Residential Ballard is generally described as extending from the 8th Avenue NW to the east and the bluff to the west, and from NW 85th Street on the north to NW 65th Street to the south. The area primarily contains single family houses, but also includes a collection of mutli-family dwellings, commercial buildings, schools, churches, and other buildings. Most of the historic buildings in Ballard are modest cottages and builder's houses, and were not architect-designed. Building styles include, but are not limited to, Victorian (primarily Queen Anne), vernacular, Craftsman, American Foursquare, Colonial Revival (including variations), Tudor Revival, Minimal Traditional, and Ranch. The historic building fabric of Ballard is threatened by a rapid pace of development.
The City of Ballard was incorporated in 1890. It was the first community to incorporate after Washington achieved statehood in 1889. Although population increased rapidly, north Ballard was still relatively rural. In 1907, primarily due to lack of adequate water for its population of 15,000, Ballard citizens voted to be annexed to Seattle to ensure a good water supply for the area.
After annexation Ballard's street names were changed to conform to Seattle's: Ship Street turned into 65th Street, Main Street became 15th Avenue. During the Great Depression and World War II, construction in Ballard nearly ground to a halt, with the exception of some houses built by Earl F. Mench. However, following World War II, fueled by the G.I. bill and the rise of the automobile, Ballard boomed again, and new housing followed. In recent years, the demand for new housing has spurred a tremendous amount of change in Ballard, with old, modest houses being replaced by large box houses and multi-family units. These changes threaten to alter the character and feeling of this historic neighborhood.
Ballard Historical Society Classic Home Tour guides.
Crowley, Walt. Seattle Neighborhoods: Ballard--Thumbnail History. HistoryLink File # 983, accessed 6/1/16.
A Field Guide to American Houses (2nd ed.). New York, NY: Alfred A Knopf Press, 2013.
Passport to Ballard: The Centennial Story. Seattle, WA: Ballard News Tribune, 1988.
sold to the nearby Free Methodist Church on Mary Avenue NW to be their parsonage. Per King County property records the sale to the church on 6.21.56 was for $12,500. It was occupied for the Reverend Forest Bush and his wife Ida. On 10.29.62 it was purchased by Lawrence R. Hilden and the following December they entered the "Christmas Trail" contest for judging as one of 300 homes in the Seattle Times' city-wide Christmas lighting contest. Current owners are Virginia and Forester Lawrence.
This 1-1/2 story house with a rectangular plan exhibits both Tudor and Craftsman influences. The steep cross-gabled roof has overhanging eaves with triangular knee braces below. Cladding consists of horizontal wood clapboard on the lower level, followed by coursed wood shingles on the upper story and stucco with decorative half-timbering on the roof gables. A wide course belt separates the wall materials. A projecting gabled entry porch houses a wood door and a multi-pane window to its right. South of the porch, a protruding front gabled square bay with a wide 9-paneled window takes up the first floor corner. A pair of centered 1-over- 1 sash windows, flanked by smaller squared windows complete the front elevation. The side (south) elevation has one sash window on the side gable, a tall exterior brick chimney and a shed projecting bay with windows on the western part of the first floor. House sits above the street on a mid-street lot facing east surrounded by a lawn and vegetation. Concrete stairs lead to the elevated porch and a paved uphill driveway on the south leads to a detached garage at the rear.


If you have tried one of Jared's ProPops at one of these locations and have enjoyed eating them as much as I enjoy making them, please take a look at all of Jared's ProPops and choose from dozens of flavors to place your next order and schedule your weekly delivery.
Due to the logistical issues of shipping frozen foods within a day, we are only able to deliver Jared's ProPop's to our friends in Oklahoma and surrounding states (Texas, Kansas, Arkansas, Missouri, and Louisiana). As an added benefit of living in Tulsa or their surrounding areas, you will receive a discounted price on your delivery order for the ProPops.
Most weeks we make deliveries to friends in Tulsa and surrounding areas on Tuesdays or Thursdays . Outside of Tulsa orders usually ship Monday through Thursday only. Orders must be received by Saturday evening to normally be included in the following week's shipment. Depending on the amount of orders we have for the week and available inventory, sometimes we will also deliver on other days as scheduling permits.
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Situated in central Montana, Lewistown is a scenic gem nestled among majestic mountains and clear alpine lakes, rivers and springs. Our peaceful position in the Judith Basin is ideal for hunting, fishing, horse riding, camping, hiking, skiing and all types of outdoor leisure. Steeped in Western tradition, Lewistown enjoys a thriving economy built primarily upon ranch, farm, agriculture, manufacturing and local commercial entrepreneurship. If you're ready to experience Big Sky culture for yourself, we cordially invite you to Lewistown, Central Montana and the state.
For a fascinating look at our region's heritage, be sure to visit the Central Montana Historical Museum in Lewistown. Browse a delectable assortment of local MT ranch and farm produce, baked goods, syrups and herbs at the Lewistown Farmers Market. Held on the first Saturday after Labor Day, the annual Chokecherry Festival is one of our community's most celebrated traditions. A tribute to the small red berry found in Lewistown, venders line the street selling specialty food items, wood crafts, art and more. Other festivities include a pancake breakfast, chokecherry culinary competition and chokecherry spitting contest.
Lewistown's rich cultural heritage began when the Metis, an aboriginal peoples from Canada, in their Red River carts followed the dwindling buffalo herds and settled here in 1879. Open range ranchers, farmers, gold miners and homesteaders soon followed. The town became an incorporated city in 1899. The beautiful golden domed Fergus County Court House was erected in 1907.
Between 1905 and 1919, Lewistown experienced a period of rapid growth and prosperity. Prominent businessmen, ranchers and farmers invested on Main Street Lewistown and created mansions for their families within walking distance of their business ventures.
Two railroads provided service to Lewistown; the Milwaukee depot and the Great Northern depot. Immigrants, in search of a better life and free land arrived by the thousands and settled in Central Montana to farm, ranch and start businesses. The railroad also made it possible for local architects and contractors to bring in materials and goods for elaborate and unique Montana architectural buildings.
Lewistown gets approximately 22 inches of rain and 115 inches of snowfall annually. The number of days with any measurable precipitation is 105. On average, there are 196 sunny days a year.
Lewistown Public School District is made up of three elementary, a junior high and senior high schools. As an alternative, Central Montana Christian School is available for private educational opportunity.
We are home to Lewistown Municipal Airport with two other public-use airports within approximately 30 miles. Residents also have access to Lewiston Public Library.
The region is home to the James Kipp Recreation Area in the C.M. Russell National Wildlife Refuge and the Judith Peak Recreation Area in the Judith Mountains. The area provides great opportunities for other activities such as hunting, fishing, horse riding, hiking, golfing, climbing, rafting and canoeing.
This is just a taste of the many amenities and recreational opportunities one will find in Lewistown, Fergus County and across the state of Montana. Come by our United Country Real Estate and Auction office and let us tell you more about the many real estate properties and auctions we offer and let us help you make your Montana real estate dreams a reality.


Featuring a barbecue, Casa Bella 108190-91488 is a villa situated in El Bedito in the Baja California Sur Region. The property boasts views of the sea and is 22 km from Cabo San Lucas.
The kitchen features a fridge. A TV with satellite channels is featured.
Los Cabos International Airport is 18 km from the property.
This entry was posted in El Bedito and tagged Baja California Sur, Casa Bella 108190-91488, El Bedito, hotel, Hoteles, México.


Enjoy top floor privacy with this two bedroom condo unit in quiet Transcona. The unit is 888 sq. ft. with dark wood flooring in all living spaces, in suite storage with shelving, a kitchen with stainless steel appliances including a dishwasher, and has a balcony off the living room. The condo is also within walking distance of a shopping center and has a bus stop within 50 feet from the front door. Utilities are included in the condo fees, along with one parking stall. No pets and non-smoking unit.
I would like more information about 43 35 Wynford Dr, Winnipeg, Manitoba, R2C3T8.


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Apple News to have human curators
June 15, 2015 at 10:32 a.m.
Take that, artificial intelligence!
Apple is hiring people to curate its upcoming Apple News in iOS 9, the next update to the operating system that underlies Apple s iPhones and iPads. Apple News will be an app.
Here s a job posting from the Jobs at Apple site on the Cupertino computer giant s web site:
Editor, Apple News
Job Number: 40934541Santa Clara Valley, California, United States
Posted: Jun. 12, 2015Weekly Hours: 40.00
Editor, Apple News The Apple News team is looking for passionate, knowledgeable editors to help identify and deliver the best in breaking national, global, and local news. These editors will help News users find the best and most timely coverage of major news events, while also managing select categories based on their areas of professional expertise. The role requires the ability to manage relationships with some of the world s finest publishers, to work seamlessly with an international team, and to craft compelling email newsletters
The posting was spotted by 9to5 Mac, which tracks developments at the company.
Apple said it s looking for people with a Bachelors in journalism, communication or a related field, with a Masters preferred.
Another requirement: 5 years of newsroom experience with an emphasis on mobile. It adds:
Successful editors will be ambitious, detail-oriented journalists with an obsession for great content and mobile news delivery. They will have great instincts for breaking news, but be equally able to recognize original, compelling stories unlikely to be identified by algorithms. Thoroughness, flexibility, technical curiosity, a deep understanding of audience metrics, and a passion for high-quality, original content are all instrumental to success in this position.
So go for it, humans.
Photo: The exterior view of Apple Computer headquarters in Cupertino, Calif., is seen in this Friday, Aug. 4, 2006 file photo. (AP Photo/Paul Sakuma, FILE)

redpajama_set_name: RedPajamaCommonCrawl


Use this SUID/SIDS Gateway to find resources for states, communities, professionals, and families to reduce sudden unexpected infant death (SUID) and Sudden Infant Death Syndrome (SIDS), promote healthy outcomes, and cope with grief when losses occur.
NCEMCH supports the U.S. Maternal and Child Health Bureau's ongoing commitment to reducing infant mortality by housing the federal investment of materials developed by the National Sudden Unexpected Infant Death (SUID) and Sudden Infant Death Syndrome (SIDS) Consortium, including the SUID/SIDS Resource Center, the Program Support Center, Project IMPACT, and the National Center for Cultural Competence. Due to cessation in funding, the Consortium was disbanded in January 2014.


