Hardware compatibility list

A hardware compatibility list (HCL) is a list of computer hardware (typically including many types of peripheral devices) that is compatible with a particular operating system or device management software. The list contains both whole computer systems and specific hardware elements including motherboards, sound cards, and video cards. In today's world, there is a vast amount of computer hardware in circulation, and many operating systems too. A hardware compatibility list is a database of hardware models and their compatibility with a certain operating system. HCLs can be centrally controlled (one person or team keeps the list of hardware maintained) or user-driven (users submit reviews on hardware they have used). There are many HCLs. Usually, each operating system will have an official HCL on its website.

Brain technology

Brain technology, or self-learning know-how systems, defines a technology that employs latest findings in neuroscience. [see also neuro implants] The term was first introduced by the Artificial Intelligence Laboratory in Zurich, Switzerland, in the context of the Roboy project. Brain Technology can be employed in robots, know-how management systems and any other application with self-learning capabilities. In particular, Brain Technology applications allow the visualization of the underlying learning architecture often coined as "know-how maps". == Research and applications == The first demonstrations of BC in humans and animals took place in the 1960s when Grey Walter demonstrated use of non-invasively recorded encephalogram (EEG) signals from a human subject to control a slide projector (Graimann et al., 2010). Soon after Jacques J. Vidal coined the term brain–computer interface (BCI) in 1971, the Defense Advanced Research Projects Agency (DARPA) first starting funding brain–computer interface research and has since funded several brain–computer interface projects. That market is expected to reach a value of $1.72 billion by 2022. Brain–computer interfaces record brain activity, transmit the information out of the body, signal-process the data via algorithms, and convert them into command control signals. In 2012, a landmark study in Nature, led by pioneer Leigh Hochberg, MD, PhD, demonstrated that two people with tetraplegia were able to control robotic arms through thought when connected to the BrainGate neural interface system. The two participants were able to reach for and grasp objects in three-dimensional space, and one participant used the system to serve herself coffee for the first time since becoming paralyzed nearly 15 years prior. And in October 2020, two patients were able to wirelessly control an operating system to text, email, shop and bank using direct thought through the Stentrode brain computer interface (Journal of NeuroInterventional Surgery) in a study led by Thomas Oxley. This was the first time a brain–computer interface was implanted via the patient's blood vessels, eliminating the need for open brain surgery. Currently a number of groups are exploring a range of experimental devices using brain–computer interfaces, which have the potential to fundamentally change the way of life for patients with paralysis and a wide range of neurological disorders. These include: as Elon Musk, Facebook, and the University of California in San Francisco. The systems. This technology is also being explored as a neuromodulation device and may ultimately help diagnose and treat a range of brain pathologies, such as epilepsy and Parkinson's disease.

