text:
Bodycam Footage of Phoenix Man's Death Reveals Disturbing Similiarites to George Floyd
Joe Jurado
PublishedJuly 9, 2020
Photo: Shutterstock (Shutterstock)
"I can't breathe" has depressingly become a common refrain as it pertains to police brutality.Eric Garner, George Floyd and countless men and women who have uttered these words before dying in police custody revealing that this is not some anomaly. And now, sadly, Muhammad Muhaymin Jr.'s is yet another name to be added to that list.
CNN reports that newly released bodycam footage shows Muhaymin repeatedly telling officers that he couldn't breathe while they detained him. On January 27, 2017, Phoenix Police responded to a call from a community center Muhaymin frequented in Maryvale, Ariz. He wanted to use the restroom in the center but they wouldn't allow him to go in with his emotional support dog. After Muhaymin argued with the manager, an employee was ordered to call 911. Initially, when the officers arrived they allowed Muhaymin to use the restroom with his dog but a background check ran on Muhaymin while he was in there showed he had an outstanding warrant for misdemeanor possession of a marijuana pipe.
From CNN:
When Muhaymin emerged from the restroom, the situation swiftly spiraled out of control. Officers tried to arrest him, and ultimately Muhaymin wound up facedown on the pavement just outside the facility with at least four officers on top of him struggling to put his hands in cuffs. Within eight minutes, Muhaymin was dead. His dog has not been seen since.
The Muhaymin family is suing the city for $10 million in a wrongful death suit that names 10 Phoenix police officers as defendants and is expected to go to jury trial early next year.
"In general, sending out armed police officers with guns to deal with somebody who's having a mental breakdown — it's a recipe for disaster," said the family's attorney, David Chami.
"We have heard the mayor of Phoenix and the police chief of Phoenix police talk about what happened in Minneapolis, say that they are ashamed, condemn the people involved as if it were something separate from us. They never acknowledge that what happened to George Floyd, their police officers did here, to Muhammad Muhaymin," said Viri Hernandez, executive director of Poder in Action, a Phoenix advocacy group told CNN.
Oswald Grenier, Jason Hobel, Ronaldo Canilao and David Head, the four officers involved, were not subject to any disciplinary action for Muhaymin's death and all four are still active on the force. The Phoenix Police force is one of the largest police forces in the country without civilian oversight, though according to AZ Central, Phoenix City Council approved almost $3 billion to go towards the formation of an oversight board last month. Phoenix PD conducted its own investigation of Muhaymin's death and found there was no wrongdoing. "The department is investigating itself here. There's an incentive to find no wrongdoing," Chami told CNN.
Muhaymin was intermittently homeless and suffered from schizophrenia and anxiety according to documents obtained by CNN. The officers can even be heard referencing Muhaymin's mental health saying, "he is a little 918," which is police code for mentally ill.
Despite Muhaymin not being physically violent during the encounter, Phoenix police nonetheless said that Muhaymin had assaulted an officer and an employee of the community center. "When did they ever retract those statements? They never did." Chami said of the police.
An autopsy conducted by Maricopa County medical examiner, Dr. Amanda Maskovyak labeled Muhaymin's death a homicide and said the primary cause of death was cardiac arrest, intensified by "coronary artery disease, psychiatric disease, acute methamphetamine intoxication and physical exertion during law enforcement subdual." Muhaymin's family brought on Dr. Bennet Omalu as an expert witness who disagrees with Maskovyak's assessment.
"Asphyxiation due to compression of his trunk and body" — not underlying conditions or drug use — was the cause of death, said Omalu, a forensic pathologist known for his research on chronic traumatic encephalopathy (CTE) injuries in football players, and who was portrayed by Will Smith in the 2015 film "Concussion."
By the time Muhaymin vomited, his body had been compressed under the weight of the officers for eight minutes, Omalu wrote in a report for the family's attorney. This "high-scale traumatic stressor" caused Muhaymin to vomit, which entered and blocked his air passageways, cutting off oxygen to his brain and leading to "sudden death," Omalu's report says.
The report, exclusively obtained by CNN, concludes: "If Muhammad did not encounter the police on Jan. 4, 2017, he would not have died."
The release of the body camera footage comes as Phoenix police are under increased pressure from protesters to release body camera footage from the shooting death of James Garcia by Phoenix Police officers on July 4th. In May, protests were held in the city when Dion Johnson died after being shot by a state trooper. In 2018, a report from the National Police Foundation showed that Phoenix Police led the nation in officer involved shooting, with 23 of the 44 incidents reported that year resulting in the victim's death.
Chami has said that, similarly to Colorado Governor Jared Polis reopening the investigation into Elijah McClain's death, he would like Muhaymin's death to be re-examined as well. Mussallina Muhaymin, Muhammad's sister, has expressed a desire for better police oversight in Phoenix.
"The police was judge, jury and executioner. And just because he wanted to use a public bathroom." Mussallina told CNN.
NewsLaw EnforcementPolice Brutality


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Nuclear Threat to Tokyo Worse Than Thought
Bad as it was, the Fukushima nuclear power plant crisis could have been much worse. According to a 15- page report delivered to then-Prime Minister Naoto Kan on March 25, two weeks after the massive earthquake and resulting tsunami in northeast Japan, the resulting meltdowns could have forced the evacuation of Tokyo.
The report was made public in early January by Goshi Hosono, Japan's minister for Fukushima recovery. It indicated that the government had drawn up a contingency plan that envisioned the possibility of having to evacuate much of Tokyo, the world's largest city, and everyone else within a 250-km radius of the damaged Fukushima Daiichi nuclear plant.
The plan was readied after the Mar. 11, 2011 earthquake and tsunami set in motion circumstances leading to fuel meltdown at three of the reactors and fears of exposed radioactive fuel stored at a fourth unit. It was prepared by Shunsuke Kondo, chairman of the Japan Atomic Energy Commission. Hosono, then a special advisor to premier Naoto Kan, asked for the plan at Kan's request.
The contingency plan envisioned a 170-km-radius mandatory evacuation zone from the stricken plant and a 250-km voluntary zone which would have brought the disaster to Tokyo's doorstep. The report said that high radiation levels might have extended beyond 250 km, and people beyond that limit might have to evacuate.
In fact, given the frantic effort to limit the radiation spewing from the plant, the government was able to hold fast to the evacuation regime imposed in the early days. It ordered full evacuation of only an area 20 km around the plant site and voluntary evacuation up to 30 km. Nonetheless, the report is a graphic indication of the potential for disaster if a nuclear facility ever goes completely out of control in a heavily populated urban area. The greater Tokyo area comprises nearly 40 million people. The idea of evacuating a major portion of that number was simply unthinkable.
Tokyo is about 250-300 km away from the plant site (depending on how one defines "Tokyo"). The government maintained in public then that Tokyo was safe and urged people and businesses to stay. Nevertheless, several foreign governments, led by France, advised their citizens to leave Tokyo. More than 30 legations moved temporarily to Osaka, and many other expatriates also left on their own accord. Japanese called them "fly-jin" a play on the word for foreigner, gaijin. The U.S. did not go so far as to move its embassy but announced a 50-mile, 80-km evacuation zone for Americans.
Hosono said in January that the report submitted on March 25 was not revealed to the public because of concern about spreading panic. Such a report if made public would have caused "excessive and unnecessary worry," Hosono said. But Kan later talked freely about his fears for Japan in interviews he made immediately after he resigned in September.
"It was a crucial moment when I wasn't sure whether Japan could function as a state," he said. Since departing office, Kan has reinvented himself as a kind of global anti-nuclear power activist. "I think we should create a world where people do not need to depend on nuclear power," he said while attending the World Economic Forum in Davos, Switzerland. He also inspected alternative energy providers in Spain and Germany.
There were several points in the early days of the crisis when things could have really spun out of control. One scenario envisioned the melting of the 1,535 fuel assemblies contained in the spent-fuel pool at Unit-4. That is the equivalent of two full reactor cores as Unit-4 was shut down for refueling when the earthquake struck. The roof had been blown off in a hydrogen explosion leaving no barrier between the pool and the atmosphere.
Another imponderable in the immediate wake of the tsunami was whether workers could have opened the vents on Unit-1 to relieve steam pressure building up. That meant releasing some radiation into the atmosphere. But to not vent steam to relieve pressure buildup might have led to further explosions, severely cracking the containment and resulting in uncontrolled released of radiation.
It was no simple matter to open the vent on Unit 1.The workers were hampered by high radiation dose rates, lack of lighting, lack of contingency plans and equipment needed to vent in the event of power blackouts and indecision by the utility head office. The vents were successfully opened about 24 hours after the earthquake struck. By then much of the fuel was exposed and melting.
The government and utility feared more hydrogen explosions and more release of radiation might cause the workers to flee and suspend recovery efforts entirely. In one worst case, the authorities estimated that the areas within the 170 km mandatory zone would have been contaminated with 1,480 kilo-Becquerels of radiation per square meter. As a last resort, the government would have smothered the damaged reactors with tons of sand dropped from aircraft or helicopters.
Eventually, after much trouble, the vents were opened, relieving pressure. It was too late to prevent a hydrogen explosion caused by the chemical reaction of steam and exposed fuel, but the damage was mainly confined to the building superstructure and not to the containment vessel itself.
There was much initial anxiety over the condition of the pools in all four plants, especially Unit-4, sparking frantic water drops by helicopters and then the use of ground-based water sprayers to keep the fuel assemblies under water. Later it was evident that the fears were exaggerated. The pools were mostly undamaged and cooling is now working properly.
However, it is probably no coincidence that the new 40-year road map to recovery, announced in mid-December, places the priority on removing fuel from the Unit-4 pool, well ahead of dealing with the "fuel debris" or melted fuel, an operation that won't even begin for another decade, according to the latest planning document.
Although the most serious worries have abated, the severely damaged reactors remain in a precarious state officially described as the "equivalent of a cold shutdown." It means that the remains of the fuel are below, in most cases well below, 100 degrees C. But it is dependent upon a jury-rigged cooling system, which is suffering from leaking frozen pipes in the cold weather that has gripped Japan.

redpajama_set_name: RedPajamaCommonCrawl


Yesterday, there was a buyer of 17,000 May 17 $123 puts of the iShares 20+ Year Treasury Bond ETF (TLT). This is a pretty aggressive bearish trade and would certainly imply a contrarian view of duration risk (short it rather than be long it). From our perspective, the risk/reward potential of May 31 Puts looks really interesting. TLT currently has a "SELL" rating according to our Quantamize models.


Our garden produces peppers and squash through the fall or until the first freeze, which ever comes first — this year it was the freeze which happened last week. Meanwhile soil vegetables, such as onions and carrots, are less affected by late fall or early winter freeze, and they continue to be available from our garden through early and sometimes late winter. We also have herbs growing year round.
Of course living near a large body of water helps to make the local climate more temperate, so temperatures tend to be a bit warmer than other parts of the country at our latitude. For example, it was rather warm today at 11°C (52°F). This all combines to increase the growing season and extends the harvest period. Making getting the vegetables for a fresh garden soup in December for us as simple as a little digging in the garden, washing and cooking.