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ZTE & Infovista Certify Interoperability of SD-WAN Solutions
Combined solution delivers single-box application aware networking on low-cost platform for enterprise customers and service providers worldwide
Shenzhen, China and Paris, France – June 26, 2017– ZTE Corporation (0763.HK / 000063.SZ), a major international provider of telecommunications, enterprise and consumer technology solutions and Infovista, the leading provider of network performance orchestration solutions for a better connected and collaborative world, announced the successful completion of interoperability testing of their combined SD-WAN (Software-Defined Wide Area Network) solution by integrating Infovista's Application Performance Orchestration Solution ("Ipanema") with ZTE's Micro Cloud Gateway (MCG) platform.
After successful verification testing, the combined SD-WAN solution provides a single box overlay networking solution for hybrid WAN connectivity, guaranteeing performance of critical business applications while also supporting a secure overlay VPN, zero touch installation and one-button recovery. This can be delivered at an extremely competitive price for enterprises looking to drastically reduce IT networking costs. Service providers can also use the new platform to deliver value-added SD-WAN services to their enterprise customers as they implement hybrid WANs and augment their enterprise networks.
The combined SD-WAN solution can replace multiple enterprise and communications equipment deployed in traditional WANs, enabling fast distribution, deployment, data backup, and disaster tolerance one-button recovery on a cloud-managed platform. The solution provides enterprises with Application Performance Orchestration capabilities that protect and guarantee the user experience of their business-critical applications essential to digital transformation strategies.
Supporting Quotes:
"The successful verification by ZTE and Infovista opens the way for hardware providers to collaborate seamlessly with application developers to drive new innovations in services and business models," said Zhu Yongxing, Vice President of ZTE Corporation. "ZTE is committed to building and supporting open systems to promote the growth of a healthy industry ecosystem. So far, there are over 20 value-added software vendors that have engaged with ZTE in the MCG project, which are all leaders in their field and industry. Now a brand-new industry alliance is taking shape."
"With this new certified interoperable SD-WAN solution, ZTE and Infovista have responded to customer demands to deliver a single-box networking platform for the digital enterprise, centrally orchestrated," said Sylvain Quartier, senior vice president, product strategy enterprises, Infovista. "Our objective is to deliver SD-WAN solutions that offer Application SLAs on top of any network, including the Internet. We have now tested our SD-WAN solution with eight network platform vendors in support of this vision and continue to grow our SD-WAN ecosystem. In the Applications Era, enterprises and service providers are looking for SD-WAN solutions that transform network architectures from complex to agile, while protecting critical business application performance."
Learn more about ZTE Micro Cloud Gateway Series based on IT&CT Integration.
Visit the "ZTE Corporation" for more information on ZTE innovation and comprehensive products and solutions.
Follow and engage with ZTE on LinkedIn, Facebook and Twitter.
Learn more about how Ipanema delivers innovative and comprehensive application-aware SD-WAN and hybrid WAN solutions to guarantee application performance across enterprises' large and complex networks.
Learn how an "Application Aware SD-WAN is a Key Enabler of the Digital Enterprise" in a white paper by Zeus Kerravala, principal analyst and founder of ZK Research.
Follow and engage with Infovista on its blog, "ProActive Network News," and on LinkedIn, Facebook, and Twitter.
About ZTE
ZTE is a provider of advanced telecommunications systems, mobile devices, and enterprise technology solutions to consumers, carriers, companies and public sector customers. As part of ZTE's M-ICT strategy, the company is committed to provide customers with integrated end-to-end innovations to deliver excellence and value as the telecommunications and information technology sectors converge. Listed in the stock exchanges of Hong Kong and Shenzhen (H share stock code: 0763.HK / A share stock code: 000063.SZ), ZTE's products and services are sold to over 500 operators in more than 160 countries. ZTE commits 10 per cent of its annual revenue to research and development and has leadership roles in international standard-setting organizations. ZTE is committed to corporate social responsibility and is a member of the UN Global Compact. For more information, please visit www.zte.com.cn.


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GM rarely weighs in on victim compensation claims
Crain News Service
By Mike Colias, Crain News Service
DETROIT (Feb. 9, 2015) — General Motors Co. has played the role of infrequent adviser in the handling of its compensation fund for victims of its defective ignition switch.
The auto maker so far has provided input in just 89 of the 4,180 claims that were submitted to attorney Ken Feinberg ahead of the Jan. 31 filing deadline, the fund's deputy administrator, Camille Biros, said last week.
Under terms of the program, both GM and the claimants "have the opportunity to say, 'Oh, I'd like to tell you something more about this claim,'" Ms. Biros said in an interview. "But GM does not get involved in the review of the claims at all."
In most of those 89 cases, GM offered engineering insight — its own interpretation of data from the car's black-box data recorder, for example. Rarely did the feedback affect the outcome of a decision, she said.
(Bloomberg News photo)
Kenneth Feinberg must sort out which victims of crashes involving recalled GM vehicles are in line for compensation.
GM hired Mr. Feinberg last spring to independently oversee the victim compensation process in the wake of its recall of 2.6 million older small cars to replace the faulty ignition switch. A GM investigation found that the many engineers and attorneys knew for years of injuries or deaths related to the switch slipping out of the run position when jarred, cutting power steering, brakes and airbags.
Through early last week, Mr. Feinberg's office had approved 51 of the 455 death claims received. Each accepted death claim will receive a minimum $1 million payout, and in some cases several times that, based on the program's parameters.
About 12 percent of the total claims had been denied. In most of those cases, the claim was for an ineligible vehicle, Ms. Biros said. One person submitted a claim for a dog, which didn't meet the program's requirements.
GM in November moved the initial filing deadline back by a month to Jan. 31 but dismissed calls from lawmakers to extend it further.
Ms. Biros said Mr. Feinberg's team would finish reviewing claims by late spring or early summer.
Last year, GM estimated the total cost of the compensation program at $400 million to $600 million. In a regulatory filing last week, GM said it had paid $93 million to eligible claimants as of Jan. 30.
This report appeared on the website of Automotive News, a Detroit-based sister publication of Tire Business.
Feinberg may seek GM switch-related cases in regulator files
GM customers demand payment for 'callous cover-up' of defect
GM accused of misleading investors over recalls
GM's confirmed ignition deaths hit 50 as claims deadline nears
GM denies ignition-switch claims extension
GM halts crossovers sale due to tire issue
Lawyers say GM documents contradict Valukas report


Recently, it's been a tough decision whether or not to get a touchpad or a laptop because they are each in pattern. People usually finish up obtaining both or one of every because of to monetary restrictions. HP has arrive up with a answer that remedies this dilemma. HP Tablet PCs were introduced to provide a gadget that serves attributes, fashion and overall performance. With a twelve stage 1 inch rotating monitors, these devices can be utilized as laptops and/or touchpads.
For 1, they cannot determine your real IP anymore. All the hackers or online virus direct attackers see your vpn 's IP and that is where their assault would first go. So, with a VPN you shield your Computer from immediate attacks. Most vpn have powerful firewall protection, they move on to their associates. So, besides streamlining bbc iplayer in the US, you get many additional benefits of using a vpn.
Speed – How quick you can access the internet should top the list of your priorities, as it would show how quick you can download things and carry out updates. These times, 3G routers are available online that allow you accessibility the Internet at speeds as high as up to 21.six Mbps. With these routers, you can divide your internet connection up to five gadgets namely laptop, Smartphone, pill etc.
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Migrants & anti-racism
Activists of MiGreat commemorate the 20.000 victims of the EU migration policy
20.000 names of fallen migrants
The MiGreat foundation brings attention to the 20.000 victims of the European migration policies during the action '20.000 names'.
The European borders are becoming more and more deadly. The percentage of migrants and refugees that die on their way to Europe is rising, and this is partly to blame on European policies. The MiGreat foundation asks attention for the anonymous victims of the refugee crisis and organizes the action '20.000 names' from the 1st till the 28th of June.
During 20.000 names, MiGreat will organize multiple memorial services throughout the country. Activists will name the 20.000 migrants that died on the European borders within the past 6 years and on the 26th and 27th of June, there will be a big memorial service on the beach of Scheveningen, where a pop-up monument will arise with thousands of memorial crosses. You can apply for this action through this form and organize a memorial in your own town or city.
The MiGreat foundation has been fighting for a humane migration policy since 2018 and pleads for a stop to the European discouragement campaign. Violent border police, unlawful deportations, and appalling conditions in refugee camps and detention centers have contributed to the rising percentage of migrants who don't survive their journey to Europe. By actively campaigning, organizing protests, and mobilizing activists, MiGreat has brought the deadly consequences of the European migration policy under international attention. For example, during their action 'Lopend Vuur', where they mobilized 5000 activists who joined a total of 60 wakes and 4 marches through the country to The Hague, or the refugee camps that got built for 24 hours in different European cities to bring attention to the devastating fire in camp Moria.
Het Actiefonds supports MiGreat, you as well? Apply through the form, let your friends know you are coming to the pop-up monument on 26 and the 27th of June, or make a donation through our website.
Help MiGreat om de 20.000 gestorven vluchtelingen te herdenken


The high top sneakers Lou Spikes have been crafted in mixed materials that bring a retro touch to an emerging energy that pays tribute to the East Village fashion of NYC. Set on a rubber sole, these audacious-style sneakers proudly show their spikes placed on the patent leather front as well as their lateral diams glitter fabric cutouts and their velour leather counter. The model is composed of flat shoelaces trimmed with the iconic black and white grosgrain, a rounded badge placed on the inner side and a plaque of the house name, as well as a pull tab on the back.