We Appreciate Power

"We Appreciate Power" is a song by Canadian musician Grimes, featuring American musician Hana. It was released on November 29, 2018, billed as the lead single from her fifth studio album Miss Anthropocene, however it is only available on the Japanese and deluxe releases. The song was written and produced by Grimes, Poppy (originally), Hana and Chris Greatti. == Background and release == The song was supposed to be one of two collaborations between Grimes and American singer Poppy, for the latter's second studio album Am I a Girl?. In an interview, Poppy mentioned that she wrote two songs with Grimes; one about "destroying things" and another about "power". The other song, "Play Destroy", was featured on the album. Grimes shared a lyric of the song with a photo of her with Poppy on Twitter in May 2018. Following feuds between the two singers, the song was released by Grimes featuring singer Hana instead. On November 26, Grimes announced she would be releasing new music on November 29. Two days later, she revealed that the single is titled "We Appreciate Power" and features Hana, and shared the artwork. The release of the song was accompanied by a lyric video directed by Grimes and her brother Mac Boucher. == Music and lyrics == "We Appreciate Power" is an industrial rock, nu metal, and techno-industrial song. The track is regarded as a further step into Grimes's experimentation with guitars that started on 2015's Art Angels. The track was compared to the works of Nine Inch Nails; Jillian Mapes of Pitchfork described the song as "an immediate onslaught of mutilated noise—distorted metal guitar chug, bloody screams, a guitar loop that conjures fear and demands worship. Flashes of Nine Inch Nails' Pretty Hate Machine reverberate through the drum programming and synths." Brendan Klinkenberg of Rolling Stone placed the song "somewhere between power pop and straightforward industrial (with an extended bridge reminiscent of the most sweeping moments in a Final Fantasy score)" and "a distinctly 2018 take on Nine Inch Nails-esque hard-edged rock." A press release stated that the song was inspired by the North Korean band Moranbong and was written "from the perspective of a Pro-A.I. Girl Group Propaganda machine who use song, dance, sex and fashion to spread goodwill towards Artificial Intelligence." In addition Grimes stated that by simply listening to the song you will be reducing your risk of ending up on any future AI overlord's hit list when it reigns supreme, mirroring the Roko's basilisk theory. Lyrically, the song touches on transhumanist ideas such as the betterment and future of the human race, the possibilities of merging consciousness with machines to extend life indefinitely through mind uploading, and the idea that reality may be simulated. The song's chorus generated a spike in interest in the word "capitulate". == Critical reception == Pitchfork critic Jillian Mapes wrote: "If "Freak on a Leash" isn't a dealbreaker, then the supervillain allure of "We Appreciate Power" might pull you in (it legitimately slaps), but it just as well may leave you weighed down by Grimes' commitment to the absolute darkest timeline." Billboard's Gil Kaufman described the song as "a dystopian, aggressive dive into a more rock-leaning sound." Similarly, Brendan Klinkenberg of Rolling Stone called it "the most aggressive single Grimes has released to date" Noisey called the song "an absolute motherfucker of a single" and opined it sounds "like a K-pop band covering nu-metal". Justin Kamp of Paste described the track as a "glitchy empowerment anthem that chugs along on screeching synths and Grimes' repeated exultations of power." == Personnel == Credits adapted from Tidal. Grimes – vocals, guitar, production, engineering Hana – vocals, guitar, additional production Chris Greatti – guitar, keyboards, production, engineering Zakk Cervini – mixing == Track listing == == Charts ==