Round stainless steel lunch box consisting of two tiers with clip closure.
Made responsibly in India from high quality 201 food grade stainless steel. Free from plastic, BPA, phthalates and lead.
Overall Dimensions - 8cm x 13cm dia. 800ml capacity.
Top Layer - 5cm x 13cm dia. 500ml capacity.
Bottom Layer - 4.5cm x 13cm dia. 300ml capacity.


text:
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eMedical Academy
CCEeXAM Webinar
Achikam (Achi) Oren-Grinberg, MD, MS
Gabriele Via, MD
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Marilyn Riley, RDCS
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James Chang, M.D.
Adam Lerner, MD.
Webinar Demo
Jordan Strom, MD.
Staff Cardiologist
Beth Israel Deaconess Medical Center (BIDMC)
Dr. Jordan Strom is Instructor in Medicine at Harvard Medical School and a non-invasive cardiologist, specializing in echocardiography and cardiovascular magnetic resonance imaging at Beth Israel Deaconess Medical Center. He received his B.S. in Biology from Yale University and an M.D. from Harvard Medical School, before subsequently training in Internal Medicine at the Massachusetts General Hospital and in Cardiovascular Medicine at Beth Israel Deaconess Medical Center where he was Chief Fellow for the 2017-2018 year. He received additional post-graduate training in Clinical Epidemiology and Biostatistics at the Harvard T.H. Chan School of Public Health, receiving an M.Sc. in Epidemiology in May 2018
Dr. Strom was the inaugural fellow and is currently a junior faculty investigator at the Richard A. and Susan F. Smith Center for Outcomes Research in Cardiology. His research involves evaluation of the relationship of cardiac structure and function to health outcomes. In particular, he specializes in the use of large administrative, registry, and clinical trial data to develop tools to help clinicians assess and communicate the risk and benefit of interventions across a variety of cardiovascular diseases, particularly valvular heart disease. Additionally, his research involves evaluating the impact of public policies and structures of care on health outcomes and the comparative effectiveness of different treatment strategies. He has received grant funding for this work from the American Heart Association and the National Institutes of Health and his publications have been featured in such journals as JACC: Interventions, European Heart Journal, and the Annals of Internal Medicine. He is an upcoming member of the Editorial Board of the Journal of the American Society of Echocardiography and the ACC representative to the Board of Directors of the Joint Review Committee on Education in Diagnostic Medical Sonography. Additionally, he is a member of the inaugural class of the American Society of Echocardiography's Leadership Academy, the ASE Twitter Journal Club Task Force, and the ImageGuideEcho Research and Publications Committee
His methodological expertise spans epidemiologic and biostatistical methods, including advanced statistical regression, longitudinal analysis, supervised and unsupervised machine learning techniques, survival analysis, decision analysis, and advanced prediction modeling. He is also interested in using translational techniques such as molecular phenotyping with metabolite and proteomic profiling to discriminate risk and benefit among cardiovascular disease patients. He has expertise in prediction of cardiac disease states, assessment of treatment effect heterogeneity, comparative effectiveness research, large database analysis, epidemiologic study design, and use of validated patient-reported outcome measures.
English (US) ›


Abstract: The aim of this paper is to discuss three main critical challenges which research and policymaking in the feld of collective youth violence in urban contexts face today. This paper argues that we need to shift the focus of research in this area from "problematic" youth to the study of the ways in which violence permeates daily lives and becomes normalised through specifc local social and political conditions. The paper then suggests that, in light of recent theory and empirical research, the relationship between violence and poverty should be re-evaluated. Additionally, and in order to properly address the causes of youth collective violence, this paper argues for a change of focus in the analysis of youth violent mobilisation. The suggested focus rests on the appeal of the symbolic revenues that mark the search for a valued social status and possibilities in contexts of adversity and violence. In fact, symbolic factors associated with the involvement in drugs traffcking and other violent activities and with youths' contact with frearms are key factors, namely the search for status, power and respect, and attracting recognition from their male and female peers. The adrenaline and danger which youth experience through these activities are highly connected with gender constructions. Finally, this paper supports the progressively accepted evidence in favour of an urgent shift in how to address and prevent youth violence, claiming that repressive policies have hitherto failed to contain violence and to contribute to improving the formulation of preventive policies. This paper is based on data collection and analysis as well as reports from several studies conducted since 2006 in Rio de Janeiro (Brazil), San Salvador (El Salvador), Praia (Cape Verde) and Bissau (Guinea-Bissau).


Hi, I am interested in the COUNTRY LIVING AT ITS BEST NEXT TO EYE OF AFRICA.
This large modern family home on 1.2 hectares of prime land is situated in a boomed off area next to Eye of Africa. It offers 5 bedrooms (mes), 2 bathrooms, an open plan living area fully tiled with a granite kitchen, scullery, dining room, 2 lounges leading onto an undercover lapa and a sparkling pool with beautiful views of the valley. Established gardens with a pond and trampoline, double garage and domestic quarters. Outbuildings includes 6 bricked enclosed chicken coops, 2 horse stables and 2 workshops. Very secure.A definite on your viewing list.
STUNNING HOME WITH LOTS OF FEATURES!!!
START YOUR LIFE OF LUXURY AND TRANQUILITY IN THIS WATERFRONT SPLENDOR!
Massive Family home for sale in Raslouw!!


Heralded by reseller Digitavia as a "revolution for the meeting environment", ActivConnect is a wireless connection and videoconferencing device launching at ISE 2016.
ActivConnect is designed to connect any smartphone, phablet, tablet, PC or Mac across multiple networks, from corporate to guest networks, and to any display, along with providing full video conferencing capabilities.
Digitavia says the ActivConnect will also seamlessly connect any device wirelessly to a display; once in a video call users can then share wirelessly connected content in the call.


text:
Q: how to remove all comments from a xml file least the ones containing the "HELP" word? I need to know the xml matching code to let me clean a file of all comments least the ones containing the HELP word.
I am currently using perl to clean multiline comments (like failed tests) from a xml, least the ones I hand picked and are helpful to the end user:
perl -i -w -0777pe 's/<!--(.(?<!(HELP|TODO)))*?-->//sg' somefile.xml
But, If there is a way to obtain the same result using a xml mathing string, I would prefer as there may have some exception that regex may not handle, but for now this is what I have to use.
Obs.: I will use it with xmlstarlet on linux, so it would be better if the solution work with it too.

A: Here is XSLT based solution.
The XSLT is very simple. One single line template removes not needed comments. The rest is just a boilerplate code for a so called Identity Transform pattern.
$ xml tr path/to/transformation.xslt path/to/source.xml

Input XML
<?xml version="1.0"?>
<root>
    <!--The Sunshine State-->
    <state>FL</state>
    <!--HELP is needed-->
    <city>Miami</city>
</root>

XSLT
<?xml version="1.0"?>
<xsl:stylesheet version="1.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
    <xsl:output method="xml" encoding="utf-8" indent="yes" omit-xml-declaration="no"/>
    <xsl:strip-space elements="*"/>

    <xsl:template match="@*|node()">
        <xsl:copy>
            <xsl:apply-templates select="@*|node()"/>
        </xsl:copy>
    </xsl:template>

    <!--keeps comment if it contains HELP-->
    <xsl:template match="comment()[not(contains(., 'HELP'))]" />
</xsl:stylesheet>

xmlstarlet
xmlstarlet ed -d '//comment()[not(contains(.,"HELP"))]' path/to/source.xml
Output
<?xml version='1.0' encoding='utf-8' ?>
<root>
  <state>FL</state>
  <!--HELP is needed-->
  <city>Miami</city>
</root>

redpajama_set_name: RedPajamaStackExchange


Temple Street Eatery is your local neighborhood dumpling and noodle bar. Since 1966, our family at Christina Wan's Mandarin House brought to South Florida original Mandarin Cantonese Szechuan cuisine. Temple Street Eatery is our new concept aiming to bring new flavors to Fort Lauderdale's palate.


Got mine from Timewalker today.
Pretty happy with it. It does smell. It's not overwhelming though. Not really sticky or tacky.
Kinda silly to have a giant drill plugged onto a ball joint, it's rather floppy.
What kind of batteries does this thing need? I didn't see any instructions anywhere.
I had battery size LR44 laying around and those seem to be the correct size.
Entertainment Earth had this set on sale for $190 today...anyone get in on that sale before it sold out?
Mother****er. Really? I would have bit at that.
That's lame. No one saw it and posted here while it was up?
Finally went in on this tonight with a gift card and promo code, and will likely pre-order subject delta next.
Really hoping Threezero continues the Bioshock line, as well as delves back into Dead Space eventually. The designs from the first 2 Dead Space games are just too cool not to become 1/6 figures.
The figure is full of fumes just as many here had reported... My figure has sticky paint all over it, which would account for the fume smells. It also rubs off just to the touch in many places. Some paint came off on the protective wrapping as pictured here. I am going to let it sit out for a couple of days, but I may actually consider returning this.
Mine has been open for a month or two now. I never experienced these "fumes" ... sure it smelled kinda funky at first, but now my figure has no aroma.
Of course, I'm one of those people that has never experienced rubber bodies splitting or leather jackets cracking. My Batman never had oily abs.
Decided to give this figure another go, this is now my third one. Previous two had issues ranging from broken parts, and terrible glue application, some even on the cloth portion. The one I have currently is "close" to acceptable but still not 100%, and is again due to glue application. I included two pics where the "face" of the figure meets the body on one side it is correct and flush, the other side though was not assembled properly but they glued it anyways and now the face sits crooked on the body. Head on you can't really tell anything but for nearly $300 it is really frustrating to even have to worry about these types of issues.


ISG is a leading provider of Large Format Graphics operating out of the beautiful Interior Services building in Boca Raton.
We specialize in corporate branding, vehicle wraps and all aspects of large format imaging for promotional purposes. We can blind ship one posterboard to your client or ship a hundred banners to locations nationwide…and everything in between.
Since 1990 we have been supplying short to medium run posters, banners, vehicle graphics and a lot of custom signage to many of the areas most prominent print companies, ad agencies and the marketing/creative departments of businesses large and small.
ISG is operated by John Bonani, a seasoned professional with over twenty-five years of executive management and production experience in the large format imaging industry.
We have no customers… only partners. We have relationships going back over twenty years in part because we integrate with our clients to such a degree that many look to us as an extension of their own operation, their large format division if you will.
I would like to personally thank you for taking a few minutes to consider us as your source for all your large format printing needs. I'm often asked what makes us different from other companies offering similar products and services. Three things come to mind... experience, passion and an almost absurd attention to detail.
I have been in this industry since the early 90's so my experience is extensive.
My staff and I are passionate. We love what we do and consider every project to be the most important job we've ever done.
Finally I truly believe our attention to detail, response time and level of customer service is second to none. But most importantly, we put YOU and your clients first!
We're extremely proud of our reputation and take it quite seriously. We would be honored for the opportunity to earn your trust and welcome you to our family.