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Let me start by saying that I am definitely jumping on this bandwagon WAY late. Honestly, I had never heard of this book until I watched a vlogger talking about it on BookTube semi-recently. I finished the book pre-holidays but I haven't had the time to sit down to write a full review for it until now.
And this book deserves a full review.
I will start by saying that the story + the art = AMAZING! A Monster Calls is designed to look like a children's picture book (to an extent). The interaction of the illustrations – which are all in black and white, befitting the theme, and quite haunting – with the text evokes the same sort of feeling as reading a picture book; both nostalgic and educational (as in, it's clear that the book is trying to teach a lesson). Okay, so that last one is not a feeling, but, yeah, you get it.
The story itself is beautiful, relatable, and heartbreaking. I do NOT cry easily, and I was ugly-crying at the end of this book – in the best way possible, I promise!
The way Patrick Ness handles darker, semi-uncomfortable or unsettling themes and topics not only seems to come from an authentic place, but are also incredibly relatable even if the reader hasn't experienced these things first-hand.
The illustrations. The illustrations. The illustrations. Listen, I will be the first to admit that I love creepy. But I love creepy even more when it blends so well literally and figuratively with the text of the novel. Symbolism = life. How many of you readers/writers out there agree?
I mentioned this before, but nothing makes me cry. Well, very few things. This is actually the first time that I have ever cried due to reading a book. (And yes, this includes the seventh Harry Potter book). I know, I'm cold-hearted. It's okay, at least I am self-aware.
There are very few children's books/YA books that deal with death, grief and the repercussions of both as eloquently as this book does.
The only warning I would give about this book is this: if you have a younger child and might be interested in reading this book with them/wondering if they should read this book, just note that there is a pretty heavy amount of swearing in the book and some of the themes may be quite dark/unsettling for young readers. I also could see how the illustrations may make some kids uneasy; they are pretty spooky (but the spookiness blends so well!) This might be something you want to read first – trust me, it's worth it.
Okay, that's it! The ultimate summary: A Monster Calls is just really good.
This books receives a UUS YOU MUST READ THIS! nomination. Because, really, you gotta read it!


Solar power is not only cleaner and more environmentally friendly, but it is also cheaper and renewable. The problem we have, however, is that we have become so accustomed to our fossil fuel power that we hardly ever take other forms of energy very seriously. That and the fact that most of these other energy sources are still relatively new and in their infancy, which makes them rather expensive, at least, as far as the initial costs are concerned.
Not everyone can afford to mount solar panels on their roof,and not everyone has the $10,000 – $30, 000 initial cost they would need to have a geothermal heating system installed.
However, if you get past the initial costs, you will find that renewable energy usage such as geothermal heating systems and solar powered devices end up saving you money in the long run. Sometimes as much as 60% of what you are spending on energy now.
If you are tired of depending on the national electrical grid and would like to save a bit of money, then it is about time you took a closer look at some of the benefits that come with installing solar panels in your home.
One of the biggest concerns with people who do not believe in using non-commonplace sources of energy such as solar is that they do not know whether or not this kind of energy will be compatible with most of their devices. Just because you are using solar power does not mean that you have to stop using your iPad or your computers.
Inverters make it so you can plug in any of your devices into solar powered wall sockets and charge them as you would when using the national electrical grid.
Here are five solar-powered devices that you should have in your home in 2018.
While it is true that campfires are an integral part of any outdoor adventure, you do not necessarily have to use that to cook your meals anymore. The Portable Solar Cooker from GoSun makes cooking outdoors a breeze. It channels the sun's energy into a parabolic reflector that turns it into electricity. With this device, you can cook meals as well as boil and pasteurize water.
With this portable solar cooker, you can now carry your kitchen with you wherever you go. You can even use it around the home to save money on cooking gas.
This is one for those people who cannot survive without their handheld devices. The Creative Edge Solar Phone Charger is a tiny little solar charger that can store up to 5,000 mAH power for up to 6 hours. While it is primarily a solar charger, you can also plug into the wall to charge it up faster and use it as a power bank. When in the great outdoors, you only need to leave it out in the sun for a little while,and you can use anyone of the two available USB ports to charge your devices.
This is a wonderful little device that goes above and beyond to keep you connected to the world on your handheld devices.
Solar powered lights are some of the simplest and most convenient sources of illumination that you could even have in the homestead. None, however, come anywhere close to being as easy and as convenient to use as this one. All you have to do is peel off the backing and stick it anywhere you want. It charges using the sun's energy during the day and lights up your compound at night.
This is a simple and very efficient way to make sure that those dark corners of your driveway and backyard are well-lit.
It is all about your handheld devices. This window-mounted solar charger ensures that they stay powered throughout the day even if you are nowhere near electricity. All you have to do is put the charger on a window with direct sunlight and it can fully charge your device within 10-15 hours.
This is a tiny, solar-powered 150WH generator that can power up your laptops, lights, and cell phones. If you go for the more powerful ones, you can even power your fridge, power tools and TV. It has an inbuilt AC inverter that makes it one of the most ideal solar powered items to have around the home.
If you are looking to power more power-hungry gadgets, then you might need to get a more powerful generator that works with the same concept.
These solar powered gadgets just made life around the home more convenient and definitely much cheaper. Plus, they are all great ways to ensure that you are minimizing your carbon footprint.


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Home » Magazine » Society » How Small Were We
How Small Were We
The Indian small town has been wiped off the map, the measured hopefulness of its middle class replaced by an acquisitiveness unchecked by social concerns, propelling itself on the twin cylinders of fundamentalist politics and global capitalism
Siddhartha Deb	20 August 2001
illustration by Jayachandran
Siddhartha Deb	August 20, 2001 00:00 IST How Small Were We
In the obituaries and retrospectives that followed in the wake of R.K. Narayan's death some months ago, it became evident that what was being mourned was as much a place as a person. That the passing away of Malgudi should become an occasion for nostalgia is not surprising. After all, it was the locus of Narayan's art for the greater part of a century, and the chronicling of Malgudi was also a charting out of what it meant to be a modern Indian. Yet this exemplary role that Malgudi seems to have played in our collective imagination is unusual, if only for the reason that the small town has never been a significant entity or idea in India.
It is not difficult to see why. The terms of the debate were set well before Independence by Gandhi and Nehru, and their contending visions seized upon the opposing poles of the country and the city. In their situation, it made sense. Not only were the small towns economically and culturally insignificant, but ideologically too they were contaminated. The typical Indian small town is for the most part a colonial creation, a stronghold and outpost of the expanding network of British modernisation whose very topography was marked by the desires of the ruling class.
This was certainly true of Shillong, where I grew up, but I am sure that the features I can recall—town club, missionary school, civil hospital, cantonment area, bus station, park are immovable, defining fixtures of other towns also, dispersed across the country. In Malgudi, too, these characteristics are a given; all that its leaders can do after Independence is rename the parks and roads and then leave them to entropy.
To live in a structure created by foreign rulers makes for a very strange sort of Indian, and the middle class that inherited the colonial small town shared much of the concerns of its predecessors. The small town was, by all accounts, the sane middle ground between the anarchy of the city and the darkness of the village, and it bred an inherent distrust for big ideas and rapid change. But this insularity was countered by an awareness of being invisible in the master narrative of the nation; things happened elsewhere, in the hinterland of great votebanks and in the cities shored up with money and power.
The more attractive features of a small-town sensibility probably emerged from this play of opposing feelings, the relief at being left alone checked by the sense of being left out. When the first televisions appeared in the early eighties, and when our Assamese neighbour was the only one with a television set, it was common for nearly a dozen of us to gather there every Sunday morning. And in that drawing room, among the food and talk and cups of tea, there was both a sense of community and a grappling with the idea of progress.
That period around the Delhi Asian Games, when transmitters sprouted all around the country, was probably the culmination of small-town identity. As usual, much was made of the fact that the television had reached Indian villages, but it was the small towns that were the perfect recipients of those centralised programmes beamed out by Doordarshan. State television had something of the effect that Benedict Anderson, in his book Imagined Communities, ascribes to newspapers in emergent nations—a way of threading together disparate spaces at the same moment in time. For a while, it was possible and even inevitable for all of us in Shillong and Bokaro and Baroda to switch on our sets at the same time and watch the same programme, a community predicated on the possibility of limited choice. And unlike Bombay cinema with its faithful representation of the country-city dichotomy, Doordarshan programmes even managed to portray small town life, including a televised version of Malgudi Days.
That way of living seems quaint, even archaic, when looked back upon from such a distance. Barely twenty years later, the Indian small town has been wiped off the map, the measured hopefulness of its middle class replaced by an acquisitiveness unchecked by social concerns, propelling itself on the twin cylinders of fundamentalist politics and global capitalism. In this race, it seems very clear, there are no places or people that can afford to remain small or intimate.
This is not to say that the Indian small town needs to be mourned indefinitely, or that it should be idealised for the sake of mere nostalgia. If it needs to be commented upon, perhaps in no more than a footnote, it is only so that we have some account of the things and places left behind in our dizzying rush forward, our destination yet unknown.
(Journalist turned author Siddhartha Deb's first novel will be released next year by Picador.)
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text:
Q: Is it safe to call `Py_DecRef` on an object created before the last Py_Initialize? Assuming all participating modules correctly handle Py_Finalize, does C API allow this:
Py_Initialize();
...
PyObject* myObj = PyObject_Call(simple_class, empty_tuple, NULL);
Py_Finalize();
...
Py_Initialize();
Py_DecRef(myObj); // <--- is this safe by design?