Possibility theory

Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. == Formalization of possibility == For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function Π {\displaystyle \Pi } from 2 Ω {\displaystyle 2^{\Omega }} to [0, 1] such that: Axiom 1: Π ( ∅ ) = 0 {\displaystyle \Pi (\varnothing )=0} Axiom 2: Π ( Ω ) = 1 {\displaystyle \Pi (\Omega )=1} Axiom 3: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any disjoint subsets U {\displaystyle U} and V {\displaystyle V} . It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons: Π ( U ) = max ω ∈ U Π ( { ω } ) . {\displaystyle \Pi (U)=\max _{\omega \in U}\Pi (\{\omega \}).} Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which Π {\displaystyle \Pi } was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any subsets U {\displaystyle U} and V {\displaystyle V} . Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: Π ( U ∩ V ) ≤ min ( Π ( U ) , Π ( V ) ) ≤ max ( Π ( U ) , Π ( V ) ) . {\displaystyle \Pi (U\cap V)\leq \min \left(\Pi (U),\Pi (V)\right)\leq \max \left(\Pi (U),\Pi (V)\right).} When Ω is not finite, Axiom 3 can be replaced by: For all index sets I {\displaystyle I} , if the subsets U i , i ∈ I {\displaystyle U_{i,\,i\in I}} are pairwise disjoint, Π ( ⋃ i ∈ I U i ) = sup i ∈ I Π ( U i ) . {\displaystyle \Pi \left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}\Pi (U_{i}).} == Necessity == Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set U {\displaystyle U} , the necessity measure is defined by N ( U ) = 1 − Π ( U ¯ ) {\displaystyle N(U)=1-\Pi ({\overline {U}})} . In the above formula, U ¯ {\displaystyle {\overline {U}}} denotes the complement of U {\displaystyle U} , that is the elements of Ω {\displaystyle \Omega } that do not belong to U {\displaystyle U} . It is straightforward to show that: N ( U ) ≤ Π ( U ) {\displaystyle N(U)\leq \Pi (U)} for any U {\displaystyle U} and that: N ( U ∩ V ) = min ( N ( U ) , N ( V ) ) {\displaystyle N(U\cap V)=\min(N(U),N(V))} . Note that contrary to probability theory, possibility is not self-dual. That is, for any event U {\displaystyle U} , we only have the inequality: Π ( U ) + Π ( U ¯ ) ≥ 1 {\displaystyle \Pi (U)+\Pi ({\overline {U}})\geq 1} However, the following duality rule holds: For any event U {\displaystyle U} , either Π ( U ) = 1 {\displaystyle \Pi (U)=1} , or N ( U ) = 0 {\displaystyle N(U)=0} Accordingly, beliefs about an event can be represented by a number and a bit. == Interpretation == There are four cases that can be interpreted as follows: N ( U ) = 1 {\displaystyle N(U)=1} means that U {\displaystyle U} is necessary. U {\displaystyle U} is certainly true. It implies that Π ( U ) = 1 {\displaystyle \Pi (U)=1} . Π ( U ) = 0 {\displaystyle \Pi (U)=0} means that U {\displaystyle U} is impossible. U {\displaystyle U} is certainly false. It implies that N ( U ) = 0 {\displaystyle N(U)=0} . Π ( U ) = 1 {\displaystyle \Pi (U)=1} means that U {\displaystyle U} is possible. I would not be surprised at all if U {\displaystyle U} occurs. It leaves N ( U ) {\displaystyle N(U)} unconstrained. N ( U ) = 0 {\displaystyle N(U)=0} means that U {\displaystyle U} is unnecessary. I would not be surprised at all if U {\displaystyle U} does not occur. It leaves Π ( U ) {\displaystyle \Pi (U)} unconstrained. The intersection of the last two cases is N ( U ) = 0 {\displaystyle N(U)=0} and Π ( U ) = 1 {\displaystyle \Pi (U)=1} meaning that I believe nothing at all about U {\displaystyle U} . Because it allows for indeterminacy like this, possibility theory relates to the graduation of a many-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example. Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle. Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full. == Possibility theory as an imprecise probability theory == There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. A possibility measure can be seen as a consonant plausibility measure in the Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set of admissible probability distributions by K = { P ∣ ∀ S P ( S ) ≤ Π ( S ) } . {\displaystyle K=\{\,P\mid \forall S\ P(S)\leq \Pi (S)\,\}.} This allows one to study possibility theory using the tools of imprecise probabilities. == Necessity logic == We call generalized possibility every function satisfying Axiom 1 and Axiom 3. We call generalized necessity the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called necessity logic. In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual modus ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).