text:
Darlington County School District » About Us » Latest News » Rotary Club of Hartsville purchases new shoes for 100 DCSD students
HARTSVILLE, S.C. – For the second year in a row, the Rotary Club of Hartsville recently hosted Darlington County School District students at the Shoe Show and purchased each student a brand new pair of shoes – 100 pairs to be exact.
That's far more than the Rotary Club could fund last year. The two-day "Happy Feet" event was certainly welcomed by the students, who came from several schools in the Hartsville area. The partnership with the Shoe Show is also in its second year.
Charles Saverance, president of the Rotary Club of Hartsville, said the group's members thoroughly enjoy working to help the community.
"The Rotary Club of Hartsville adopted 'Happy Feet' as our Signature Project in 2018," Saverance said. "Seeing children take off their shoes, which may be a couple of sizes too small or that may have holes in them, and watching them walk out of the store with a brand new pair makes it all worthwhile. Sometimes this is the first new pair of shoes they have ever owned. Rotary Club exists to do good in the world, and that always starts at home."
Student volunteers from Hartsville High School (HHS), who are members of Ready 2 Interact (R2I), also helped out with the event. R2I is the Rotary Club's branch organization at the high school. R2I members guided the younger students around to try on different pairs of shoes and helped them check out.
Cierra Hillian, president of R2I and a senior at HHS, said helping other Darlington County School District students proved rewarding.
"Happy Feet is such a wonderful opportunity that R2I looks forward to each year," Hillian said. "It is such a fun and heartwarming chance to be able to help children find a pair of shoes. For many, it is just a simple part of life to get a new pair of shoes, but to other it can mean the world. We are so thankful to be a part of that."


text:
Q: Centos8 image creation through packer on azure results in to errors Can somebody tell me what I am doing wrong here as I am unable to create the image? actually the commands, when I run on my host Centos machine, works fine but when I execute the same commands through packer result into error.
JSON file:
{
  "builders": [{
    "type": "azure-arm",

    "client_id": "{{user `azure-client-id`}}",
    "client_secret": "{{user `azure-client-secret`}}",
    "tenant_id": "{{user `azure-tenant-id`}}",
    "subscription_id": "{{user `azure-subscription-id`}}",

    "managed_image_resource_group_name": "{{user `azure-resource-group`}}",
    "managed_image_name": "CentOS8-Packer",

    "os_type": "Linux",
    "image_publisher": "OpenLogic",
    "image_offer": "CentOS",
    "image_sku": "8_2-gen2",

    "location": "{{user `azure-region`}}",
    "vm_size": "{{user `vm-size`}}"
  }],

  "provisioners": [
    {
      "execute_command": "chmod +x {{ .Path }}; {{ .Vars }} sudo -E sh '{{ .Path }}'",
      "script":"ami-script.sh"
      "type": "shell"
    }
  ]
}

ami-script.sh content:
#!/bin/bash -e
dnf update -y
dnf install terminator xrdp -y

Error:
Updated part:
dnf update -y error:

dnf install terminator xrdp error (resolved)


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text:
Micoud – dystrykt w Saint Lucia. Stolicą dystryktu jest Micoud. Dystrykt zajmuje powierzchnię 78 km² i jest jednym z 11 dystryktów wchodzących w skład państwa Saint Lucia. Liczba ludności to 17 153, a gęstość zaludnienia wynosi 219,9 osób/km².

Podział administracyjny Saint Lucia

redpajama_set_name: RedPajamaWikipedia


Rachel did my cut and highlights exactly how I wanted them!
A wonderful first time experience at That French Salon. The moment you enter you are greeted with genuine warmth and friendliness. Rachel was my stylist and created a look that I love . She did highlights, cut and style explaining the products she was using. Everyone went above and beyond to make me feel comfortable and relaxed. Thank you Brigitte , Tony and especially Rachel. I shall return !
Brigitte has been cutting my hair for many years. I have thick, coarse hair. Brigitte is a stylist who can control it with a great cut and she keeps me "updated." The salon is friendly and welcoming. Both my husband and son have been clients of hers as well because of the wonderful haircuts.
That French Salon is fabulous! My stylist always knows exactly what I want and makes me feel amazing. I love my style because it is so versatile - just look at one haircut styled 3 different ways.
I've been coming to That French Salon since 1994 and I love it! The staff is professional, the atmosphere is comfortable, and of course, my hair looks great! Brigitte listens to what I want and makes suggestions as to how I can achieve it. I also REALLY like their high quality wigs!


Kendall Jenner looked casual chic while leaving the Kanye West's church service in Los Angeles. Kendall Jenner rocked straight fit white denim high rise ankle length jeans with side pockets. She paired it with white basic tank top. She finished her look with black lace up sneakers.


text:
\section{Introduction}
Clustering aims to partition data $y_1,\ldots, y_n$ into disjoint groups. There is a large literature ranging from various algorithms such as K-means and DBSCAN \citep{macqueen1967classification,10.5555/3001460.3001507,frey2007clustering} to mixture model-based approaches [reviewed by \cite{fraley2002model}]. In the Bayesian community, model-based approaches are especially popular. To roughly summarize the idea, we view each $y_i$ as generated from a distribution $\mathcal K(\cdot\mid \theta_i)$, where the parameters $(\theta_1,\ldots,\theta_n)$ are drawn from a discrete distribution $ \sum_{k=1}^K w_k \delta_{\theta^*_k}(\cdot)$, with $w_k$ as the probability weight, and $\delta_{\theta^*_k}$ as a point mass at $\theta^*_k$. With prior distributions, we could estimate all the unknown parameters  ($\theta^*_k$'s, $w_k$'s, and $K$) from the posterior.

The model-based clustering has two important advantages. First, it allows important uncertainty quantification such as the probability for cluster assignment $c_i$, $\text{Pr}(c_i=k\mid y_i)$, as a probabilistic estimate that $y_i$ comes from the $k$th cluster ($c_i=k \Leftrightarrow \theta_i =\theta^*_k$). Different from commonly seen asymptotic results in statistical estimation, the clustering uncertainty does not always vanish even as $n\to \infty$. For example, in a two-component Gaussian mixture model with equal covariance, for a point $y_i$ at nearly equal distances to two cluster centers, we would have both $\text{Pr}(c_i=1\mid y_i)$ and $\text{Pr}(c_i=2\mid y_i)$ close to $50\%$ even as $n\to \infty$. For a recent discussion on this topic as well as how to quantify the partition uncertainty, see \cite{wade2018bayesian} and the references within. Second, the model-based clustering can be easily extended to handle more complicated modeling tasks. Specifically, since there is a generative process associated with the clustering, it is straightforward to modify it to include useful dependency structures. We list a few examples from a rich literature: \cite{ng2006mixture} used a mixture model with random-effects to cluster correlated gene-expression data,  \cite{muller2010random,park2010bayesian,ren2011logistic} allowed the partition to be varying according to some covariates, \cite{guha2016nonparametric} proposed to simultaneously cluster the predictors and use them in a high-dimensional regression.

On the other hand, model-based clustering has its limitations. Primarily, one needs to carefully specify the density/mass function $\mathcal K$, otherwise, it will lead to unwanted results and difficult interpretation. For example, \cite{coretto2016robust} demonstrated the sensitivity of the Gaussian mixture model to non-Gaussian contaminants, \cite{miller2018robust} and \cite{cai2021finite} showed that when the distribution family of $\mathcal K$ is misspecified, the number of clusters would be severely overestimated. It is natural to think of using more flexible parameterization for $\mathcal K$, in order to mitigate the risk of model misspecification. This has motivated many interesting works, such as modeling $\mathcal K$ via skewed distribution \citep{fruhwirth2010bayesian,lee2016finite}, unimodal distribution \citep{rodriguez2014univariate}, copula \citep{kosmidis2016model}, mixture of mixtures \citep{malsiner2017identifying}, among others. Nevertheless, as the flexibility of $\mathcal K$ increases, the modeling and computational burdens also increase dramatically.

In parallel to the above advancements in model-based clustering, spectral clustering has become very popular in machine learning and statistics.  \cite{von2007tutorial} provided a useful tutorial on the algorithms and a review of recent works. On clustering point estimation, spectral clustering has shown good empirical performance for separating non-Gaussian and/or manifold data, without the need to directly specify the distribution for each cluster. Instead, one calculates a matrix of similarity scores between each pair of data, then uses a simple algorithm to find a partition that approximately minimizes the total loss of similarity scores across clusters (adjusted with respect to cluster sizes). This point estimate is found to be not very sensitive to the choice of similarity score, and empirical solutions have been proposed for tuning the similarity and choosing the number of clusters \citep{zelnik2005self,shi2009data}. There is a rapidly growing literature of frequentist methods on further improving the point estimate [\cite{chi2007evolutionary,rohe2011spectral,NIPS2011_31839b03,lei2015consistency,han2021eigen,lei2022bias}; among others], although, in this article, we focus on the Bayesian perspective and aim to characterize the generative distribution.

Due to the algorithmic nature, spectral clustering cannot be directly used in model-based extension, or produce uncertainty quantification. This has motivated a large Bayesian literature. There have been several works trying to quantify the uncertainty around the spectral clustering point estimate. For example, since the spectral clustering algorithm can be used to estimate the community memberships in a stochastic block model, one could transform the data into a similarity matrix, then treat it as if generated from a Bayesian stochastic block model \citep{snijders1997estimation,nowicki2001estimation,mcdaid2013improved,geng2019probabilistic}. Similarly, one could take the Laplacian matrix (a transform of the similarity used in spectral clustering) or its spectral decomposition, and model it in a probabilistic framework \citep{socher2011spectral,duan2019spiked}.

Broadly speaking, we can view these works as following the recent trend of robust Bayesian methodology, in conditioning the parameter of interest (clustering) on an insufficient statistic (pairwise summary statistics) of the data. See \cite{lewis2021bayesian} for recent discussions.
Pertaining to Bayesian robust clustering, one gains model robustness by avoiding putting any parametric assumption on within-cluster distribution $\mathcal K(\cdot \mid \theta^*_k)$; instead, one models the pairwise information that often has an arguably simple distribution. Recent works include the distance-based P\'olya urn process
\citep{blei2011distance,socher2011spectral},
Dirichlet process mixture model on Laplacian eigenmaps \citep{banerjee2015bayesian},
Bayesian distance clustering \citep{duan2021dist},
generalized Bayes extension of product partition model
\citep{rigon2020generalized}.

This article follows this trend. Instead of modeling $y_i$'s as conditionally independent (or jointly dependent) from a certain within-cluster distribution $\mathcal K(\cdot \mid \theta_k^*)$, we choose to model $y_i$ as dependent on another point $y_j$ that is close by, provided $y_i$ and $y_j$ are from the same cluster. This leads to a Markov graphical model based on a spanning forest, a graph consisting of multiple disjoint spanning trees (each tree as a connected subgraph without cycles). The spanning forest itself is not new to statistics. There has been a large literature on using spanning trees and forests for graph estimation, such as \cite{meila2000learning,meilua2006tractable,edwards2010selecting,byrne2015structural,duan2021bayesian,luo2021bayesian}. Nevertheless, a key difference between graph estimation and graph-based clustering is that --- the former aims to recover both the node partition and the edges characterizing dependencies, while the latter only focuses on estimating the node partition alone (equivalent to clustering). Therefore, a distinction of our study is that we will treat the edges as a nuisance parameter/latent variable, while we will characterize the node partition in the marginal distribution.

Importantly, we formally show that by marginalizing the randomness of edges, the point estimate on the node partition is provably close to the one from the normalized spectral clustering algorithm. As the result, the spanning forest model can serve as the generative model for the spectral clustering algorithm --- this relationship is analogous to the one between the Gaussian mixture model and the K-means algorithm \citep{macqueen1967classification}. Further, we show that treating the spanning forest as random, as opposed to a  fixed parameter (that is unknown), leads to much less sensitivity in clustering performance, compared to cutting the minimum spanning tree algorithm \citep{gower1969minimum}.

