById('hide').value = concatno; console.log(concatno)" />

<input type="text" id="hide" name="hide" value=concatno disabled />

It's also worth noting though, that you should generally avoid using obtrusive event handlers. Instead, delegate event handling to external Javascript. This way, your designer doesn't need to understand or even worry about the JS.
Here's an example using unobtrusive handlers:

document.getElementById('number').addEventListener('blur', function() {
  document.getElementById('hide').value = this.value.concat('0001');
});
<input type="text" name="number" id="number" maxlength="4" />
<input type="text" id="hide" name="hide" placeholder="concatno" disabled />

A: Try using name/id instead;

<input type="text" name="number" id="number" maxlength="4" oninput='hide.value=(this.value + "0001")'  autofocus=''/>
<input type="hidden" id="hide" name="hide" />

without inline scripts:

document.addEventListener('DOMContentLoaded', function() {
  let hide = document.querySelector('#hide');
  document.querySelector('#number').addEventListener('input', function() {
    hide.value = this.value + '0001';
  });
});
<input type="text" name="number" id="number" maxlength="4" autofocus='' />
<input type="hidden" id="hide" name="hide" />


New findings on autoantibodies targeting synaptic proteins have been recently published in Neuron and BRAIN journals by researchers from the Institute of Neurosciences and IDIBAPS. Both articles have been leaded by the team of Josep Dalmau in collaboration with researchers from the Neurophysiology group, David Soto and Xavier Gasull, and the group of Christian Geis in the Jena University Hospital (Germany). The paper entitled 'LGI1 antibodies alter Kv1.1 and AMPA receptors changing synaptic excitability, plasticity and memory' was published in BRAIN 2018 and focus on the effects of patient-derived immunoglobulin G (IgG) antibodies against Leucine-rich glioma-inactivated 1 (LGI1) protein. LGI1 forms a trans-synaptic complex with two proteins – ADAM23 and ADAM22 –, that interact presynaptically with Kv1.1 potassium channel and postsynaptically with AMPA receptors, respectively. Antibodies disrupt the interaction of LGI1 with these proteins and produce a decrease of synaptic levels of Kv1.1 and AMPA receptors, causing an increase in hippocampal neuronal excitability and glutamatergic transmission and severe impairment of long-term potentiation. Overall, these findings demonstrate that patient-derived IgG disrupt presynaptic and postsynaptic LGI1 signaling, causing neuronal hyperexcitability, decreased plasticity, and reversible memory deficits.
In a second paper published in Neuron (Human Autoantibodies against the AMPA Receptor Subunit GluA2 Induce Receptor Reorganization and Memory Dysfunction), the same authors demonstrate how specific autoantibodies found in patients with autoimmune encephalitis react against a specific subunit (GluA2) of the neuronal AMPA receptor severely affecting synaptic transmission. Anti-GluA2 antibodies induce receptor internalization with the concomitant reduction of synaptic GluA2-containing AMPARs, which impairs synaptic plasticity and memory in animals treated with patient antibodies. This work also links the disease symptoms to a specific immune-neuronal rearrangement of AMPA receptor subunit.
This two works published in prestigious Neuroscience journals shed light on the molecular mechanisms underlying autoimmune encephalitis that affect synaptic proteins providing a framework to explain disease symptoms.


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...does that mean 3 of the 15 diseases are incurable?
I love the way that the Holy Father waits a year to tell the Curia what the cures are for the ailments diagnosed last year. Now, that's what I call professionalism.
But seriously, I think in his modest gentleness, the Holy Father has kindly overlooked the shortcomings of his Curia as diagnosed last year (for this is the Year of Mercy) and gone for a different approach this year simply guiding his team towards virtue.
Hmm...nothing quite says "Happy Christmas" like pointing out 'cures' for workplace failings, does it?
The inclusion of that ''prayer'' by Cardinal Dearden is also revealing.
It was Dearden who prepared the vast consultation for - and chaired - the 1976 Call to Action meeting in Detroit.
Apropos of this "Year of Mirthy," let's have some fun and compare these necessary virtues with the personal praxis of Our Pope and Our Cross.
Look at the virtues and then ask your own self which of these virtues are possessed by and actualised by Franciscus.
We have already been told that those who are younger than us (Adherents of Rabbinical Judaism, formed after the destruction of jerusalem in 70 A.D.) are our Elder Brothers so, clearly, this Year of Mirthy is not intended to be taken seriously but to be experienced as a light-hearted romp through reality.
Christ is Born; and Our Lady's fiat cannot be seweraged by any amount of Novus Ordo Protestant plumbing.
With his call back to primal essentials, this insufferable, rambling nag of a prelate once again hoisted his true flag, that of the classical Reformation, by imposing the slogan of that same revolt--"Ecclesia semper reformanda"--upon the Church which he wants to remake in his own slovenly image. Kasper will be pleased to know that his augury of Bergoglio was once again vindicated: he's a radical in the tradition of Luther.
"He 'wants to initiate a new beginning for the church,' Kasper said, but not by destroying tradition. Rather, 'Pope Francis stands in a great tradition, reaching back to the earliest beginnings.' He does not represent a liberal position, but a radical position, understood in the original sense of the word as going back to the roots, the radix.' By reaching back through time, he is, in fact, "constructing a bridge to the future."
"[T]he Pope's emphasis on the centrality of the proclamation of the Gospel message and the life of charity places him inside a 'great tradition' that includes, in various ways, St. Augustine, St. Francis and St. Dominic, St. Thomas Aquinas, Martin Luther, and the Second Vatican Council."
I read through that speech with perplexity. It was like wading through verbal blancmange. Some aide needs to explain to the Holy Father the difference between holy virtues and the servile character traits preferred by an old-fashioned employer: e.g. 'Innocuousness' (or 'harmlessness' in a better translation) is surely not a virtue (nor even a trait to be recommended to any Vatican prelate).
knowing that others will watch over them."
That is certainly not the language of a Catholic prayer. More like a vaguely Christian-tinged hymn to the revolution. It sounds so harmless - but as I'm sure many of us have already discovered in our lives, 'harmlessness' is sometimes the most dangerously harmful thing of all.
Yes when i heard him say "ecllessia semper reformanda" all I could think of was Orson Welles portraying Cardinal Wolsey in A Man for All Seasons. He couldn't even get the empty slogan out in the face of Thomas More's withering stare.
Who will abstain from the 'fiat' of Protestants?