SCIgen

SCIgen is a paper generator that uses context-free grammar to randomly generate nonsense in the form of computer science research papers. Its original data source was a collection of computer science papers downloaded from CiteSeer. All elements of the papers are formed, including graphs, diagrams, and citations. Created by scientists at the Massachusetts Institute of Technology, its stated aim is "to maximize amusement, rather than coherence." Originally created in 2005 to expose the lack of scrutiny of submissions to conferences, the generator subsequently became used, primarily by Chinese academics, to create large numbers of fraudulent conference submissions, leading to the retraction of 122 SCIgen generated papers and the creation of detection software to combat its use. == Sample output == Opening abstract of Rooter: A Methodology for the Typical Unification of Access Points and Redundancy: Many physicists would agree that, had it not been for congestion control, the evaluation of web browsers might never have occurred. In fact, few hackers worldwide would disagree with the essential unification of voice-over-IP and public/private key pair. In order to solve this riddle, we confirm that SMPs can be made stochastic, cacheable, and interposable. == Prominent results == In 2005, a paper generated by SCIgen, Rooter: A Methodology for the Typical Unification of Access Points and Redundancy, was accepted as a non-reviewed paper to the 2005 World Multiconference on Systemics, Cybernetics and Informatics (WMSCI) and the authors were invited to speak. The authors of SCIgen described their hoax on their website, and it soon received great publicity when picked up by Slashdot. WMSCI withdrew their invitation, but the SCIgen team went anyway, renting space in the hotel separately from the conference and delivering a series of randomly generated talks on their own "track". The organizer of these WMSCI conferences is Professor Nagib Callaos. From 2000 until 2005, the WMSCI was also sponsored by the Institute of Electrical and Electronics Engineers. The IEEE stopped granting sponsorship to Callaos from 2006 to 2008. Submitting the paper was a deliberate attempt to embarrass WMSCI, which the authors claim accepts low-quality papers and sends unsolicited requests for submissions in bulk to academics. As the SCIgen website states: One useful purpose for such a program is to auto-generate submissions to conferences that you suspect might have very low submission standards. A prime example, which you may recognize from spam in your inbox, is SCI/IIIS and its dozens of co-located conferences (check out the very broad conference description on the WMSCI 2005 website). Computing writer Stan Kelly-Bootle noted in ACM Queue that many sentences in the "Rooter" paper were individually plausible, which he regarded as posing a problem for automated detection of hoax articles. He suggested that even human readers might be taken in by the effective use of jargon ("The pun on root/router is par for MIT-graduate humor, and at least one occurrence of methodology is mandatory") and attribute the paper's apparent incoherence to their own limited knowledge. His conclusion was that "a reliable gibberish filter requires a careful holistic review by several peer domain experts". === Schlangemann === The pseudonym "Herbert Schlangemann" was used to publish fake scientific articles in international conferences that claimed to practice peer review. The name is taken from the Swedish short film Der Schlangemann. In 2008, in response to a series of Call-for-Paper e-mails, SCIgen was used to generate a false scientific paper titled Towards the Simulation of E-Commerce, using "Herbert Schlangemann" as the author. The article was accepted at the 2008 International Conference on Computer Science and Software Engineering (CSSE 2008), co-sponsored by the IEEE, to be held in Wuhan, China, and the author was invited to be a session chair on grounds of his fictional Curriculum Vitae. The official review comment: "This paper presents cooperative technology and classical Communication. In conclusion, the result shows that though the much-touted amphibious algorithm for the refinement of randomized algorithms is impossible, the well-known client-server algorithm for the analysis of voice-over-IP by Kumar and Raman runs in _(n) time. The authors can clearly identify important features of visualization of DHTs and analyze them insightfully. It is recommended that the authors should develop ideas more cogently, organizes them more logically, and connects them with clear transitions." The paper was available for a short time in the IEEE Xplore Database, but was then removed. The entire story is described in the official "Herbert Schlangemann" blog, and it also received attention in Slashdot and the German-language technology-news site Heise Online. In 2009, the same incident happened and Herbert Schlangemann's latest fake paper PlusPug: A Methodology for the Improvement of Local-Area Networks was accepted for oral presentation at the 2009 International Conference on e-Business and Information System Security (EBISS 2009), also co-sponsored by IEEE, to be held again in Wuhan, China. In all cases, the published papers were withdrawn from the conferences' proceedings, and the conference organizing committee as well as the names of the keynote speakers were removed from their websites. === List of works with notable acceptance === ==== In conferences ==== Rob Thomas: Rooter: A Methodology for the Typical Unification of Access Points and Redundancy, 2005 for WMSCI (see above) Mathias Uslar's paper was accepted to the IPSI-BG conference. Professor Genco Gulan published a paper in the 3rd International Symposium of Interactive Media Design. A 2013 scientometrics paper demonstrated that at least 85 SCIgen papers have been published by IEEE and Springer. Over 120 SCIgen papers were removed according to this research. ==== In journals ==== Students at Iran's Sharif University of Technology published a paper in Elsevier's Journal of Applied Mathematics and Computation. The students wrote under the surname "MosallahNejad", which translates literally from Persian language (in spite of not being a traditional Persian name) as "from an Armed Breed". The paper was subsequently removed when the publishers were informed that it was a joke paper. Mikhail Gelfand published a translation of the "Rooter" article in the Russian-language Journal of Scientific Publications of Aspirants and Doctorants in August 2008. Gelfand was protesting against the journal, which was apparently not peer-reviewed and was being used by Russian PhD candidates to publish in an "accredited" scientific journal, charging them 4,000 Rubles to do so. The accreditation was revoked two weeks later. (See Dissernet for related information.) Springer Science+Business Media and IEEE were also the subject of similar pranks. === Spoofing Google Scholar and h-index calculators === Refereeing performed on behalf of the Institute of Electrical and Electronics Engineers has also been subject to criticism after fake papers were discovered in conference publications, most notably by Labbé and a researcher using the pseudonym of Schlangemann. Cyril Labbé from Grenoble University demonstrated the vulnerability of h-index calculations based on Google Scholar output by feeding it a large set of SCIgen-generated documents that were citing each other, effectively an academic link farm, in a 2010 paper. Using this method the author managed to rank "Ike Antkare" ahead of Albert Einstein for instance. === 2013 retractions === In 2013, over 122 published conference papers created by SCIgen were retracted by Springer and the IEEE. Unlike previous submissions that were intended to be pranks, this submission were largely made by Chinese academics, who were using SCIgen papers to boost their publication record. === SciDetect === In 2015, SciDetect was released by Springer. This software, developed by Cyril Labbé, is designed to automatically detect papers generated by SCIgen. === 2021 report === In 2021, a study was published on 243 SCIgen papers that had been published in the academic literature. They found that SCIgen papers made up 75 per million papers (< 0.01%) in information science, and that only a small fraction of the detected papers had been dealt with.