On the prior specification on the node and edges, we take a Bayesian non-parametric approach by considering the forest model as realized from a ``forest process'' --- each cluster is initiated with a  point from a root distribution, then gradually grown with new points from a leaf distribution. We characterize the key differences in the partition distribution between the forest and classic P\'olya urn processes. This difference also reveals that extra care should be exerted during prior specification, when using graphical models for clustering.

Lastly, by establishing the generative model counterpart for spectral clustering, we show how such models can be easily extended to incorporate other dependency structures. We demonstrate several extensions, including a multi-subject clustering of the brain networks, and a high-dimensional clustering of photo images.

\section{Method}

\subsection{Background on Spectral Clustering Algorithms}

We first provide a brief review of spectral clustering algorithms. For data $y_1,\ldots, y_n$, let $A_{i,j}\ge 0$ be a similarity score between $y_i$ and $y_j$, and denote the degree $D_{i,i}= \sum_{j\neq i} A_{i,j}$. To partition the data index $(1,\ldots, n)$ into $K$ sets, $\mathcal V= (V_1,\ldots, V_K)$, we want to solve the following problem:
\begin{equation\label{eq:normalized_graph_cut}
\min_{\mathcal V} \sum_{k=1}^K \frac{\sum_{i\in V_k, j \not \in V_k} A_{i,j}}{
\sum_{i\in V_k} D_{i,i} }.
 \end{aligned
This is known as the minimum normalized cut loss. The numerator above represents the across-cluster similarity due to cutting $V_k$ off from the others; and the denominator prevents trivial solution of forming tiny clusters with small $ \sum_{i\in V_k} D_{i,i}$.

This optimization problem is a combinatorial problem, hence has motivated approximate solutions such as spectral clustering. To start, using the Laplacian matrix $L=D-A$ with $D$ the diagonal matrix of $D_{i,i}$'s, and the normalized Laplacian $N=D^{-1/2} L D^{-1/2}$, we can equivalently solve the above problem via:
\begin{equation*
\min_{\mathcal V}  \text{tr} (Z'_{\mathcal V} N Z_{\mathcal V}),
 \end{aligned
where $Z_{{\mathcal V}: i,k} =   1(i\in V_k) \sqrt{D_{i,i}} / \sqrt{\sum_{i\in V_k}D_{i,i}}$. It is not hard to verify that $Z'_{\mathcal V}Z_{\mathcal V}=I_K$. We can obtain a relaxed minimizer of $Z:Z'Z=I_K$, by simply taking $\hat Z$ as the bottom $K$ eigenvectors of $N$ (with the minimum loss equal to the sum of the smallest $K$ eigenvalues). Afterward, we cluster the rows of $\hat Z$ into $K$ groups (using algorithms such as the K-means), hence producing an approximate solution to \eqref{eq:normalized_graph_cut}.

To clarify, there is more than one version of the spectral clustering algorithms. An alternative version to \eqref{eq:normalized_graph_cut} is called ``minimum ratio cut'', which replaces the denominator $\sum_{i\in V_k} D_{i,i} $ by the size of cluster $|V_k|$. Similarly, continuous relaxation approximation can be obtained by following the same procedures above, except for clustering the eigenvectors of the unnormalized $L$. Details on comparing those two versions can be found in \cite{von2007tutorial}. In this article, we focus on the one based on \eqref{eq:normalized_graph_cut} and the normalized Laplacian matrix $N$. This version is also commonly referred to as ``normalized spectral clustering''.

\subsection{Generative Model via Bayesian Spanning Forest}
The next question is if there is some partition-based generative model for $y$, that has the maximum likelihood estimate (or, the posterior mode in the Bayesian framework) almost the same as the point estimate from the normalized spectral clustering.

We found an almost equivalence in the spanning forest model. A spanning forest model is a special Bayesian network that describes the conditional dependencies among $y_1,\ldots, y_n$. Given a partition $\mathcal V=(V_1,\ldots, V_K)$ of the data index $(1,\ldots,n)$, consider a forest graph $\mathcal F_{\mathcal V}= (T_1,\ldots, T_k)$, with each $T_k=(V_k, E_k)$ a component tree (a connected subgraph without cycles), $V_k$ the set of nodes and $E_k$ the set of edges among $V_k$. Using $\mathcal F_{\mathcal V}$ and a set of root nodes $\mathcal R_{\mathcal V}=(1^*,\ldots, K^*)$ with $k^*\in V_k$, we can form a graphical model with likelihood:
\begin{equation\label{eq:dag_likelihood}
\mathcal L(y ; \mathcal V, \mathcal F_{\mathcal V}, \mathcal R_{\mathcal V}, \theta) =  \prod_{k=1}^{K}
\bigg [ r(y_{k^*} ; \theta) \prod_{(i,j) \in T_k }f(y_i \mid y_j; \theta)
\bigg ],
 \end{aligned
where we refer to $r( \cdot ; \theta)$ as a ``root'' distribution, and $f(\cdot\mid y_j;\theta)$ as a ``leaf'' distribution; and we use $\theta$ to denote the other parameter; and we use simplified notation $(i,j)\in G$ to mean that $(i,j)$ is  an edge of the graph $G$.

 \begin{figure}[H]
        \begin{subfigure}[t]{.3\textwidth}
        \centering
           \includegraphics[width=.9\linewidth]{illustration1}
       \end{subfigure}
               \begin{subfigure}[t]{.3\textwidth}
                       \centering
           \includegraphics[width=.9\linewidth]{illustration2}
       \end{subfigure}
               \begin{subfigure}[t]{.3\textwidth}
                       \centering
           \includegraphics[width=.9\linewidth]{illustration3}
       \end{subfigure}
        \caption{Three examples of clusters that can be represented by a spanning forest.\label{fig:illustration}}
\end{figure}
 Figure \ref{fig:illustration} illustrates the high flexibility of a spanning forest in representing clusters. It shows the sampled $\mathcal F$ based on three clustering benchmark datasets. Note that some clusters are not in an elliptical or convex shape. Rather, each cluster can be imagined as if it were formed by connecting a point to another nearby.
\begin{remark}
To clarify, the point estimation on a spanning forest (as some fixed and unknown graph) has been studied  as early as \cite{gower1969minimum}. However, a distinction here is that we only consider $\mathcal V$ as the parameter of interest, but the edges (as in $\mathcal F_{\mathcal V}$ given $\mathcal V$) as a nuisance parameter / latent variable. The practical performance difference will be illustrated in Section \ref{sec:compare_vs_mst_cut}.
\end{remark}
The stochastic view of $\mathcal F_\mathcal V$ is important, as it allows us to incorporate the uncertainty of edges and avoids the sensitivity issue in point graph estimate. Equivalently, our clustering model is based on the marginal likelihood that varies with the node partition $\mathcal V$:
\begin{equation\label{eq:marginal_lik}
\mathcal L(y ; \mathcal V, \theta)
=\sum_{\mathcal F_{\mathcal V}, \mathcal R_{\mathcal V}}\mathcal L(y ; \mathcal V, \mathcal F_{\mathcal V}, \mathcal R_{\mathcal V}, \theta).
 \end{aligned
Further, by viewing $\mathcal F_{\mathcal V}$ as random, we can quantify the marginal connecting probability for each potential edge $(i,j)$:
\begin{equation\label{eq:marginal_connecting}
M_{i,j}:=\text{Pr}[F_\mathcal V \ni (i,j)]
\propto \sum_{\mathcal V}\sum_{\mathcal F_{\mathcal V}, \mathcal R_{\mathcal V}}
1[(i,j)\in  F_\mathcal V]
\mathcal L(y ; \mathcal V, \mathcal F_{\mathcal V}, \mathcal R_{\mathcal V}, \theta).
 \end{aligned
Similar to the normalized graph cut, there is no closed-form solution for directly maximizing \eqref{eq:marginal_lik}. However, closed-form does exist for  \eqref{eq:marginal_connecting} (see Section 4). Therefore, an approximate maximizer of \eqref{eq:marginal_lik}, $\hat {\mathcal V}$, can be obtained via computing the matrix $M$ and searching for $K$ diagonal blocks (after row and column index permutation) that contain the highest total values of $M_{i,j}$'s. Specifically, we can extract the top leading eigenvectors of $M$ and cluster the rows into $K$ groups.

This approximate marginal likelihood maximizer produces almost the same estimate as the normalized spectral clustering does. This is because the two sets of eigenvectors are almost the same. To provide some numerical evidence,
Figure \ref{fig:spectral_equiv}  compares the eigenvectors of the matrix $M$ and the normalized Laplacian $N$ (that uses $f$ and $r$ to specify $A$, with details provided in Section 4). Clearly, these two are almost identical in values. Due to this connection, the clustering estimates from spectral clustering can be viewed as an approximate estimate for $\hat{\mathcal V}$ in \eqref{eq:marginal_lik}. And the spanning forest graph provides a generative model for the spectral clustering algorithm.

\begin{figure}[H]
        \centering
           \includegraphics[width=.8\linewidth]{evs}
               \caption{Comparing the eigenvectors of a marginal connecting probability matrix $M$ and the ones of normalized Laplacian $N$.\label{fig:spectral_equiv}}
\end{figure}

We now begin to fully specify the Bayesian forest model.
For simplicity, we now focus on continuous $y_i \in \mathbb{R}^p$. For ease of computation, we recommend choosing $f$ as a symmetric function $f(y_i\mid y_j; \theta) = f(y_j\mid y_i;\theta)$, so that the likelihood is invariant to the direction of each edge; and choose $r$ as a diffuse density, so that the likelihood is less sensitive to the choice of a node as root.

In this article, we choose a Gaussian density for $f$ and Cauchy for $r$:
\begin{equation\label{eq:choice_density}
& f(y_i \mid y_j;\theta) =  {(2\pi\sigma_{i,j})^{-p/2}} \exp \left \{ - \frac {\|y_i-y_j\|_2^2}{2\sigma_{i,j}} \right\},\\
&r(y_i; \theta) =  \frac{\Gamma[(1+p)/2]}{\gamma^p \pi^{(1+p)/2}} \frac{1}{ (1+ \|y_{i}-\mu\|_2^2/\gamma^2)^{(1+p)/2}}.
 \end{aligned
where  $\sigma_{ij} >0$ and $\gamma>0$ are scale parameters.