Narrowed down example that crashes
Py_Initialize();
PyObject* builtins = PyEval_GetBuiltins();
PyObject* iter = PyDict_GetItemString(builtins, "iter");

Py_IncRef(iter);

Py_Finalize();

// ----- new run starts, iter should still be alive

Py_Initialize();

Py_DecRef(iter);

Py_Finalize(); // fails inside PyGC_Collect -> validate_list

Background (OLD, before narrowing down the problem)
I am debugging 3.9+ support in https://github.com/pythonnet/pythonnet
The issue I originally got is validate_list from gcmodule.c fails at
assert(prev == GC_PREV(head));

In debugging this, I sprinkled my code with calls to validate_list, and found
that the assertion fails in the following scenario:
Py_Initialize();
...
PyObject* myObj = PyObject_Call(simple_class, args, NULL);
Py_Finalize();
...
Py_Initialize();
...
validate_list(gc.generations[2],
              collecting_clear_unreachable_clear); // OK
Py_DecRef(myObj);
validate_list(gc.generations[2],
              collecting_clear_unreachable_clear); // EXPLOSION

Now I have not yet tried to narrow down the steps further, because I realized
that I am unsure about the question in the title. E.g. is this use of myObj
supported by embedding API in general, or is this expected, that after
reinitializing the runtime Py_DecRef on old objects would break GC?
In my scenario simple_class is defined in Python as inheriting from object, e.g. it is a heap type created by Python runtime from Python code:
class SimpleClass(object):
  def __init__(self, type, tb):
    self.type = type
    self.traceback = tb

type and tb are values you get from PyErr_Fetch when list ends.
The reason why I have to Py_DecRef after Py_Finalize + Py_Initialize is that myObj actually gets collected by a different GC (.NET).

redpajama_set_name: RedPajamaStackExchange


text:
What Are the Different Ethical Issues in Business?
Sheri Cyprus
One issue of ethics in business is child labor.
Ethical issues in business include concerns of quality, value and honesty as well as the category of corporate responsibility. In addition to proper treatment of all workers, business ethics basically boil down to being honest about what products and services are in relation to their origin and price. Corporate responsibility involves a sense of protection in offering safe products for consumers that are created through decent working environments for the people who manufacture or create them. To be dishonest in what ingredients are included, where they originated or where a good was made is an example of poor business ethics and a lack of corporate responsibility.
Corporate social responsibility is an important aspect of business ethics.
The supply chain is the path of getting a product to the end user. Different ethical issues in business that involve the supply chain involve the workplace, workers and the marketing mix. The marketing mix includes the product, packaging, price, place or distribution and promotion. Each of these components can create a different ethical issue. For instance, unethical advertising would promote an unsafe product as being safe. Fruit that was actually grown in Mexico, yet stamped as being imported from Canada is an example of unethically representing the distribution part of the supply chain in the marketing mix strategy to sell the goods to unsuspecting consumers.
Charging a price far above the suggested retail one is another of the ethical issues in business. Using packaging that doesn't contain warnings such as for a cleaner with harmful fumes to be used only in a well-ventilated area is a different kind of unethical business issue. Most countries have strict regulations and laws on what manufacturers and marketers must reveal about a product created for sale. Consumer advocate groups are another source for advocating ethical issues in business. If a company knows that unethical issues surround its products or services, yet doesn't take any action to correct it, this can in itself be a breach of ethics.
Workplaces and the workers involved in manufacturing products relate to ethical issues revolving around being paid fairly for their work as well as being treated humanely. Ethical issues in business that are relevant to these concerns are often categorized under the term corporate social responsibility (CSR). They include slave and child labor or wages that are so low that healthy survival is close to impossible. Dangerous workplaces or work environments in which staff are beaten or treated violently are other types of CSR ethical issues.
What Is Moral Disengagement?
What Is Ethical Marketing?
What is an Ethical Fund?
What are Basic Business Ethics Theories?
What are Ethical Business Practices?
What are Basic Business Ethics Concepts?
What are the Most Common Ethics Issues?
oasis11
@Icecream17 -Oh I remember that story. It was so sad. I also think that companies have an obligation to sell products that are safe for consumers. For example, many toys were found to be laced with lead when they were made overseas in a factory in China. If the company was aware of the situation and had not done anything about it, it would have clearly been an ethical violation because they would be putting profits ahead of the general public's safety. Thank goodness that they did a voluntary recall.
This happened recently with a Japanese car manufacturer after it was proven that they tried to hide problems with the sudden acceleration and faulty brakes that have killed several people. I think that once a company's reputation is damaged this way, their business will continue to suffer long after they resolved the problem because the public will have a hard time trusting the company again.
icecream17
@Suntan12 -I agree and wanted to say that some companies do their own recalls in order to protect their customers once they know that there are problems with some of their products. These are honest companies that value their clients.
I also wanted to say with respect to honesty that the many of the recent business ethics issues revolved around disclosing poor accounting practices and misleading investors.
Some companies in the past have lied about their earnings in order to pump up their stock price. One of the most devastating examples was the Enron collapse in which all of the employees lost their life savings because the company not only lied about its finances but it knowingly encouraged employees to invest even more money.
suntan12
@Sunny27 -I know that there will always be controversy regarding the cigarette companies, but I also think that the financial sector also requires proper disclosures so people know what they are investing in.
As a matter of fact, only a licensed professional can talk to you about investing in securities. In fact many banks have to have a designated section for investments and have to have a sign that states that certain investments are not FDIC insured and can lose value.
I think that a lot of business ethics issues in the news are resolved through additional governmental regulation. Although businesses should do this on their own at least the government steps in to protect consumers and keep businesses honest.
When I think of business ethics issues I always think of a company's responsibility to tell consumers of the potential risks associated with their product or service so that consumers could make an informed decision.
For example, cigarette smoking has long been proven to cause cancer I read that it wasn't until 1996 that the first warning labels came out on the packaging. When you consider that one of the biggest cigarette companies began manufacturing cigarettes in the United States in the early 1900's it really was almost one hundred years before we started warning the public about cigarette smoke.
Some say that we did not know the hazards of cigarette smoke, but I know that if the Food and Drug Administration would have known then what it knows now I don't think that cigarettes would have ever been approved for sale in the United States.
I saw the movie, "The Insider" with Al Pacino and Russell Crowe that was a true account of the tobacco companies and their prior knowledge of the dangers of cigarette smoking. I believe that ethics in business really requires that you tell the truth about your product no matter how it impacts your sales.
By: poco_bw
By: onurakgul


text:
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text:
\section{Introduction}
Hidden Markov Models (HMMs),\cite{murphy2012machine}, are a well established and widely used method for modeling sequential data,  with applications in a variety of fields, such as speech recognition \cite{yu2016automatic} and economical time series \citep{hamilton2020time,kaufmann2019hidden}, to name a few.
In all of the above applications, it is common to have \emph{missing observations}. That is, we assume that a certain system, after $N$ time steps, produces a full sequence of observations $U_i = (u_1, \ldots, u_N)$, but there is a \emph{known} subset of times,  $1\leq t_1, \ldots, t_m \leq N$, $m \leq N$, such that the observation values at these times, $\Set{u_{t_j}}_{j=1}^m$, are not known to us. This situation is illustrated in the first and second lines of Figure \ref{fig:illProcess}. We refer to this type of missing values as missing values with \emph{known locations} (or \emph{known gaps}, Interchangeably), since we know which of the values are missing.

\begin{figure}[h]

    \begin{subfigure}[b]{\textwidth}
    \begin{center}
        \includegraphics[scale=0.5]{Figures/section_1/ill_process.png}
        \caption{Omission Process Illustration}
        \label{fig:illProcess}
    \end{center}
    \end{subfigure}

    \begin{subfigure}[b]{\textwidth}
    \begin{center}
        \includegraphics[scale=0.5]{Figures/section_1/figure_one_per_exp_two_plots.png}
        \caption{Ignoring the Gaps}
        \label{fig:NaiveIntro}
    \end{center}
    \end{subfigure}
    \caption{
\textbf{(a)} The gaps location may alter the sequence considerably.
\textbf{(b)} The \emph{Naive} approach may perform poorly, highlighting the need for a more careful treatment of gaps. (s.d $<$ .062)}
    \label{fig:Intro}

\end{figure}

\subsection{Motivation And Previous works} \label{Intro:part2}
It is commonly assumed that observations are provided with time stamps, and that a single time unit corresponds to a step of the underlying chain. Such assumptions naturally imply that when there are missing values their location is known, however, this assumption do not hold for many applications. For example consider the case of irregular time series \cite{IrrTimeSe}, also known as unevenly spaced time series, where the system is observed at varying time intervals that do not correspond to the steps of the underlying chain. For example, in the context of cyber-attack detection, it is often assumed that the attacker sequence of actions can be modeled as a Markov process \cite{Chadza2020-mz}. However, in practice, the observations of these malicious activities are irregular and incomplete. Another scenario is when the sampling clock is different from the underlying process clock. An example of this is the monitoring of sepsis patients, as the disease can progress in an unpredictable manner, resulting in inconsistent and possibly incomplete measurements of vital signs \cite{pmlr-v106-moor19a}.

One domain where missing observations in unknown locations is frequently encountered is computational biology, where data is often only ordered and lacks time stamps\citep{pmid29107741,liu2017reconstructing,orr2018learning}.
For that reason, Silent-state HMM were developed. In SHMM, some states are not associated with any observation, and the model can transition between these states without emitting any observation. These states are referred to as "silent" states. Notice that silent states are incorporated explicitly into the assumed chain structure.
For example, consider the wide use of Profile hidden Markov models \cite{10.1093/bioinformatics/14.9.755} for DNA sequence searching, where the goal is to find optimal matches between a model input and a database of sequences \citep{10.1093/nar/gkr367,Wheeler2013}. In this context, PHMMs are used to match the input model with sequences in the database that are assumed to come from the same model. However, it is also assumed that some states (nucleotides) may have been "deleted" through the course of evolution and are therefore not observable, or "silent".