CinePlayer

CinePlayer is a software based media player used to review Digital Cinema Packages (DCP) without the need for a digital cinema server by Doremi Labs. CinePlayer can play back any DCP, not just those created by Doremi Mastering products. In addition to playing DCPs, CinePlayer can also playback JPEG2000 image sequences and many popular multimedia file types. There are two versions of CinePlayer available, standard and Pro. The standard version supports playback of non-encrypted, 2D DCP's up to 2K resolution. The Pro version supports playback of encrypted, 2D or 3D DCP's with subtitles up to 4K resolution. == Supported formats == === Containers === AVI MOV MXF MPG TS WMV M2TS MTS MP4 MKV === Video codecs === JPEG2000 ProRes 422 DNxHD YUV Uncompressed 8-10 bits DIVX XVID MPEG4 AVC / H-264 VC-1 MPEG2 === Supported image sequences === BMP TIFF TGA DPX JPG J2C === Supported audio files === WAV MP3 WMA MP2

Model collapse

Model collapse, also known by other names such as "AI inbreeding", "AI cannibalism", "Habsburg AI", and "model autophagy disorder" or "MAD" is a phenomenon noted in artificial intelligence studies, where machine learning models gradually degrade due to errors coming from uncurated synthetic data, or due to training on the outputs of another model such as prior versions of itself. It is unclear to what extent the phenomenon threatens the long-term development of such models, and some techniques have been proposed to mitigate the effect. == Characteristics == Shumailov et al. coined the term to describe two specific stages to the degradation of machine learning models: early model collapse and late model collapse: In early model collapse, the model begins losing information about the tails of the distribution – mostly affecting minority data. Later work highlighted that early model collapse is hard to notice, since overall performance may appear to improve, while the model loses performance on minority data. In late model collapse, the model loses a significant proportion of its performance, confusing concepts and losing most of its variance. == Mechanism == Using synthetic data as training data can lead to issues with the quality and reliability of the trained model. Model collapse occurs for three main reasons: functional approximation errors sampling errors learning errors Importantly, it happens in even the simplest of models, where not all of the error sources are present. In more complex models the errors often compound, leading to faster collapse. == Disagreement over real-world impact == Some researchers and commentators on model collapse warn that the phenomenon could fundamentally threaten future generative AI development: As AI-generated data is shared on the Internet, it will inevitably end up in future training datasets, which are often crawled from the Internet. If training on "slop" (large quantities of unlabeled synthetic data) inevitably leads to model collapse, this could therefore pose a difficult problem. However, recently, other researchers have disagreed with this argument, showing that if synthetic data accumulates alongside human-generated data, model collapse is avoided. The researchers argue that data accumulating over time is a more realistic description of reality than deleting all existing data every