As the magnitudes of distances between neighboring points may differ significantly from cluster to cluster, we use a local parameterization $\sigma_{i,j} =\tilde \sigma_i \tilde \sigma_j$, and will regularize $(\tilde \sigma_1,\ldots, \tilde \sigma_n)$ via a hyper-prior.

\begin{remark}
        In \eqref{eq:choice_density}, we are effectively using Euclidean distances $\|y_i-y_j\|_2$ to build a forest model. We find the model in \eqref{eq:choice_density} working well in a low-to-moderate dimension $p\le 50$.
      For high dimensional clustering, we develop a model-based extension inspired by the success of subspace clustering \citep{vidal2011subspace,wu2014spectral}, where we model the latent representation $z_i$ using a forest model, based on the sparse von Mises-Fisher density. The details are provided in the Supplementary Materials.
 \end{remark}

\subsection{Forest Process and Product Partition Prior}

To simplify notations as well as to facilitate computation, we now introduce an auxiliary node $0$ that connects to all roots $(1^*,\ldots, K^*)$. As the result, the model can be equivalently represented by a spanning tree rooted at $0$:
\begin{equation*
& \mathcal T=(V_{\mathcal T}, E_{\mathcal T}), \\
& V_{\mathcal T}=\{0\} \cup V_1 \cup \ldots \cup V_K, \; E_{\mathcal T}= \{ (0,1^*),\ldots, (0,K^*)\} \cup E_1\cup\ldots \cup E_K.
 \end{aligned
In this section, we focus on the prior specification for $\mathcal T$, denoted by $\Pi_0(\mathcal T)$. The prior $\Pi_0(\mathcal T)$ can be factorized according to the following hierarchies: picking the number of partitions $K$, partitioning the nodes into $(V_1,\ldots, V_K)$, forming edges $E_k$ and picking one root $k^*$ for each $V_k$.
\begin{equation\label{eq:hierarchical}
\Pi_0(\mathcal T)= \underbrace{
\bigg\{
\Pi_0(K)\Pi_0(V_1,\ldots, V_K \mid K)}_{\Pi_0(\mathcal V)
}
\bigg\}
\underbrace{\prod_{k=1}^K
 \bigg \{\Pi_0(E_k\mid V_k) \Pi_0( k^* \mid E_k, V_k) \bigg\}}_{
\Pi_0(\mathcal F_{\mathcal V}, \mathcal R_{\mathcal V})}.
 \end{aligned
\begin{remark}
In Bayesian non-parametric literature, $\Pi_0(K)\Pi_0(V_1,\ldots, V_K \mid K)$ is known as the partition probability function, which plays the key role in controlling cluster sizes and cluster number in model-based clustering. However, when it comes to graphical model-based clustering (such as our forest model), it is important to note the difference  --- for each partition $V_k$, there is an additional probability $\Pi_0(E_k, k^* \mid V_k)$ due to the multiplicity of all possible subgraphs formed between the nodes in $V_k$.
\end{remark}

For simplicity, we will use discrete uniform distribution for $\Pi_0(E_k, k^* \mid V_k)$. Since there are $n_k^{(n_k-2)_+}$  possible spanning trees for $n_k$ nodes [$(x)_+=x$ if $x>0$, otherwise $0$], and  $n_k$ possible choice of roots. We have $\Pi_0(E_k, k^* \mid V_k) = n_k^{-(n_k-1)}$.

We now discuss two different ways to complete the prior construction. We first take a ``ground-up'' approach by viewing  $\mathcal T$ as from a stochastic process where the node number $n$ could grow indefinitely. Starting from the first edge $e_1=(0,1)$, we sequentially draw new edges and add to $\mathcal T$, from
\begin{equation\label{eq:fp}
& e_i \mid e_1,\ldots e_{i-1} \sim  \sum_{j=1}^{i-1} \pi^{[i]}_j \delta_{(j,i)}(\cdot) +  \pi^{[i]}_i \delta_{(0,i)}(\cdot),\\
& y_i \mid (j,i) \sim 1(j \ge 1) f(\cdot \mid y_j)+ 1(j=0) r(\cdot),
 \end{aligned
with some  probability vector $(\pi^{[i]}_1,\ldots, \pi^{[i]}_i)$  that adds up to one.
 We refer to \eqref{eq:fp} as a forest process.

The forest process is a generalization of the P\'olya urn process \citep{blackwell1973ferguson}. For the latter, $e_i=(j,i)$ would make node $i$ take the same value as node $j$, $y_i =y_j$ [although in model-based clustering, one would use notation $\theta_i =\theta_j$ , and $y_i\sim \mathcal K(\cdot \mid \theta_i)$]; $e_i=(0,i)$ would make node $i$ draw a new value for $y_i$ from the base distribution.  Due to this relationship, we can borrow popular parameterization for $\pi^{[i]}_j$ from the urn process literature. For example, we can use the Chinese restaurant process parameterization $\pi^{[i]}_j= 1/(i-1+\alpha)$ for $j=1,\ldots, (i-1)$, and $\pi^{[i]}_i= \alpha/(i-1+\alpha)$ with some chosen $\alpha>0$. After marginalizing over the order of $i$ and partition index [see \cite{miller2019elementary} for a simplified proof on the partition function], we obtain:
\begin{equation\label{eq:crp}
\Pi_0(\mathcal T)=\frac{\alpha^{K} \Gamma(\alpha)}{\Gamma(\alpha+n)} \prod_{k=1} ^K\Gamma(n_k) n_k^{-(n_k-1)}.
 \end{aligned
Compared to the partition probability prior in the Chinese restaurant process, we have an additional $n_k^{-(n_k-1)}$ term that corresponds to the conditional prior weight of for each possible $(k^*,E_k)$ given a partition $V_k$.

To help understand the effect of this additional term on the posterior, we can imagine two extreme possibilities in the conditional likelihood given a $V_k$. If the conditional  $\mathcal L(y_i: i\in V_k \mid k^*,E_k)$ is skewed toward one particular choice of tree $(\hat k^*,\hat E_k)$ [that is, $\mathcal L(y_i: i\in V_k \mid k^*,E_k)$ is large when $(k^*,E_k)= (\hat k^*,\hat E_k)$, but is close to zero for other values of $(k^*,E_k)$], then $n_k^{-(n_k-1)}$ acts as a penalty for a lack of diversity in trees.
 On the other hand, if $\mathcal L(y_i: i\in V_k \mid k^*,E_k)$ is  equal for all possible $(k^*,E_k)$'s, then we can simply marginalize over $(k^*, E_k)$ and be not be subject to this penalty [since  $\sum_{(k^*, E_k)} n_k^{-(n_k-1)}=1$].

 Therefore, we can form an intuition by interpolating those two extremes: if a set of data points (of size $n_k$) are ``well-knit'' such that they can be connected via many possible spanning trees (each with a high conditional likelihood), then it would have a higher posterior probability of being clustered together, compared to some other points (of the same size $n_k$) that have only a few trees with high conditional likelihood.

With the ``ground-up'' prior construction useful for understanding the difference from the classic urn process, the prior \eqref{eq:crp} itself is not very convenient for posterior computation. Therefore, we also explore the alternative of a ``top-down" approach. This is based on directly assigning a product
partition probability \citep{hartigan1990partition,barry1993bayesian,crowley1997product,quintana2003bayesian} as
\begin{equation\label{eq:simple_prior}
& \Pi_0(V_1,\ldots, V_K \mid K) =  \frac{\prod_{k=1}^K  n_k^{(n_k-1)}}{
\sum_{\text{all }(V^*_1,\ldots, V^*_K) }\prod_{k=1}^K  |V^*_k|^{(|V^*_k|-1)}
},
 \end{aligned
where the cohesion function $n_k^{(n_k-1)}$ effectively cancels out the prior probability for each $(k^*,E_k)$. To assign a prior for $K$, we assign a probability $\Pi_0(K) = \lambda^K$ with $\lambda>0$, similar to a truncated geometric distribution support on $\{1,\ldots,n\}$, multiplying the terms according to \eqref{eq:hierarchical} leads to
\begin{equation\label{eq:trunc_poisson}
\Pi_0(\mathcal T) \propto  \lambda^K,
 \end{aligned
which is easy to handle in posterior computation, and we will use this from now on.

\subsection{Hyper-prior for Other Parameters}
In model-based clustering, it is a common practice to use a data-dependent empirical prior for the parameters in $\mathcal K(\mid \theta^*_k)$
\citep{richardson1997bayesian,miller2018mixture}. For example, one could set the location prior to center at the sample mean of the observed data, and the scale prior to distribute around the covering radius of the data. Using such empirical priors is usually optional for the theory, but is shown to be very effective in leading to good mixing of Markov chains in posterior estimation.

Following this practice, we propose a set of empirical priors for the Bayesian forest model. To make the root density $r(\cdot)$ close to a small constant in the support of the data, we set $\mu=\bar y$ (with prior probability 1)
and $\gamma^2 \sim \text{Inverse-Gamma}(2, \hat \sigma^2_y)$, with $\bar y= \sum_{i=1}^n y_i/n$ and $\hat \sigma^2_y = \sum_{i=1}^n \|y_i- \bar y\|_2^2/(np)$.
For the leaf density $f(y_i \mid y_j; \sigma_{i,j})$, we want it to more likely pick up an edge of small distances $\|y_i -y_j\|_2$, by making $\sigma_{i,j}=\tilde\sigma_i \tilde\sigma_j$ small. Therefore, we use a gamma prior centered around the shortest positive distance, $\mu_{\tilde \sigma,i}=\min_{j: y_j\neq y_i}\|y_i-y_j\|_2/\sqrt{p}$, $\tilde \sigma_i \sim \text{Gamma}( \alpha_{\tilde \sigma}, \mu_{\tilde \sigma,i})$, and we use $\alpha_{\tilde \sigma}=0.5$. Our choice resembles the heuristics proposed by \cite{zelnik2005self}, who fixes $\tilde \sigma_i$ to be a low order statistic of the distances to $y_i$. For the partition concentration parameter $\lambda$, we set $\lambda=0.5$. An alternative is to follow \cite{ascolani2022clustering} by putting a hyper-prior such as generalized Gamma  $\Pi_0(\lambda)\propto \lambda^{d_\lambda-1} \exp[- (\lambda/ a_\lambda)^{p_\lambda}]$.

\subsection{Model-based Extensions}

Compared to algorithms, a major advantage of generative models is the ease of building useful model-based extensions. We demonstrate three directions for extending the Bayesian forest model.  Due to the page constraint, we present the main ideas in this section, and provide the modeling details and numeric results in the Supplementary Materials.

\noindent\textbf{Latent Forest Model:} First, one could use the realization of the forest process as latent variables in another model $\mathcal M$ for data $(y_1,\ldots,y_n)$,
\begin{equation*
 z_1,\ldots, z_n \sim \t{Forest Model} (\mathcal T;\theta_z),\qquad y_1,\ldots, y_n \sim \mathcal M(z_1,\ldots, z_n;\theta_y),
 \end{aligned
where $\theta_z$ and $\theta_y$ denote the other needed parameters.
For example, for clustering high-dimensional data such as images, it is often necessary to represent each high-dimensional observation $y_i$ by a low-dimensional coordinate   $z_i$ \citep{wu2014spectral,chandra2020escaping}. In the Supplementary Materials, we present a high-dimensional clustering model, using an autoregressive matrix Gaussian for $\mathcal M$ and a sparse von Mises-Fisher for the forest model.

\noindent\textbf{Informative Prior in Forest Model:} Second, in applications it is sometime desirable to have the clustering dependent on some external information $x$, such as covariates  \citep{muller2011product} or an existing partition \citep{paganin2021centered}. From a Bayesian view, this can be achieved via taking an $x$-informative prior. In our framework, we use:
\begin{equation*
\mathcal T \sim \Pi_0(\cdot\mid x), \qquad
 y_1,\ldots, y_n \sim \t{Forest Model} (\mathcal T;\theta).
 \end{aligned
In the Supplementary Materials, we illustrate an extension by adapting a covariate-dependent product partition model [PPMx, \cite{muller2011product}] into the prior of $\mathcal T$.