Despite their potential usefulness and popularity, the uses of PHMMs and SHMMs are limited to a small set of problems. Currently, the focus of PHMM use is on inference, where it is assumed that the transition matrix and omitting probabilities are known (known as the profile HMM with known profiles).
However, there is a significant need for many applications of SHMM to be able to directly learn the model parameters from data\citep{Wheeler2013,setty2019characterization, ye2019circular,orr2018learning,Pattabiraman2021-ol}. There are currently several limitations that have hindered this goal.
\textbf{First}, while the use of Expectation Maximization algorithms for learning the parameters of SHMM has been proposed, it has not yet been thoroughly studied and has not been shown to be effective in practice. In addition, it has been suggested in recent years that PHMMs are not identifiable \cite{Pattabiraman2021-ol}. In this paper, we address this limitation and examine the reasons for the failure of these algorithms, providing a new perspective on the identifiability issue. \textbf{Second}, while partial knowledge that is crucial for learning is often available, current frameworks are not flexible enough to incorporate this information.
\textbf{Most importantly}, SHMMs assume that the underlying chain is acyclic (i.e the underline chain forms a DAG), which is frequently violated in the field of computational biology \citep{ye2019circular,deconinck2021recent,LummertzdaRocha2018}. This assumption can be particularly problematic in areas such as pseudo-time analysis \cite{ye2019circular} in a variety of biological systems, \cite{campbell2018uncovering}, with particularly extensive applications in single-cell trajectory analysis which is crucial in understanding the dynamics of a cell (see for instance \citep{pmid29107741, van2020trajectory} and a survey \cite{deconinck2021recent}).
In this type of applications, one is given a \emph{set} of observations, usually corresponding to information about the state of a cell. However, due to the nature of the data collection process, it is often impossible to obtain timestamps for each individual observation.

In the interest of providing a comprehensive overview, we would like to highlight the following previous works related to this topic: In \cite{IrrTimeSe}, the authors formally described the concept of irregular time series in HMMs, but did not propose an algorithm for reconstruction. \cite{NIPS2013_32b30a25} dealt with the sub-problem of flow network analysis where the positions of a few observed nodes are fixed and known in advance. The effort closest to our own, as far as we are aware, is the work of \cite{orr2018learning}, in which the authors developed an Expectation Maximization-based algorithm for reconstructing HMMs from data with similar properties, but under the restriction of acyclic chains and a strict assumption about the underlying Markov process.

\subsection{Missing Observations In Unknown Locations And The Need For Special Methods}
\label{Intro:MissingObsIntro}
Let $O= \Brack{o_1, \ldots, o_k}$ be an \emph{order} set of observations, that is, we assume that a system passed through some sequence of states $S = \Brack{s_1, \ldots, s_N}$, \emph{some} of which have generated the observations $O$. This situation is illustrated in the third line of Figure \ref{fig:illProcess}, where different subsequences correspond to different possible values of $O$. Note in particular that the same $O$ can result from different deletions of the full observation sequence. In this situation, we do not apriori know the positioning of the observations $O$ (and, equivalently, of the observations missing from $O$) with respect to the state sequence $S$, and therefore we refer to such observations as observations with \emph{unknown gaps}.
The goal of the reconstruction (or learning) algorithm is to recover the underlying dynamics of the state sequence $S$ from the observations $O$.

It is natural to ask whether one indeed needs to treat the unknown gaps situation specially. For example, a possible approach to reconstruction, termed \emph{``Naive"} in this paper, is to simply ignore the missing observations and to reconstruct from sequences $O$ as if these were the full observation sequences $U$. See for instance \citep{liu2017reconstructing,setty2019characterization}. We now show that the \emph{Naive} method can indeed introduce extremely large errors into estimation. First, we denote $\Psi(s)$ to be the probability of observation emitting from state $s$ to be omitted (see section \ref{sec:background}). For a fixed benchmark HMM and a fixed $\Psi(s)= 1-p_c : \forall s$, we generate synthetic observation trajectories with gaps based on $p_c$, and compare three reconstruction methods: (a) The \emph{Naive} approach, where we ignore the missing observations, (b) The Random approach, where we use the standard reconstruction algorithm with \emph{known} gaps, but the gap locations given to the algorithm are chosen at random, and (c) The ``Known'' reconstruction, where the gap locations provided to the algorithm are the \emph{true} gap locations. This algorithm is used as an \textbf{ideal benchmark}, as its performance is the best one can hope for, for any unknown gaps algorithm.
The experiment is repeated for different benchmark HMM (see section \ref{sec:experiments} for full details), and for a range of values of $p_c$. For each run, we measure the quality of the reconstruction by the $L_1$ distance between the reconstructed and the ground truth transition matrices.

As shown in Figure \ref{fig:NaiveIntro}, for intermediate values such as $p_c = 0.5$ the performance of the \emph{Naive} and random approaches are poor, and for some (MultiPartite HMM) this is especially pronounced, with error of $1.6$ even at $p_c = 0.8$ (the maximal error of $L_1$ is $2$).
This experiment emphasizes the need for reconstruction algorithms that model the unknown gap locations much more carefully than the \emph{Naive} and ``random'' methods. Appendix \ref{Supp:wrognW} contains additional benchmark HMMs.

\subsection{Paper Summary and Contributions}\label{sec:SummeryContribution}
This paper aims to propose a comprehensive solution for learning HMMs in scenarios where observations are missing in unknown locations. Those solutions do not impose restrictive assumptions on the underlying chain, with the exception of the "Naive" method, which is shown to be ineffective. We demonstrate the effectiveness of our approach through various methods, which correspond to varying levels of prior knowledge about the process. These methods enable reconstruction in cases where prior knowledge is partially known, which is not possible with previous methods. The rest of the paper is organized as follows:\\

\noindent\textbf{Model definition}. section \ref{sec:background} defines the HMMOP (HMM with Omission Process) model. In this model, a set $U$ of full sequences of observations  $U_i = (u_{i1},\ldots,u_{iN})$ is generated by a standard HMM. Then, the sequence $O_i$ is formed by deleting (equivalently,  omitting) entries $u_{ij} \in U_i$ at random with a probability $\Psi$ which depends on the state, independently of the other $u_{i'j'}$. This type of state dependent omitting process in known as \textbf{Non- Ignorable Omitting Process}. section \ref{sec:nonign} examines the non-ignorable setup for missing observations, while section \ref{ssec:noiid} emphasizes the significance of taking $\Psi(\cdot)$ into account for accurate reconstruction.\\
\textbf{Analytical Analysis}.
section \ref{sec:analytic} presents an analytical analysis for the case when the probability of missing observations is constant across all states, $\Psi(s) = 1-p_c \forall s$. This analysis highlights the limitations of previous attempts to reconstruct HMMs with missing values using PHMMs.\\
\textbf{Reconstruction methods}. section \ref{sec:gibbs} introduces two approaches to reconstructing HMMOPs. One approach assumes that in addition to the observed trajectories $\mathcal{O} = \Set{O_i}$ the length of the original sequence is known. The second approach assumes that only the omitting probabilities ($\Psi$) are known. Additionally, we present a method for imputing $\Psi$ when only partial knowledge of it is available.\\
\textbf{Experiments and evaluations}. section \ref{sec:experiments} evaluates the unknown gaps reconstruction algorithms on a number of synthetic and semi-synthetic benchmarks, where it is possible to compare the reconstructed transition matrices to the ground truth matrices that generated the data.
The experiments are performed for a range of $\Psi$, allowing us to study the effect of different percentages of missing observations on the performance. It is worth noting that given $\Psi$ is known, our reconstruction method (term ``Gaps sampler") \textbf{match the performance of an ideal benchmark for which the locations are known}. This result is particularly remarkable, as it demonstrates that our method is able to achieve the best possible performance. Indeed, observe that the reconstruction algorithm only takes $\Psi$ (or a single number $p_c$ for the ignorable case), as information about missing samples. Given an observation sequence $O_i$, there is an exponential number of possible placings of gaps within that sequence, and we know from the experiment described in Figure \ref{fig:NaiveIntro} that a random placement of gaps yield poor results. Yet, the algorithms manage either to get close to, or match the performance of the sampler that knows the locations of the missing values. To the best of our knowledge, this is the first time such results have been achieved. \\
\textbf{Experiments and evaluations under miss-specifications}. In section \ref{sec:pcMiss}, we evaluate the robustness of the algorithms with respect to various \emph{model miss-specifications}. First we study the effect of providing a perturbed $\Psi$ to the algorithm. Second, we study the ability to reconstruct when $\Psi$ is unknown for some states. Third, we evaluate other miss-specifications w.r.t the omitting process: We tested the performance for a non-constant $\Psi$ (that is, changing per sentence) and a scenario where the omitting process is a Markov process. For all of the above miss-specifications we show that the results are quite stable. \\
\textbf{Code}. To the best of our knowledge, our implementation is the first publicly available Gibbs sampling-based HMM learning implementation for Python, and the first to handle non-ignorable missing observations in general. The code is provided in the supplementary material and will be made publicly available with the final version of the paper.