year, and that the real-world impact of model collapse may not be as catastrophic as feared. An alternative branch of the literature investigates the use of machine learning detectors and watermarking to identify model generated data and filter it out. == Mathematical models of the phenomenon == === 1D Gaussian model === In 2024, a first attempt has been made at illustrating collapse for the simplest possible model — a single dimensional normal distribution fit using unbiased estimators of mean and variance, computed on samples from the previous generation. To make this more precise, we say that original data follows a normal distribution X 0 ∼ N ( μ , σ 2 ) {\displaystyle X^{0}\sim {\mathcal {N}}(\mu ,\sigma ^{2})} , and we possess M 0 {\displaystyle M_{0}} samples X j 0 {\displaystyle X_{j}^{0}} for j ∈ { 1 , … , M 0 } {\displaystyle j\in {\{\,1,\dots ,M_{0}\,{}\}}} . Denoting a general sample X j i {\displaystyle X_{j}^{i}} as sample j ∈ { 1 , … , M i } {\displaystyle j\in {\{\,1,\dots ,M_{i}\,{}\}}} at generation i {\displaystyle i} , then the next generation model is estimated using the sample mean and variance: μ i + 1 = 1 M i ∑ j X j i ; σ i + 1 2 = 1 M i − 1 ∑ j ( X j i − μ i + 1 ) 2 . {\displaystyle \mu _{i+1}={\frac {1}{M_{i}}}\sum _{j}X_{j}^{i};\quad \sigma _{i+1}^{2}={\frac {1}{M_{i}-1}}\sum _{j}(X_{j}^{i}-\mu _{i+1})^{2}.} Leading to a conditionally normal next generation model X j i + 1 | μ i + 1 , σ i + 1 ∼ N ( μ i + 1 , σ i + 1 2 ) {\displaystyle X_{j}^{i+1}|\mu _{i+1},\;\sigma _{i+1}\sim {\mathcal {N}}(\mu _{i+1},\sigma _{i+1}^{2})} . In theory, this is enough to calculate the full distribution of X j i {\displaystyle X_{j}^{i}} . However, even after the first generation, the full distribution is no longer normal: It follows a variance-gamma distribution. To continue the analysis, instead of writing the probability density function at each generation, it is possible to explicitly construct them in terms of independent random variables using Cochran's theorem. To be precise, μ 1 {\displaystyle \mu _{1}} and σ 1 {\displaystyle \sigma _{1}} are independent, with μ 1 ∼ N ( μ , σ 2 M 0 ) {\displaystyle \mu _{1}\sim {\mathcal {N}}\left(\mu ,{\frac {\sigma ^{2}}{M_{0}}}\right)} and ( M 0 − 1 ) σ 1 2 ∼ σ 2 Γ ( M 0 − 1 2 , 1 2 ) {\displaystyle (M_{0}-1)\,\sigma _{1}^{2}\sim \sigma ^{2}\,\Gamma \left({\frac {M_{0}-1}{2}},{\frac {1}{2}}\right)} , following a Gamma distribution. Denoting with Z {\displaystyle Z} Gaussian random variables distributed according to N ( 0 , 1 ) {\displaystyle {\mathcal {N}}(0,1)} and with S i {\displaystyle S^{i}} random variables distributed with 1 M i − 1 − 1 Γ ( M i − 1 − 1 2 , 1 2 ) {\displaystyle {\frac {1}{M_{i-1}-1}}\Gamma \left({\frac {M_{i-1}-1}{2}},{\frac {1}{2}}\right)} , it turns out to be possible to write samples at each generation as X j 0 = μ + σ Z j 0 , {\textstyle X_{j}^{0}=\mu +\sigma Z_{j}^{0},} X j 1 = μ + σ M 0 Z 1 + σ S 1 Z j 1 , {\textstyle X_{j}^{1}=\mu +{\frac {\sigma }{\sqrt {M_{0}}}}Z^{1}+\sigma {\sqrt {S^{1}}}Z_{j}^{1},} and more generally X j n = μ + σ M 0 Z 1 + σ M 1 S 1 Z 2 + ⋯ + σ M n − 1 S 1 × ⋯ × S n − 1 Z n + σ S 1 × ⋯ × S n Z j n . {\displaystyle X_{j}^{n}=\mu +{\frac {\sigma }{\sqrt {M_{0}}}}Z^{1}+{\frac {\sigma }{\sqrt {M_{1}}}}{\sqrt {S^{1}}}Z^{2}+\dots +{\frac {\sigma }{\sqrt {M_{n-1}}}}{\sqrt {S^{1}\times \dots \times S^{n-1}}}Z^{n}+\sigma {\sqrt {S^{1}\times \dots \times S^{n}}}Z_{j}^{n}.