\noindent\textbf{Multi-view Clustering:}
Third, for multi-subject/group data $(y^{(s)}_1,\ldots, y_n^{(s)})$ for $s=1,\ldots, S$, we want to find a clustering for every $s$. At the same time, we can borrow strength among subjects/groups, by allowing some of them to share the same partition of $(1,\ldots,n)$. This is known as multi-view clustering. To achieve this effect, we assign each subject/group with an individual forest $\mathcal T^{{(s)}}$, while imposing a discrete prior on $\mathcal T^{{(s)}}$.
\begin{equation\label{eq:np_forest_model}
& \mathcal{T}^{(s)} \sim \sum_{k=1}^{\tilde K} v_k \delta_{\mathcal T^{(k)}_*} (\cdot) , \qquad
& \{ y_{1}^{(s)}, \ldots, y_{n}^{(s)}\} \sim \t{Forest Model} (\mathcal T^{{(s)}}),
 \end{aligned
where $v_k$'s are the weights adding up to one, $\mathcal T^{(k)}_*$ denotes a unique forest. We illustrate this model in Section 6 of clustering brain regions for multiple subjects.
\section{Posterior Computation}

\subsection{Gibbs Sampling Algorithm}

We now describe the Markov chain Monte Carlo (MCMC) algorithm. For ease of notation, we use an $(n+1)\times (n+1)$ matrix $S$, with $S_{i,j}= \log f (y_i \mid y_j ; \sigma_{i,j})$, $S_{0,i}=S_{i,0}= \log r (y_i)  +\log \lambda$ (for convenience, we use $0$ to index the last row/column), $S_{i,i}=0$, and $A_{\mathcal T}$ to represent the adjacency matrix of $\mathcal T$. We have the posterior distribution
\begin{equation\label{eq:comp_posterior}
\Pi( \mathcal T, \theta \mid y) \propto  \exp \big\{ \t{tr}[S(\theta) A_{\mathcal T}]/2\big\}\Pi_0(\theta).
 \end{aligned
Note the above form conveniently include the prior term for the number of clusters, $\lambda^K$, via the number of edges adjacent to node $0$.

Our MCMC algorithm alternates in updating $\mathcal T$ and $\theta$, hence is a Gibbs sampling algorithm. To sample the tree $\mathcal T$ given $\theta$, we take the random-walk covering algorithm for weighted spanning tree \citep{mosbah1999non}, which is an extension of the Andrei--Broder algorithm for sampling uniform spanning tree \citep{broder1989generating,aldous1990random}. For this article to be self-content, we describe the algorithm below.

\begin{algorithm}
\begin{algorithmic}
\State Start with $V_{\mathcal T}=\{0\}$ and $E_{\mathcal T}=\varnothing$, and set $i\leftarrow0$:
\While{$|V_{\mathcal T}|\neq n+1$}
    \State Take a random walk from $i$ to $j$ with probability
    $\text{Pr}(j \mid i) = \frac{\exp[S_{i,j} (\theta)]}{\sum_{j:j\neq i} \exp[S_{i,j} (\theta)]}.$
\If {$j\not \in V_{\mathcal T}$ } \State Add $j$ to $V_{\mathcal T}$. Add $(i,j)$ to $E_{\mathcal T}$.
\EndIf
\State Update $i \leftarrow j$.
\EndWhile
\end{algorithmic}
\caption{Random-walk covering algorithm for sampling the augmented tree $\mathcal T$.}
\end{algorithm}
The above algorithm is known to produce a random sample $\mathcal T$ following the full conditional $\Pi(\mathcal T \mid \theta,y)$ proportional to \eqref{eq:comp_posterior}. It has an expected finish time of $O(n\log n)$. There are various algorithms proposed with faster computing speed, such as \cite{schild2018almost}, although we choose to use this random-walk covering algorithm for its simplicity.

To sample $\theta$ given the spanning tree $\mathcal T$, we sequentially update each $\tilde\sigma_i$ for $i=1,\ldots, n$ from the generalized inverse Gaussian (GIG) distribution,
\[
(\tilde\sigma_i \mid \cdot) \sim \text{GIG} \bigg \{ \frac{2}{\mu_{\tilde \sigma,i}},\sum_{j:(i,j)\in \mathcal T} \frac{\|y_i-y_j\|^2_2}{\tilde \sigma_j},  -\frac{p{\sum_{j} 1[(i,j)\in \mathcal T] }}{2}+\alpha_{\tilde \sigma} \bigg  \}.
\]

To update
 $\gamma$, we use the form of the multivariate Cauchy as a scale mixture of $\text{N}(\mu, \gamma^2 u_{\gamma,i} I_p )$ over $u_{\gamma,i} \sim \text{Inverse-Gamma}(1/2,1/2)$.  We can update these parameters via
\begin{equation*
& u_{\gamma,i} \sim \text{Inverse-Gamma}( \frac{1+p}{2},  \frac{1}{2} +\frac{\|y_i-\mu\|_2^2}{2\gamma^2}),\\
& \gamma^2 \sim  \text{Inverse-Gamma}(2 + \frac{Kp}{2},  \hat\sigma^2_y + \sum_{i:(0,i)\in \mathcal T}\frac{\|y_i-\mu\|_2^2}{2 u_{\gamma,i}}).\\
 \end{aligned

We run the MCMC algorithm iteratively for many iterations. And we discard the first half of iterations as burn-ins. We find the algorithm enjoying good mixing of Markov chains, and we provide a few example diagnostics plots in the Supplementary Materials.

\subsection{Posterior Point Estimate on Clustering}
To produce a point estimate on clustering, $(\hat c_1,\ldots, \hat c_n)$, one could take advantage of our discovered relationship between the Bayesian forest model and the spectral clustering algorithm. This involves first using a posterior point estimate on $\hat\theta$ (such as the posterior mean), then applying a spectral clustering algorithm on a similarity matrix formed by $A_{i,j}=\exp(S_{i,j})$ to produce $K$ clusters.

On the other hand, a more preferable approach that incorporates the randomness of $\theta$ is to use the posterior sample of $(c_1,\ldots, c_n)$ collected from the MCMC, and minimizes some loss function over all possible clusterings. For example, \cite{lau2007bayesian,wade2018bayesian} proposed to minimize the Binder's loss or the variation of information loss. Both losses are defined based on the  posterior co-assignment probability $\text{Pr}(c_i=c_j\mid y)$ (also known as the posterior similarity matrix). Despite the theoretical appeals, for those two losses, we find the associated search algorithms lack a performance guarantee to produce the near-optimal solution. More importantly, they empirically tend to produce point estimates with only one cluster, which is difficult to interpret.

 Inspired by these posterior-summary approaches, we also use pairwise posterior co-assignment  $\text{Pr}(c_i=c_j\mid y)$ but minimize the normalized graph cut loss. This is defined as in \eqref{eq:normalized_graph_cut} except $A_{i,j}:=\Pr(c_i=c_j\mid y)$. Then we use the spectral clustering algorithm and obtain $(\hat c_1,\ldots, \hat c_n)$ corresponding to $\hat K$ clusters, which is guaranteed to be close to the minimizer of normalized graph cut loss. Lastly, to choose $\hat K$ for producing the above point estimate, we use the empirical posterior mode of $K$ based on the MCMC sample.

 \section{Theoretical Properties}

\subsection{ Convergence of Eigenvectors \label{sec:spectral}}
We now formalize the closeness of the eigenvectors of matrices $N$ and $M$ (shown in Section 2.2), by establishing the convergence of the two sets of eigenvectors as $n$ increases.

To be specific, we focus on the normalized spectral clustering algorithm using the similarity  $A_{i,j}=\exp(S_{i,j})$, as used in \eqref{eq:comp_posterior}. On the other hand, for the specific form of $S_{i,j}$, $f(y_i \mid y_j)$ can be any density satisfying $f(y_i\mid y_j, \theta) = f(y_j\mid y_i, \theta)$, $r(y_i; \theta)$ can be any density satisfying $r(y_i;  \theta)>0$. For the associated normalized Laplacian $N$, we denote the first $K$ bottom  eigenvectors by  $\phi_1,\ldots, \phi_K$, which correspond to the smallest $K$ eigenvalues.

Let $M$ be the matrix with $M_{i,j}=\t{pr}[\mathcal T \ni  (i,j)  \mid y, \theta ]$ for $i\neq j$ and $M_{i,i}=0$. The Kirchhoff's tree theorem \citep{chaiken1978matrix} gives an enumeration of all $\mathcal T\in \mathbb{T}$,
\begin{equation*
\sum_{\mathcal T\in \mathbb{T}} \prod_{(i,j)\in \mathcal T} \exp(S_{i,j}) = (n+1)^{-1}\prod_{h=2}^{n+1}\lambda_{(h)} (L)
 \end{aligned
where $L$ is the Laplacian matrix transform of the similarity matrix $A$; $\lambda_{(h)}$ denotes the $h$th smallest eigenvalue. Differentiating its logarithmic transform with respect to $S_{i,j}$,
\begin{equation*
M_{i,j} & =  \t{Pr}[ \mathcal T \ni  (i,j)\mid y ]
  =
  \frac{\sum_{\mathcal T\in \mathbb{T},(i,j)\in \mathcal T} \prod_{(i',j')\in \mathcal T} \exp(S_{i',j'})}{\sum_{\mathcal T\in \mathbb{T}} \prod_{(i',j')\in \mathcal T} \exp(S_{i',j'})}
  =
   \frac{\partial \sum_{i=2}^{n+1} \log\lambda_{(i)} (L)}{\partial S_{i,j}}.
 \end{aligned
Let $\Psi_1, \ldots, \Psi_K$ be the top $K$ eigenvectors of $M$, associated with eigenvalues $\xi_1\ge \xi_2 \ge \ldots \ge \xi_K$, and $\xi_K > \xi_{K+1}\ge \xi_{K+2}\ge \ldots \ge \xi_{n+1}$. And we can compare with the $K$ leading eigenvectors of $(-N)\in \mathbb{R}^{n\times n}$,  $\phi_1,\ldots, \phi_K$. Using $\Psi_{1:K}$ and $\phi_{1:K}$ to denote two $(n+1)\times K$ matrices, we now show they are close to each other.

\begin{theorem}
There exists an orthonormal matrix $R\in\mathbb{R}^{K\times K}$ and a finite constant $\epsilon>0$,
\begin{equation*
\|\Psi_{1:K}-\phi_{1:K} R \|_F \le \frac{
  40 \sqrt{K (n+1)}
  }{\xi_{K} -\xi_{K+1}}  \max_{i,j} \left \{ (1+\epsilon)(D^{-1/2}_i-D^{-1/2}_j)^2
A_{i,j} \right\},
 \end{aligned
        with probability at least $1- \exp(-n).$
\end{theorem}
\begin{remark}
To make the right hand side go to zero, a sufficient condition is to have   all $ A_{i,j}/D_{i,i}= O(n^{-\kappa})$ with $\kappa>1/2$.
We provide the detailed definition of the bound constant $\epsilon$ in the Supplementary Materials.
\end{remark}
Therefore, under mild conditions, as $n\to \infty$, the two sets of leading eigenvectors converge. In the Supplementary Materials, we show that the convergence is very fast, with the two sets of leading eigenvectors becoming almost indistinguishable starting around $n\ge 50$.