\section{Background and Model Definition}
\label{sec:background}
A Hidden Markov Model  $(\mathbb{X},T,\Theta)$ is defined by a finite state space $\mathbb{X}$ and a transition probability matrix $T \in R^{\Abs{\mathbb{X}} \times \Abs{\mathbb{X}}}$ which defines a Markov chain on $\mathbb{X}$.
The observations of the model take values in a set $\mc{E}$, and for every state $X \in \mathbb{X}$, $\Theta_X$ is a distribution on $\mc{E}$ corresponding to $X$.
Let $X_i = [X_{i0},\ldots,X_{iN}]$ be a sequence of states, and let $U_i = [u_{i0},\ldots,u_{iK}]$ be a sequence of observations (also referred to as \emph{sentence}),
$u_{ij} \in \mc{E}$, $K\leq N$.\\
The sequence $W_i$ encodes the locations of the non-missing observations. Define $W_i = [w_0,\ldots,w_K] $ as a sequence of indexes $w_k \in \{0,..,N\}$,$w_{k}<w_{k+1}$, such that $w_k \in W_i$ if observation $u_{ik}$
was generated at time $w_k$.
We denote by $U_i^{W_i} = [u_{iw_0},\ldots,u_{iw_K}]$ the restriction of $U_i$ to $W_i$.

\subsection{HMMOP - Unknown Gaps Model Definition}\label{ProblemSetup}
Let $M' = (\mathbb{X},T,\Theta)$ be an HMM. Define a random mapping $\Phi_{\Psi}(\cdot)$ of a full observation sequence $U_i$ to a partial observation sequence $O_i$ as follows:
Given a full sequence of states, $X_i$, corresponding to an observation sequence, $U_i$, an indicator vector, $C_i = [c_{i0},\ldots,c_{iN}]$ is sampled where $c_{ij}$ are independent Bernoulli variables with success probability $\Psi(X_{ij})$. The set $W_i = \Set{j \vert c_{ij} = 1}$ is defined, and the partial observation sequence, $O_i$, is set as $O_i = \Phi_{\Psi}(U_i) := U_i^{W_i}$.
Then HMMOP is defined to be the generative model where a state sequence $X_i$ and a full observation sequences $U_i$ are generated by an HMM, and the corresponding observation sequences $O_i$ are then produced as $O_i = \Phi_{\Psi}(U_i)$. The joint distribution of the HMMOP is :
\begin{equation*}
\begin{split}
    P(O_i, W_i, X_i|\Theta) = P(X_0|\Theta) P(O_0|X_0) \prod_{t=1}^{N} T_{X_t,X_{t-1}} \cdot \begin{cases}
    (1-\Psi(X_t)) \cdot P(O_t|X_t,\Theta) ,  \text{if}\ t \in W_i \\
  \Psi(X_t),  \text{otherwise}
\end{cases}
\end{split}
\end{equation*}

\subsection{Non Ignorable omitting process} \label{sec:nonign}
Non Ignorable omitting process in HMMs, also known as State-Dependent Missingness (SDM), was first introduced by \cite{doi:10.1080/03610918.2011.581778} and later developed by \cite{https://doi.org/10.48550/arxiv.2109.02770}.
Reconstruction with non-ignorable missing observations refers to the problem of reconstructing HMMs when the probability of an observation to be missing depends on the state of the system, in contrast to the ignorable case where $\Psi(s)$ is constant and simply translate to the percentage of missing observations in the data. Reconstruction of dynamical process with SDM (not necessarily HMMs) is considered a challenging problem and it is an active area of research. In this paper we generally consider the non-ignorable case and we carefully examine the impact of the SDM assumption on our reconstruction results. Notably, we show that we can still achieve ideal results for known omitting probabilities ($\Psi$) even in the SDM case. section \ref{sec:experiments} presents comprehensive evaluation of the ignorable case.

\subsection{Analytical Analysis}\label{sec:analytic}
For analytical analysis we consider the case of ignorable missing observations, that is, $\Psi(s) = 1-p_c \forall s$. More, let us first discuss the \emph{non-hidden} case. That is, we assume for the moment that the observation given a state $X \in \mathbb{X}$ is the state $X$ itself. Note that the unknown gaps reconstruction is interesting even in this simpler case. \\

\begin{proposition} \label{MMM}
Let $M' = (\mathbb{X},T,\Theta)$ be an non-hidden Markov model. $\mathbb{I}$ the identity matrix. Then the sequence of observations $O_i = \Phi_{\Psi = p_c}(U_i)$ is a Markov chain with transition matrix :
\begin{equation}
\label{eq8}
T_r =   p_c \cdot T \cdot [\mathbb{I} - (1-p_c) \cdot T]^{-1}.
\end{equation}
\end{proposition}

\begin{proof}
By definition, in the non-hidden case the sequence $U_i$ coincides with the state sequence $X_i$, and $O_i$ is a sub-sequence of $U_i$. Note that the waiting time $d$ between two occurrences of $1$ in $C_i$ is geometrically distributed with mean $p_c$, which implies that
\begin{equation}
\label{eq:T_r_sum}
T_r = \sum_{d=1}^{\infty} p_c (1-p_c)^{d-1} T^{d}.
\end{equation}
For every stochastic matrix $T$ we have $\|T\|\leq 1$, \cite{UpperBoundsStochastic}, where $\|\cdot\|$ is the operator norm, $\|(1-p_c) \cdot T\| < 1$, and thus the series in \eqref{eq:T_r_sum} are summable, with sum given by \eqref{eq8}.
\end{proof}
Now, note that given the partial observations $O_i$, we can directly learn the transition matrix $T_r$ for $O_i$ by counting the co-occurrences of the states in $O_i$. This is the instantiation of the \emph{Naive} approach. Next, observe that if $p_c$ is known, using \eqref{eq8} we can \emph{invert} the omission process to obtain $T$:
\begin{equation} \label{eq:eqbackward}
    T =  [\mathbb{I} \cdot p_c + (1-p_c) \cdot T_{r}]^{-1} \cdot T_{r}.
\end{equation}
We refer to this as the \emph{backward transformation}.

As we can see, $T$ is a transition matrix of a Markov model for any $p_c \in (0, 1]$. Therefore, given a set of missing observations $O$, it is not possible to infer any "original"  T, but rather only a backward version of T that is conditioned on a specific value of $p_c$.
As Supplementary material \ref{suppFB} shows, the difference between reconstructed T for different $p_c$ might be significant.\\
 The current approach to reconstructing Profile-HMMs involves guessing a random $p_c$, then using an Expectation-Maximization (EM) procedure to learn $T$ given the guessed $p_c$, and vice-versa. However, as we demonstrated, guessing the $p_c$ is equivalent to randomly selecting a backward transformation of $T$, which has no practical meaning. This we believe explain the lack of successful PHMMs reconstruction studies, despite its importance.
Detailed evaluations of \eqref{eq8},\eqref{eq:eqbackward}, together with additional results and characteristics of the backward transformation, are given in Supplementary Material section \ref{suppFB}

Note that the ``backward transformation" can be seen as a reconstruction method for the case of ignorable missing observations. \textbf{For the proper hidden case}, we can apply any standard HMM learning algorithm to learn $T_r$ using the \emph{Naive} approach from $\mathcal{O}$. Then, we can use \eqref{eq:eqbackward} in the same way as in the non-hidden case to obtain $T$. Since this approach requires some HMM learning, which is typically non-analytic, we refer to this approach as semi-analytic. We evaluate the results of this method in compare to the other methods in Supplementary material \ref{sec:experiments}.

\section{Reconstruction Methods}
\label{sec:gibbs}
\begin{figure}[t]
    \centering
    \begin{subfigure}[b]{0.42\textwidth}
        \includegraphics[scale=0.28]{Figures/section_2/illus_smapling_W.png}
        \vspace*{2mm}
        \caption{Illustration of W Sampling given $\Psi = p_c$.}
        \label{fig:illusWSample}
    \end{subfigure}
    \hspace{3pc}
    \begin{subfigure}[b]{0.42\textwidth}
        \includegraphics[scale=0.28]{Figures/section_3/illus_process_with_d.png}
        \caption{Illustration of HMMOP given $\Psi = p_c$.}
        \label{fig:IllKnownPC}
    \end{subfigure}

    \caption{ Reconstruction methods. }
\label{fig:sampleIll}
\end{figure}

In this section, we present the reconstruction methods. As discussed in sections \ref{sec:SummeryContribution} and \ref{sec:analytic}, these methods rely on different levels of knowledge about the underlying latent dynamic process or the omitting process. \textbf{The first method}, named the "Matching Sampler," requires knowledge of the full unknown sequence length (N) as input. For the case of ignorable missing observations, this method does not require any additional information. However, in the case of non-ignorable missing observations, knowledge of the omitting probabilities ($\Psi$) is required. We show that even if this information is only partially known, it can be inferred as part of the algorithm and the method is highly robust to misspecifications regarding $\Psi$. \textbf{The second method}, named the "Gaps Sampler," only requires knowledge of $\Psi$ as input. Although it does not have access to information about N, the "Gaps Sampler" is also somewhat robust to misspecifications regarding $\Psi$.
Often, knowledge about $N$ can be derived from prior knowledge on the underlying process $M'$, for instance, when there are distinctive "start" and "end" states with knowledge about the latent process rate \citep{Saelens2019,Herring2018}. On the other hand, knowledge about $\Psi$ exists for cases in which prior knowledge on the sampling method ($\Phi_{\Psi}$) exists \cite{LummertzdaRocha2018}. In addition, in cases where the reconstruction is a first step for a prediction task \cite{10.1371/journal.pone.0221245}, $\Psi$ can be partially inferred as an hyper parameter using the labeled data (Supplementary Material section \ref{Supp:Inference} presents a prediction algorithm for HMMOP).

\textbf{Gibbs sampling.} Both of the methods relays on the Gibbs sampler \citep{gelman2013bayesian,10.1214/08-BA326}. A Gibbs sampler samples the HMM parameters (i.e., $T$ and $\Theta$) and the latent state sequences $\mathcal{X}$ conditioned on $\mathcal{O}$. More specifically, one interchangeably samples $P(X_i|T,\Theta,\mathcal{O})$, and then samples $P(\Theta|X_i,\mathcal{O})$ and $P(T|X_i)$.
Here we extend this procedure by incorporating the uncertainty of the unknown gaps into the sampling process using an additional set of latent helper variables.
Let $ \mathcal{X} = \Set{X_i}$ be the set of all latent states sequences as describe above, and let $O_i\in \mathcal{O}$ be the observed sequences corresponding to $X_i$.