} Note, that these are not joint distributions, as Z n {\displaystyle Z^{n}} and S n {\displaystyle S^{n}} depend directly on Z j n − 1 {\displaystyle Z_{j}^{n-1}} , but when considering X j n {\displaystyle X_{j}^{n}} on its own the formula above provides all the information about the full distribution. To analyse the model collapse, we can first calculate variance and mean of samples at generation n {\displaystyle n} . This would tell us what kind of distributions we expect to arrive at after n {\displaystyle n} generations. It is possible to find its exact value in closed form, but the mean and variance of the square root of gamma distribution are expressed in terms of gamma functions, making the result quite clunky. Following, it is possible to expand all results to second order in each of 1 / M i {\displaystyle 1/M_{i}} , assuming each sample size to be large. It is then possible to show that 1 σ 2 Var ⁡ ( X j n ) = 1 M 0 + 1 M 1 + ⋯ + 1 M n − 1 + 1 + O ( M i − 2 ) . {\displaystyle {\frac {1}{\sigma ^{2}}}\operatorname {Var} (X_{j}^{n})={\frac {1}{M_{0}}}+{\frac {1}{M_{1}}}+\dots +{\frac {1}{M_{n-1}}}+1+{\mathcal {O}}\left(M_{i}^{-2}\right).} And if all sample sizes M i = M {\displaystyle M_{i}=M} are constant, this diverges linearly as n → ∞ {\displaystyle n\to \infty } : Var ⁡ ( X j n ) = σ 2 ( 1 + n M ) ; E ( X j n ) = μ . {\displaystyle \operatorname {Var} (X_{j}^{n})=\sigma ^{2}\left(1+{\frac {n}{M}}\right);\quad \mathbb {E} (X_{j}^{n})=\mu .} This is the same scaling as for a single dimensional Gaussian random walk. However, divergence of the variance of X j n {\displaystyle X_{j}^{n}} does not directly provide any information about the corresponding estimates of μ n + 1 {\displaystyle \mu _{n+1}} and σ n + 1 {\displaystyle \sigma _{n+1}} , particularly how different they are from the original μ {\displaystyle \mu } and σ {\displaystyle \sigma } . It turns out to be possible to calculate the distance between the true distribution and the approximated distribution at step n + 1 {\displaystyle n+1} , using the Wasserstein-2 distance (which is also sometimes referred to as risk): E [ W 2 2 ( N ( μ , σ 2 ) , N ( μ n + 1 , σ n + 1 2 ) ) ] = 3 2 σ 2 ( 1 M 0 + 1 M 1 + ⋯ + 1 M n ) + O ( M i − 2 ) , {\displaystyle \mathbb {E} \left[\mathbb {W} _{2}^{2}\left({\mathcal {N}}(\mu ,\sigma ^{2}),{\mathcal {N}}(\mu _{n+1},\sigma _{n+1}^{2})\right)\right]={\frac {3}{2}}\sigma ^{2}\left({\frac {1}{M_{0}}}+{\frac {1}{M_{1}}}+\dots +{\frac {1}{M_{n}}}\right)+{\mathcal {O}}\left(M_{i}^{-2}\right),} Var ⁡ [ W 2 2 ( N ( μ , σ 2 ) , N ( μ n + 1 , σ n + 1 2 ) ) ] = 1 2 σ 4 ( 3 M 0 2 + 3 M 1 2 + ⋯ + 3 M n 2 + ∑ i ≠ j 4 M i M j ) + O ( M i − 3 ) . {\displaystyle \operatorname {Var} \left[\mathbb {W} _{2}^{2}\left({\mathcal {N}}(\mu ,\sigma ^{2}),{\mathcal {N}}(\mu _{n+1},\sigma _{n+1}^{2})\right)\right]={\frac {1}{2}}\sigma ^{4}\left({\frac {3}{M_{0}^{2}}}+{\frac {3}{M_{1}^{2}}}+\dots +{\frac {3}{M_{n}^{2}}}+\sum _{i\neq j}{\frac {4}{M_{i}M_{j}}}\right)+{\mathcal {O}}\left(M_{i}^{-3}\right).} This directly shows why model collapse occurs in this simple model. Due to errors from re-sampling the approximated distribution, each generation ends up corresponding to a