\subsection{Consistent Clustering of Separable Sets}
\label{sec:clustcon}

We show that clustering consistency is possible, under some separability assumptions. Specifically,
we establish posterior ratio consistency, as the ratio between the maximum posterior probability assigned to other possible clustering assignments to the posterior probability assigned to the true clustering assignments converges to zero almost surely under the true model \citep{cao2019posterior}.

To formalize the above statements, we denote the true cluster label used for generating $y_i$ by $c^0_i$ (subject to label permutation among clusters), and we define the enclosing region for all possible $y_i:c^0_i=k$ as $R_k^0$ for $k=1,\ldots, K_0$ for some true finite $K_0$. And we refer to $R^0=(R_1^0,\ldots, R_{K_0}^0)$ as the ``null partition''.  By separability, we mean the scenario that $(R_1^0,\ldots, R_{K_0}^0)$ are disjoint and there is a lower-bounded distance between each pair of sets. As alternatives, regions $R=(R_1,\ldots, R_K)$  could be induced by $\{c_1,\ldots,c_n\}$ from the posterior estimate of $\mathcal T$. For simplicity, we assume the scale parameter in $f$ is known and all equal $\sigma_{i,j}=\sigma^{0,n}$.

\vspace{5mm}

{\noindent \underline{Number of clusters is known.}}
We first start with a simple case when we have fixed $K=K_0$. For regularities, we consider  data as supported in a compact region $\mathcal{X}$, and satisfying the following assumptions:
\begin{itemize}
   \item (A1, diminishing scale) $\sigma^{0,n}=C'(1/\log n)^{1+\iota}$ for some $\iota>0$ and $C'>0$.
    \item (A2, minimum separation) $\inf_{x\in R_k^0,y\in R_{k'}^0}\|x-y\|_2>M_n$, for all $k\neq k'$ with some positive constant $M_n>0$ such that $M_n^2/\sigma^{0,n}=8\tilde m_0\log(n)$  for all $(i,j)$ and is known for some constant $\tilde  m_{0}>p/2+2$.
\item (A3, near-flatness of root density) For any $n$, $\epsilon_1<r(y)<\epsilon_2$ for all $y\in\mathcal{X}$.
\end{itemize}

Under the null partition, $\Pi(\mathcal{T}|y)$ is maximized at $\mathcal T=\mathcal{T}_{\textrm{MST},R^0}$, which contains $K_0$ trees with each $T_k$ being the minimum spanning tree (denoted by subscript ``MST'') within region $R_k^0$.
Similarly, for any alternative $R$, $\Pi(\mathcal{T}|y)$ is maximized at the $\mathcal{T}=\mathcal{T}_{\textrm{MST},R}$.

\begin{theorem}
Under  (A1,A2,A3), we have ${\Pi(\mathcal{T}_{\textrm{MST},R}|y)}/{\Pi(\mathcal{T}_{\textrm{MST},R^0}|y)}\rightarrow 0$ almost surely, unless $R_{i}^0\subseteq R_{\xi(i)}$ for some permutation map $\xi(\cdot)$.
\label{knownclust}
\end{theorem}

{\noindent\underline{Number of clusters is unknown:}} Next, we relax the condition by having a $K$ not necessarily equal to $K_0$.
We show the consistency in two parts for 1)$K<K_0$, and 2) $K>K_0$ separately. In order to show posterior ratio consistency in the second part, we need some finer control on $r(y)$:
\begin{itemize}
    \item (A3') The root density satisfies $\tilde m_1e^{-M/2\sigma^{0,n}}\leq r(y)\leq \tilde  m_2 e^{-M/2\sigma^{0,n}}$ for some $\tilde  m_1<\tilde  m_2$.
\end{itemize}
In this assumption, we essentially assume the root distribution to be flatter with a larger $n$. Then we have the following results.
\begin{theorem}
1) If $K<K_0$, under the assumptions (A1,A2,A3), we have \\${\Pi(\mathcal{T}_{\textrm{MST},R}|y)}/{\Pi(\mathcal{T}_{\textrm{MST},R^0}|y)}\rightarrow 0$ almost surely.

2) If $K>K_0$, under the assumptions (A1,A2,A3'), we have ${\Pi(\mathcal{T}_{\textrm{MST},R}|y)}/{\Pi(\mathcal{T}_{\textrm{MST},R^0}|y)}\rightarrow 0$ almost surely.
\label{unknownclust}
\end{theorem}

\section{Numerical Experiments}

\subsection{Clustering Near-Manifold Data}
To illustrate the capability of uncertainty quantification, we carry out clustering tasks on those near-manifold data commonly used for benchmarking clustering algorithms.

\begin{figure}[H]
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{sim2_map}
            \caption{Posterior point estimate.}
    \end{subfigure}
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[width=.8\linewidth]{sim2_uncertainty}
            \caption{$\text{Pr}(c_i=c_j \mid y)$.}
    \end{subfigure}
    \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{sim1_map}
            \caption{Posterior point estimate.}
    \end{subfigure}
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[width=.8\linewidth]{sim1_uncertainty}
            \caption{$\text{Pr}(c_i=c_j \mid y)$.}
    \end{subfigure}
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{s3_data0}
            \caption{Posterior point estimate.}
    \end{subfigure}
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[width=.8\linewidth]{s3_uq0}
            \caption{$\text{Pr}(c_i=c_j \mid y)$.}
    \end{subfigure}
    \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{s3_data1}
            \caption{One posterior sample.}
    \end{subfigure}
     \begin{subfigure}[t]{.24\textwidth}
        \includegraphics[width=.8\linewidth]{s3_uq1}
            \caption{$\text{Pr}(c_i=c_j \mid y)$.}
    \end{subfigure}
        \caption{Uncertainty quantification in clustering data generated near three manifolds. When  data are close to the manifolds (Panels a,e), there is very little uncertainty on clustering in low $\text{Pr}(c_i=c_j\mid j)$ between points from different clusters (Panels b,f). As data deviate more from the manifolds (Panel c,g), the uncertainty increases (Panels d,h). And in Panel g, the point estimate shows a two-cluster partitioning, while there is about $20\%$ of probability for three-cluster partitioning.
         \label{fig:uq_3rings}
        }
        \end{figure}

In the first simulation, we start with $300$ points drawn from three rings of radii $0.2$, $1$ and $2$, with $100$ points from each ring. Then we add some Gaussian noise to each point to create a coordinate near a ring manifold. We present two experiments, one with noises from $\t{N}(0,  0.05^2 I_2)$, and one with noises $\t{N}(0,  0.1^2 I_2)$. As shown in Figure \ref{fig:uq_3rings}, when these data are well separated (Panel a, showing Posterior point estimate), there is very little uncertainty on the clustering (Panel b), with the posterior co-assignment $\text{Pr}(c_i=c_j\mid y)$ close to zero for any two data points near different rings. As noises increase, these data become more difficult to separate. There is a considerable amount of uncertainty for those red and blue points: these two sets of points are assigned into one cluster with probability close to $40\%$ (Panel d). We conduct another simulation based on an arc manifold and two point clouds (Panels e-h), and find similar results. Additional experiments are described in the Supplementary Materials.

\subsection{Comparison with Minimum Spanning Tree-based Cut}\label{sec:compare_vs_mst_cut}
Since our Bayesian forest model uses spanning trees, it is natural to compare with the clustering algorithm based on cutting the minimum spanning tree (MST). To formalize, the minimum spanning tree-based cut (MST-Cut) algorithm first finds the MST:
\begin{equation*
\hat T = {\arg\min}_{T\in \mathbb{T}_n}\sum_{(i,j)\in T} \|y_i-y_j\|,
 \end{aligned
where $\mathbb{T}_n$ denotes all spanning trees that connect $n$ nodes, with $\|y_i-y_j\|$ as some distance between the two points. Then, one removes the longest $(K-1)$ edges (with  length defined as $\|y_i-y_j\|$) to create $K$ clusters.
This algorithm is shown to be equivalent to the single-linkage clustering algorithm \citep{Hartigan1981}.

As we could imagine, such MST-Cut algorithms work well when clusters are well separated. In that case, those clusters will more likely be connected by the longest few  edges. However, such algorithms will suffer sensitivity issues, when any one or more of the following happens: 1) when clusters are close to each other; 2) when there are a few isolated points lying between two clusters; 3) when one or more clusters are from a heavy-tailed distribution, with a few points away from the bulk of the cluster. As a result, the longest edges in the MST may not be ideal for partitioning data.

\begin{figure}[H]
     \begin{subfigure}[t]{.3\textwidth}
        \includegraphics[width=1\linewidth, trim=20cm 10cm 15cm 10cm,clip]{mst_problem1}
            \caption{Partitioning the data by cutting the longest edge in the minimum spanning tree.}
    \end{subfigure}\;
         \begin{subfigure}[t]{.3\textwidth}
        \includegraphics[width=1\linewidth, trim=20cm 10cm 15cm 10cm,clip]{mst_problem2}
            \caption{Partitioning the data by cutting the top $10\%$ longest edges in the minimum spanning tree.}
    \end{subfigure}\;
     \begin{subfigure}[t]{.3\textwidth}
        \includegraphics[width=1\linewidth, trim=20cm 10cm 15cm 10cm,clip]{mst_problem1_fixed}
            \caption{Partitioning the data using the Bayesian spanning forest model.}
    \end{subfigure}
        \caption{
            Comparing point estimates from the minimum spanning tree-based cut (MST-Cut) algorithms and the Bayesian spanning forest model. \label{fig:sensitivity_mst_cut}
        }
        \end{figure}

To illustrate this problem, we use a simulation with data from a two-component $t$-distribution in $\mathbb{R}^2$ with $3$ degrees of freedom. One component has the mean $(0,0)$ and the other has $(4,4)$, and both have the scale parameter equal to $I_2$. And we generate $n=200$ data points. As shown in Figure \ref{fig:sensitivity_mst_cut}(a), due to the heavy-tail and closeness of the two clusters, cutting the longest edge in the MST (using Euclidean distances) yields a trivial and sub-optimal partition. Further, cutting the top $10\%$ longest edges still does not produce the desired result (Panel b).

Fundamentally, the reason is that relying on the minimum spanning tree (that is, one tree) leads to an underestimation of the graph uncertainty. Different from the MST-Cut algorithms, the Bayesian forest model effectively uses the marginal distribution  \eqref{eq:marginal_lik} incorporating the multiplicity of those likely trees (with edges shown in Panel c). As the result, it leads to better performance than the MST-Cut algorithm.

\subsection{Uncertainty Quantification for Data from Mixture Model}
In the Supplementary Materials, we present some uncertainty quantification results, for clustering data that are from mixture models. We compare the estimates with the ones from Gaussian mixture models, which could correspond to correctly/erroneously specified component distribution. Empirically, we find that the uncertainty estimates on $\text{Pr}(c_i=c_j\mid y)$ and $\text{Pr}(K\mid y)$ from the forest model are close to the ones based on the true data-generating distribution; whereas the Gaussian mixture models  suffer from sensitivity in model specification, especially when $K$ is not known.

\section{Application: Clustering in Multi-subject Functional Magnetic Resonance Imaging Data}

In this application, we conduct a neuroscience study for finding connected brain regions under a varying degree of impact from Alzheimer's disease. The source dataset is resting-stage functional magnetic resonance imaging scan data, collected from $S=144$ subjects at different stages of Alzheimer's disease. Each subject has scans over $n=116$ regions of interest and over $p=119$ time points. We denote the observation for the $s$th subject in the $i$th region by $y_{i}^{(s)}\in \mathbb{R}^{p}$.