\subsection{The Matching Sampler}\label{MatchingSampler}
The Matching Sampler builds upon $\mathcal{W} = \Set{W_i}$ (see section \ref{sec:background}) as an helper set of latent variables, hence the challenge mainly lies in sampling the helper variables $P(W_i|X_i, N_i, O_i, T, \Theta)$. To address this challenge, we first note that the conditional distribution of $W_i$ given $X_i$ is independent of $T$. Therefore, our problem reduces to sampling $P(W_i = [w_{i0},\dots, w_{iK}]|\Theta,X_i,O_i,N_i,\Psi).$\\
For $k\leq K$, let $W^k$ denote the sequence $W$ with $w_k$ omitted. Also let  $I_{[w_{k-1},w_{k+1}]}(w)$ be the indciator function.
The sampling of $W_i$ will be done using its own Gibbs sampler. That is, instead of sampling $W_i$, we iteratively sample its components,$w_{ik}$, conditioned on the rest of the matching $P(w_{ik}|W_i^k,\Theta,X_i,O_i,\Psi).$
The Markov property of the sequence $X$ implies that $w_{ik}$ depends on the sequences $W_i^k,O_i,X_i$ only through ``local varaibles'' at $k$,  that is, it depends on $X_{i,w_{i,k-1}}$ up to $X_{i,w_{i,k+1}}$, and on the single observation $O_{ik}$ :
\begin{equation} \label{PWk_2}
\begin{split}
    P(w_{ik}|W_i^k,\Theta,X_i ,O_i,\Psi) \propto P(O_{k}|  X_{w_k}) (1-\Psi(X_{w_k}))
  \cdot I_{[w_{k-1},w_{k+1}]}(w_k)
\end{split}
\end{equation}
This situation is illustrated in Figure \ref{fig:illusWSample}.

One can see a similarity between $W_i$ to the ``wrapping" vector from the known Stochastic Dynamic Time Wrapping (DTW) \cite{doi:10.1080/03772063.1988.11436710}. Notice that while variations of SDTW can infer optimal W from $P(W|\ldots)$, our challenge is \textbf{sampling} $P(W|\ldots)$. Moreover, not every state in $X_i$ is part of $W_i$ as the SDTW algorithm requires.

In practice we use the Metropolis-Hasting \cite{doi:10.1063/1.1699114} for better convergence. Full details of the derivation of \eqref{PWk_2} and the complete algorithm, including the sampling of $T$, $X$, and $W$, as well as initialization considerations, can be found in section \ref{GibbsSamplerKnownN} of the appendix.

\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
         \includegraphics[width=\textwidth]{Figures/section_2/known_N_PC.png}
         \caption{Reconstruction for ignorable setup.}
         \label{fig:knownN}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
         \includegraphics[width=\textwidth]{Figures/section_2/known_Psi.png}
         \caption{Reconstruction for nonignorable setup.}
         \label{fig:nonIgnknownN}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
          \includegraphics[width=\textwidth]{Figures/section_2/imputing_vs_not.png}
          \caption{The impact of $\Psi$ imputation.}
    \label{fig:imputation}
\end{subfigure}
\caption{\textbf{(a, c)} Both Samplers significantly improves on the \emph{Naive} method. While the "Gaps" is better. \textbf{(b)} The Gap sampler converge to a better placement, faster.}
\label{fig:knownstuff}
\end{figure*}

\subsection{The Gaps sampler}
The Gaps Sampler presents a different representation for the missing observations locations by introducing variables $d = [d_0, \ldots, d_K]$. Those variables represent the number of omitted observations between any two observed observations, which we denote as gaps. Figure \ref{fig:IllKnownPC} illustrates the Gaps sampler for the ignorable case for which $\Psi(\cdot) = p_c$. Our goal is then sample $P(d_k |X_{k,k+1})$. Lets define $s = [s_0,\ldots,s_{d-1}] \in S^d$ as the set of all $d$ long states trajectories, also, lets assume $d < \mathcal{S}$ for a predefined $\mathcal{S}$. We say that a sequence is \emph{fully omitted} if we didn't observed any of the sequence states, we denote the case of a fully omitted $s$ as $\Bar{s}$. Given $\Phi_{\Psi}$ independent, $P(\Bar{s}|s) = \prod_{i=0}^{d-1} \Psi(s_i) $.
By definition, $P(d_k |X_{k,k+1})$ is the probability for $s \in S^d$ been fully omitted:
\begin{multline*}
 P(d_k |X_{k,k+1}) =
 \frac{\sum\limits_{s \in S^d} T_{[X_k,s_0]} \cdot T_{[s_{d-1}, X_{k+1}]} \cdot \Psi(s_0) \cdot \prod\limits_{i=1}^{d-1} T_{[s_{i-1}, s_{i}]} \cdot \Psi(s_i)
  }{\sum\limits_{\tau = 0}^{\mathcal{S}} \sum\limits_{s' \in S^{\tau}} P(s'|X_{k,k+1})\cdot P(\Bar{s'}|s')} \\  \propto \sum_{s \in S^d}
  T_{[X_k,s_0]} \cdot T_{[s_{d-1}, X_{k+1}]} \cdot \Psi(s_0) \cdot \prod\limits_{i=1}^{d-1} T_{[s_{i-1}, s_{i}]} \cdot \Psi(s_i)
\end{multline*}
And given that summing over $s \in S^d$ is not feasible, we suggest the following formulation:
\begin{multline*}
=\sum\limits_{x \in \mathcal{X}} \sum\limits_{j \in \mathcal{X}} T_{[j,x]} \cdot \Psi(x) \cdot P(d-1|X_{k,k+1},s_{d-2}=j)
P(d-1|(X_{k, k+1})) = \sum\limits_{j \in \mathcal{X}} P(d-1|X_{k,k+1},s_{d-2} = j)
\end{multline*}

This form the \emph{forward algorithm} \cite{18626} with $P(d|X_{k,k+1},s_{d-1} = j)$ as the forwarding elements.
Notice that $P(\ldots, d-1)$ are intermediate steps for calculating $p(d)$, hence the computational complexity of $P(1), \ldots, P(\mathcal{S})$ and $P(\mathcal{S})$ are equal.
Full details of the Gaps sampler are given in section \ref{supp:samplingd} of the appendix.

\subsubsection{Sampling $\Psi$}\label{subsec:SamplingPsi}
Both methods include the ability to sample and impute $\Psi$ for cases where $\Psi$ is partially known. Note that $\Psi(s)$ follows a Bernoulli distribution, so the conjugate prior for $\Psi(s)$ for each $s$ is the Beta distribution. Hence, the posterior distribution of $\Psi(s)$ given $X$ and $W$ is $\Psi(s)|X,W \sim Beta \left(\mu(s \in X^W), \mu(s \in X) - \mu(s \in X^W)\right)$ where $\mu(\cdot)$ is the counting measure. section \ref{sec:experiments} extensively evaluate the ability of our methods to impute partial $\Psi$.

\section{Experiments}
\label{sec:experiments}

This section evaluates the performance of the following reconstruction methods:
1. The Matching sampler (\textbf{\textcolor{Purple}{Purple}});
2. The Gaps sampler (\textbf{\textcolor{Green}{Green}});
3. The \emph{Naive} reconstruction as described in section \ref{Intro:MissingObsIntro} (\textbf{\textcolor{BurntOrange}{Orange}}); And,
4. The ``ideal benchmark'' (\textbf{\textcolor{Blue}{Blue}}) where the true gaps locations are known to the sampler as described in section \ref{Intro:part2}; Additionally, for the ignorable case, we evaluate the semi analytical approach (\textbf{\textcolor{Brown}{Brown}}) which is a byproduct of the analytical analyses; \\
The following models were used to generate the data (full details are given in Supplementary Material section \ref{Supp:EvaluationsModel}):
1,2. \textbf{``Synthetic Degree$\{d\}$" } A synthetic chain on 10 states, where each state has $d$ outgoing  transitions, chosen at random;
3. \textbf{``Cyclic MultiPartite"} Synthetic chain with 25 states. The states are divided into 5 groups ranging from 1 to 5, transitions are only possible between states of consecutive groups and in a cyclic way. For example, a state from group 4 only moves to a state in group 0, etc;
4. \textbf{``Part Of Speech"} process. Transitions and emissions (part of speech and words respectively) probabilities were extracted from the Brown NLP corpus \cite{BrownDS}. The trajectories were then sampled based on those parameters. \\
With the exceptions of the ``Part Of Speech" chains, state emissions are distributed with Normal distribution $N(\mu_i,0.1)$ for $\mu_i \in [0,n_{state}-1]$;
Unless specified otherwise, 1500 sentence($U$) of length 80($N$) were sampled. For all figures, Y-axis is the $L_1$ distance between the reconstructed and the ground truth transition matrices. Supplementary Material section \ref{Supp:SD} presents the standard deviations (s.d) of this section results.

\subsection{Reconstruction Under Correct Specification}\label{exp:KnownNPC}
This section evaluates the reconstruction performance of the methods with no miss-specification. Figure \ref{fig:knownN} presents the reconstruction results for the ignorable case, that is, only a single number is provided to the methods which is the true percentage of missing observations (or the true trajectory length $N$ for the Matching sampler). In this experiment, we pick varying $p_c$'s (X-axis), i.e. $\Psi = 1-p_c$ percentage of observations are deleted.
As the figure shows, the performance of all the algorithms is significantly better than the \emph{Naive} algorithm, which for most cases, is the only applicable one (\ref{Intro:part2}). Moreover, for the Gaps sampler, the reconstruction performance \textbf{matches that of the ideal benchmark} (\ref{Intro:part2}). As discussed in section \ref{sec:SummeryContribution}, this result is remarkable, due to the exponential number of possible placements $W$.