Our goal is to find and distinguish between distinct clustering patterns for these subjects, while allowing some subjects to share the same clustering of brain regions. To achieve this purpose, we use the model in \eqref{eq:np_forest_model}.
Since we do not know the number of clustering patterns, we set $\tilde K$ as unbounded and use a stick-breaking parameterization, $\nu_k\sim \t{Beta}(1,0.1)$ with $0.1$ to encourage having fewer patterns, $v_1=\nu_1$, and $v_k=\nu_k\prod_{l<k}(1-\nu_l)$ for $k\ge 2$; and we model each ${\mathcal T^{(k)}_*}$ as drawn from a  uniform base measure.

        \begin{figure}[H]
        \centering
     \begin{subfigure}[t]{.22\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{brain1}
            \caption{Clustering pattern 1, mostly corresponding to subjects in the healthy group.}
    \end{subfigure}\;
     \begin{subfigure}[t]{.22\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{brain2}
            \caption{Clustering pattern 2, mostly corresponding to subjects in the diseased group.}
    \end{subfigure} \;
      \begin{subfigure}[t]{.22\textwidth}
        \includegraphics[height=4cm,width=1\linewidth]{brain3}
            \caption{Clustering pattern 3, mostly corresponding to subjects in the diseased group.}
    \end{subfigure} \;
     \begin{subfigure}[t]{.22\textwidth}
        \includegraphics[width=1\linewidth,height=4cm,]{brain_subjects.png}
            \caption{The observed healthy or diseased labels versus the estimated for subjects.}
    \end{subfigure}
        \caption{Multi-view clustering of the brain regions (viewed from the right side of the head) for multiple subjects: each subject is assigned to a latent group label, with each group corresponding to one clustering pattern parameterized by a forest graph (shown in Panels a-c, colors representing different clusters, and white representing singletons). The estimated group labels at the Posterior point estimate are very similar to the observed labels of diseased or healthy subjects, as an external validating variable (Panel d).
        \label{fig:application}
        }
        \end{figure}

         We ran MCMC for $5,000$ iterations and discarded the first $2,500$ as burn-in. Among the posterior samples, most correspond to having three distinct values in $\{T^{(s)}\}_{s=1}^S$. In Figure \ref{fig:application} we plot three forests at the empirical posterior mode. By grouping subjects $s$ and $s'$ with $\mathcal T^{{(s)}}=\mathcal T^{{(s')}}$ together, we obtain three-group assignments for these subjects. By comparison, the first one (Panel a) has nodes more connected in $6$ clusters (along with a few isolated data points), whereas the other two (Panels b and c) have more small clusters that make the brain regions more fragmented. Using an external validating variable about the disease status of each subject, we find our result to be clinically interpretable: the first clustering pattern mostly corresponds to healthy subjects, while the second and third do to subjects that are in stages of Alzheimer's disease (Panel d). Combining the last two estimated groups, we obtain accuracy of $78\%$ when compared against the healthy/diseased labels. To provide a baseline, a simple K-means on the raw scan time series data with $K=2$ yields a match rate of only 56\%.

        We find that most of the across-hemisphere connections (red and orange) from Panel a are missing in Panels b and c, supporting the fact that inter-hemispheric connectivity decreases among the diseased population \citep{qiu2016inter}.
Additionally, connections associated with the temporal lobe, occipital gyrus, angular gyrus, fusiform, cingulate anterior, and frontal cortex from Panel a are also found to be much sparser in the other two. These regions are primarily responsible for  cognitive tasks and memory management, and are known to be affected by Alzheimer's disease \citep{galton2001temporal, cai2015altered}.

        \section{Discussion}
        In this article, we present our discovery of a generative model for the popular spectral clustering algorithms. This enables straightforward uncertainty quantification and model-based extensions through the Bayesian framework. There are several directions worth exploring. First, our consistency theory is conducted under the condition of separable sets, similar to \cite{ascolani2022clustering}. For general cases with non-separable sets, clustering consistency (especially on estimating $K$) is challenging to achieve; to our best knowledge, existing consistency theory only applies to data generated independently from a mixture model \citep{miller2018mixture,zeng2020quasi}. For data generated dependently via a graph, this is still an unsolved problem. Second, in all of our forest models, we have been careful in choosing densities with tractable normalizing constants. One could relax this constraint by using densities $f(y_i\mid y_j,\theta)= \alpha_f g_f(y_i\mid y_j;\theta)$ and $r(y_i;\theta)= \alpha_r g_r(y_i;\theta)$, with $g$ some similarity function, and $(\alpha_f,\alpha_r)$ potentially intractable. In these cases, the forest posterior becomes
        $\Pi(\mathcal T\mid .)\propto  (\lambda\alpha_r/\alpha_f)^K  \prod_{(0,i)\in \mathcal T} g_r(y_i;\theta)
        \prod_{(i,j)\in \mathcal T} g_r(y_i\mid y_j;\theta)$. Therefore, one could choose an appropriate $\tilde \lambda=\lambda\alpha_r/\alpha_f$ (equivalent to choosing some value of $\lambda$), without knowing the value of $\alpha_f$ or $\alpha_r$; nevertheless, how to calibrate $\tilde \lambda$ still requires further study.

redpajama_set_name: RedPajamaArXiv


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Think of a pivotal moment within your favorite sports drama. More likely than not, the team's ability and willingness to execute is more about the trust, loyalty or respect between players and coach than which play to run. Yet, most coaches have little training and spend significantly less time on the fundamentals of leadership than they do searching for and designing drills and activities.
Kentucky coach and leadership consultant DeAngelo Wiser is an exception to that rule. While he admits to owning stacks of books, DVDs and practice plans, he is equally concerned with creating an environment in which athletes can succeed.
Wiser's recently released book, Winning Your Players through Trust, Loyalty, and Respect, gets to the heart of what it takes to be a successful coach in the modern game. It offers accumulated wisdom, insight and solutions garnered from years of developing outstanding players and working with them in competitive environments at key moments. Wiser is a highly regarded coach with more than 20 years of experience during which he won many competitions and honors but also also experienced heartbreaking losses.
"There are few resources for coaches, athletic directors or administrators on how to apply leadership principles to the art of coaching. I was fortunate to have a varied management career that gave me a background beyond sports," said Wiser. "There is so much pressure to win these days but surprisingly little support for the leadership skills needed to build a winning program and culture. That is a gap I want to help fill with this book."
Through real-life stories and outlines of practical steps, Wiser offers a formula for every coach to have a positive impact on their players during practice, in games, and throughout their lifetimes.
Winning Your Players offers a clear pathway for coaches who want to develop and nurture talent to the best of their abilities and gives insight into situations that require strong leadership at key team moments. In those moments, coaches need every resource possible to clearly do what's best for their teams. Winning Your Players is a must-read during those times.
Published by renowned British sports publisher Bennion Kearny, Winning Your Players through Trust, Loyalty, and Respect is available in print and ebook formats, and review copies are available.
About the Author: Coach Wiser's leadership skills were developed first on the basketball court and next honed by United States Marine Corps where he learned what it takes to motivate, inspire and earn respect from others. In the competitive automobile industry, he earned Salesman of the Year status and led a young sales team to set records. He then left behind the corporate world to achieve his lifelong dream of being a teacher and a coach. Focusing on integrity and character more than wins and championships, he established winning soccer programs at multiple schools. His teams were recognized by game officials for sportsmanship and his win total is in Kentucky's top 10. Wiser teaches leadership skills to players, coaches and academic systems as well as coaching at the collegiate level. His passion for excellence is obvious as he assists administrators, athletic directors, coaches, and players achieve their personal, school and team goals.
Book details:https://www.bennionkearny.com/Coach-DeAngelo-Wiser-Winning-Your-Players-Soccer-Coaching-Book.htm ISBN: 9781910773307 | Paperback 148pp | $12.99 | Multiple eBook formats available. For inspection, copies contact Adam Walters, Adam@BennionKearny.com
Tweets by CoachWiser
March 19, 2020 10 QUESTIONS PLAYERS MUST CONQUER
November 18, 2019 Leadership...what's your vision?
April 1, 2018 The Coaching Interview, Are You Ready?
March 29, 2018 CAN YOU LEAD YOUR PLAYERS THROUGH THE DOUBT AND ANXIETY?
"Coach for, and live for moments, not trophies and titles. Moments live forever with your players; trophies and titles are forgotten and gather dust. Make those moments special."
- DeAngelo Wiser
For inquiries: Email Us
2250 Mackey Pike, Nicholasville, KY 40356
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Copyright © 2020 Wiser Sports Leadership.


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Q: SUM the result of a group by query I have a following query running in SQL Server that returns the count and the city:
SELECT
    COUNT(*), city
FROM
    address
WHERE
    address1 LIKE '%[0-9]%'
    AND phone1 LIKE '%[0-9]%'
GROUP BY
    city
HAVING
    COUNT(*) > 250
ORDER BY
    COUNT(*) DESC

I get the following result:
1232 Atlanta
345 Chicago

How can I write a query to get the sum of the result of the counts (1232 + 345 in the above example?)
I have tried subqueries, but seems to cause some errors because of the group.. New to SQL and my apologies for a poor question.

A: Try this:
select sum(c)
from (
    select COUNT(*) c
    from address
    where address1 like '%[0-9]%'
        and phone1 like '%[0-9]%'
    group by city
    having count(*) > 250
    );

A: You can define a CTE and then use it:
WITH cte AS (
    SELECT city, COUNT(*) AS total
    FROM address
    WHERE address1 LIKE '%[0-9]%' AND
          phone1 LIKE '%[0-9]%'
    GROUP BY city
    HAVING COUNT(*) > 250
)

SELECT t.city,
       t.total,
       (SELECT SUM(total) FROM cte) AS grand_total
FROM cte t
ORDER BY t.total DESC

If you just want the grand total then use the following query:
SELECT SUM(total) AS grand_total
FROM cte

A: Here is another way using SUM Over() window aggregate function
SELECT city,
       Count(*) AS total,
       Sum(Count(*))OVER() grand_total
FROM   address
WHERE  address1 LIKE '%[0-9]%'
       AND phone1 LIKE '%[0-9]%'
GROUP  BY city
HAVING Count(*) > 250

A: you can use ROLLUP of GROUP,For example:
    ;WITH address(address1,phone1,city) AS
    (
        SELECT '1','2','Atlanta' UNION ALL
        SELECT '2','2','Atlanta' UNION ALL
        SELECT '3','2','Atlanta' UNION ALL
        SELECT '4','2','Atlanta' UNION ALL
        SELECT '5','2','Chicago' UNION ALL
        SELECT '6','0','Chicago' UNION ALL
        SELECT '7','0','Chicago'
    )
    SELECT COUNT(*), ISNULL(city,'Summary') FROM address where
    address1 LIKE '%[0-9]%'
    and phone1 LIKE '%[0-9]%'
    group by city WITH ROLLUP
    HAVING count(*) > 2
    order BY CASE WHEN city IS NULL THEN 1 ELSE 0 END , COUNT(*) DESC

----------- -------
4           Atlanta
3           Chicago
7           Summary