To gain a better understanding of the performance distinctions between the samplers, we will conduct a more in-depth examination of them. To that end, we fix an ground-truth $T$, and learn the placement parameters $W$ (or, equivalently,  $\mathbf{d}$ ). That is, $T$ is fixed, and in a single iteration of the sampler, we sample $P(X \vert W,T,O)$ and then $P(W \vert X,T,O)$. The performance was measured by evaluating the average log likelihood of the trajectories, $\frac{1}{|\mathcal{O}|} \sum_{i}^{\mathcal{O}} log \left( P(O_i,X_i \vert T,W_i) \right)$ after each iteration of the sampler. We have also varied the number of times $P(W|X,O)$ is sampled, that is, for a fixed $X$, the matching sampler had from 30 to 300 steps to find the ``right'' $W$, before the next $X$ was sampled. For the Gaps sampler, only one step of $W$ sampling is required by design per $X$ sample.
As shown in Figures \ref{fig:convergenceSpeed}, while on the degree $d=5$ the performance is comparable, on the Cyclic MultiChain data the Gaps sampler finds more likely trajectories dramatically faster.
\begin{figure}[h]
\begin{center}
          \includegraphics[width=0.6\textwidth]{Figures/section_2/pc_vs_d_speed.png}
          \vspace*{-3mm}
          \caption{Convergence of the placements parameters.  }
    \label{fig:convergenceSpeed}
\end{center}
\end{figure}

Figure \ref{fig:nonIgnknownN} presents the evaluation results for the non-ignorable case. In this experiment, for each $s$ we sample $\Psi(s) \sim Uniform[0.5 - \epsilon , 0.5 + \epsilon]$ for varying $\epsilon$ (X-axis), which control the variance of $\Psi(s)$. After omitting observations according to $\Psi(s)$, we provide it to the methods.
As the figure shows, the results are consistent with those of Figure \ref{fig:knownN} even for drastic variance in $\Psi(s)$. Previous works has demonstrated a decrease in performance when dealing with non-ignorable missing observations \cite{https://doi.org/10.48550/arxiv.2109.02770}, but our methods appear to be effective in this case, likely due to the use of Gibbs sampling which impute missing observations as an intermediate step.

Figure \ref{fig:imputation} evaluates the performance for when only limited knowledge of $\Psi$ is available. This experiment highlights a key advantages of sampling-based methods, which can infer missing values of $\Psi$.
Specifically, we randomly generate a single $\Psi(s) \sim Uniform[0.35 , 0.65]$, and then delete a proportion, $\epsilon$, of entries. The partial $\Psi(s)$ is then provided to the methods. In the "Sampling imputation" case (dashed lines), the missing entries are inferred using the sampling procedure described in section \ref{subsec:SamplingPsi}, while in the "Random imputation" case, they are filled in with random values.\\
As shown in Figure \ref{fig:imputation}, the imputation step leads to a significant improvement in performance. The Matching Sampler, which is provided with knowledge of $N$, demonstrates improved results across all benchmarks and levels of $\epsilon$. Additionally, the Gaps Sampler, which does not have access to this information, also shows notable improvement for some benchmarks. These results are noteworthy, as knowledge of $N$ is common in many applications of PHMMs, and obtaining prior knowledge about $\Psi$ can be costly or difficult to infer. Furthermore, the fact that the Gaps Sampler does not require knowledge of $N$, making this result even more impressive.

\subsection{Robustness under Misspecification}\label{sec:pcMiss}

\begin{figure*}[t]
\centering
\begin{subfigure}{0.45\textwidth}
  \includegraphics[width=\textwidth]{Figures/section_3/Wrong_PC.png}
  \caption{Wrong $\mathbf{p_c}$.}
    \label{fig:wrongPC}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{Figures/section_3/non_constent_PC.png}
    \caption{Non-constant $\mathbf{p_c}$.}
    \label{fig:robuNonConsPC}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
       \includegraphics[width=\textwidth]{Figures/section_3/non_iid_PC.png}
       \caption{$\mathbf{\Phi_{p_c}}$ with memory.}
    \label{fig:robuNoniisPC}
\end{subfigure}
\hfill
\caption{Robustness Under Misspecification.}
\label{fig:missps}
\end{figure*}

From a practical point of view, it is reasonable to assume that $\Psi$ will be known to the reconstruction algorithms only up to some error. Therefore in this section we evaluate the robustness of the algorithms under different misspecifications.
In view of the results of the previous section, in this section we focus on the "Gaps sampler". Additionally, we presents the results of the semi-analytical approach as a comparison to the ignorable case.

Figure \ref{fig:wrongPC} presents the case of ``wrong $p_c$''. Here we address the ignorable case where the $p_c$ estimate provided to the algorithm differs from the ground truth value $p_c = \frac{1}{2}$. As the figure shows, \textbf{both of the algorithms are relatively robust with respect to wrong $p_c$}. Moreover, note that even for sizable error in $p_c$ both algorithms are superior to the \emph{Naive} approach. In addition, the sampler based algorithm tends to be better then the semi analytical algorithm.

Figure \ref{fig:robuNonConsPC} presents the case of "non-constant" $p_c$, where there is no single constant ground truth $p_c$. Instead, for each sentence, $p_c$ is sampled independently from a normal distribution $p_c \sim N(\frac{1}{2},\sigma)$. In this case, the $p_c$ provided to the algorithm is the average one ($p_c = \frac{1}{2}$). Notice that the larger the $\sigma$, the noisier the data is, and hence the harder the problem. Never the less, the Gaps sampler give great results even for high values of  $\sigma$, outperforming the semi-analytic method in a more pronounced way.
Given that the case of non constant $p_c$ is the most realistic one, this result showcases the \textbf{benefit in sampling based methods}.

\textbf{Robustness under Observations Removal Process Misspecifications.}\label{ssec:noiid}
Previous sections assume that the omission process,$\Phi_{\Psi}$ (section \ref{ProblemSetup}), is modeled as a series of  independent Bernoulli trials. We now consider a case where instead the omission process is a Markov process with two states: $R_s="seen"$ and $R_m="missing"$, with transition probabilities given by
$P(R_s|R_s) = p_c + \epsilon$ (and hence  $P(R_m|R_s) = p_c - \epsilon$), and  $P(R_s|R_m) = (1-p_c) - \epsilon$ (and hence $P(R_m|R_m) = (1-p_c) + \epsilon$).
Thus, $\epsilon$ represents the bias of staying at the same state, with $\epsilon = 0$ case being equivalent to the i.i.d Brenoulli process above.
Figure \ref{fig:robuNoniisPC} compares the algorithms for different values of the bias $\epsilon$.
As the figure shows, the sampler results are mostly better then the ones of the \emph{Naive} solution, and constantly better then the semi-analytical ones. Nevertheless, the results of the \emph{Naive} solution improved as $\epsilon$ increases, and after some threshold of $\epsilon$, the \emph{Naive} algorithm become somewhat better then the sampler.
As discussed in Proposition \ref{MMM}, the \emph{Naive} reconstruction can be seen as the weighted mean of the ground-truth matrix T and wrong transition matrices $T^d$ derived from sequences of length d of omitted observations only. As $\epsilon$ increases, the probability of longer consecutive sequences increases (that is, long sequences of $P(R_s|R_s)$ or $P(R_m|R_m)$). But, while longer sequences of consecutive observed states ($R_s$ ) are beneficial to the $Naive$ solution, the solution is negatively effected by the number of omitted observations sequences, and not their length.

\textbf{Robustness under Ignorability Assumption Misspecifications.}
Figure \ref{fig:nonignorableMiss} examines the impact of considering the non-ignorable setup on the performance of the methods.
For this, for each $s$ we sample $\Psi(s) \sim Uniform[0.5 - \epsilon , 0.5 + \epsilon]$ for varying levels of $\epsilon$ (X-axis), and provide the algorithms with only the empirical $p_c$, or $N$ for the Matching Sampler (dashed lines). We also compare the results to those of the \emph{``ideal benchmark,"} which assumes fully observed missing locations but does not take into account the non-ignorability assumption. As the figure shows, while the performance is similar for low values of $\epsilon$, it becomes more pronounced for higher variance for both the Gaps Sampler and the \emph{``ideal benchmark."} This demonstrates the importance of considering the non-ignorability assumption, which is often overlooked in HMM reconstruction methods.
\begin{figure}[h]
\begin{center}
       \includegraphics[width = 0.55\textwidth]{Figures/section_3/ignorable_missp.png}
       \caption{Assume ignorable for non-ignorable $\Phi$. }
    \label{fig:nonignorableMiss}
\end{center}
\end{figure}

\section{Conclusion }
\label{sec:discussion}
This study addresses the challenge of reconstructing hidden Markov models from ordered trajectories with missing observations at unknown locations, which is a common issue in fields such as computational biology. A novel and general approach was proposed, which overcomes many of the limitations of existing methods and opens the way for new applications. Additionally, the approach addresses the non-ignorable missing observations setting, which is less commonly addressed in literature. Through evaluations, the robustness of the algorithms to different misspecifications was demonstrated. Two notable results were presented: 1) the reconstruction performances are comparable to those of known missing locations, even when accounting for the non-ignorable case, and 2) partially known omitting probabilities can be inferred from data. Further improvement is possible by incorporating prior knowledge of the latent process structure into the Bayesian framework and by using specialized omitting process tailored to specific problems.

\bibliographystyle{unsrtnat}

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