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Huawei Mobile Services

Huawei Mobile Services (HMS) is a collection of proprietary services and high level application programming interfaces (APIs) developed by Huawei Technologies Co., Ltd. Its hub known as HMS Core serves as a toolkit for app development on Huawei devices. HMS is typically installed on Huawei devices on top of running HarmonyOS 4.x and earlier operating system on its earlier devices running the Android operating system with EMUI including devices already distributed with Google Mobile Services. Alongside, HMS Core Wear Engine for Android phones with lightweight based LiteOS wearable middleware app framework integration connectivity like notifications, status etc. HMS consists of seven key services and the HMS Core. The key services are Huawei ID, Huawei Cloud, AppGallery, Themes, Huawei Video, Browser, and Assistant. The web browser is based on Chromium. Huawei Quick Apps is the alternative to Google Instant Apps. By January 2020, over 50,000 apps had been integrated with HMS Core. Its rival, Google Mobile Services has 3 million apps on Google's Play Store. The AppGallery claimed 180 billion downloads in 2019. In March 2020, HMS was used by 650 million monthly active users across 170 countries. A Chinese phone manufacturer, LeTV, hosted a smartphone business communication meeting in Beijing on September 27, 2021, to demonstrate its phone, the LeTV S1. This was the first smartphone from a third-party manufacturer to include Huawei Mobile Services (HMS). == HMS on Android and HarmonyOS == Huawei Mobile Services on Android goes all the way back to August 2016 as Huawei ID services for phones, basic functionalities for Huawei P9 series. However, in May 2019 proved to be a significant change to HMS when Google was prohibited from working with Huawei on any new devices extending ecosystem for AppGallery store front launched in April 2018, year prior. This also included bundling Google's Apps, including Gmail, Maps and YouTube. Any new Huawei devices launched after 16 May 2019 were unable to receive updates from Google services and would be considered 'uncertified' meaning Huawei's only solution at the time was to turn HMS into a genuine competitor to Google and incentivize app developers to utilize the platform. Huawei officially launched Huawei Mobile Services in China on December 24, 2019, as a beta. Huawei expanded Huawei Mobile Services in Europe in February 2020 and other markets in Asia, Latin America, Middle East & Africa, Canada, Mexico followed outside banned US market. HMS is available on the Honor 9X Pro, View 30 Pro, Huawei Mate XS. HMS is also available, alongside GMS, on many other Huawei models launched before the ban. Huawei promised developers it would take, “less than 10 minutes", to port their app over to HMS - to illustrate the ease of portability between Google's Play Store and the HMS AppGallery. On January 15, 2020, HMS Core 4.0 (Huawei Mobile Services Core 4.0) was officially launched. Huawei announced that at this time, there were already 1.3 million developers and 55,000 applications on board. The next day, Huawei held a developer day event in London and invested £20 million to encourage developers in the United Kingdom and Ireland to use HMS. On July 15, 2021, Huawei expanded HMS with classic HarmonyOS dual-framework that provided Java support and eventually with JavaScript and ArkTS (eTS) language support with HMS Core 6.0 for app development with primarily Android apps, alongside limited HAP imperative developed based apps that shares AOSP file system libraries in all types of devices from smartphones, tablets, smart screens, smartwatches, and car machines. Including various third-party development frameworks, such as React Native, Cordova, etc. At HDC 2023, Huawei unveiled HarmonyOS 5, marking a total break from the hybrid Android derived platform. This shift replaced the legacy Android and classic HarmonyOS-based HMS SDK with a full native API developer kit SDK built solely on OpenHarmony. The architecture moved from middleware services to vertical integration path. In this new model, HMS Core libraries are no longer external add-ons but are bundled directly into the system and DevEco Studio as native HarmonyOS Kits. == HMS Core == HMS Core is a hub for Huawei Mobile Services and serves as a toolkit for app development on Huawei devices. The core comprises Development, Growth and Monetizing and was created as a replacement for Google Mobile Services (GMS) Core. HMS core services were available in more than 55,000 apps in June 2020; HMS Core 5.0 debuted in September 2020. HMS Core 6.0 was launched in June 2021 with extended support for Huawei Cloud services. In June 2021, the number of registered developers within the HMS ecosystem was 4 million, and the number of apps integrated with the HMS Core had reached 134,000. As of July 2022, registered developers within HMS ecosystem had grown to 5 million, and the number of apps integrated with the HMS Core reached 203,000. The number of apps had grown to 220,000 by 30 September 2022. == AppGallery == The AppGallery has a key rival, Google's Play Store on Android. The AppGallery is available in 170 countries, across 78 languages. == Reception == The reception of HMS is mixed, with the majority of discussion based around the key Google/Android apps which are not yet present on the AppGallery and whether or not this presents a significant problem to users. The open development of HMS Core has been regarded by some as benefiting the Android project as a whole, "If Huawei continues to invest in a holistically open approach ... the result could be that we could all end up a bit less beholden to Google".
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Pachinko allocation

In machine learning and natural language processing, the pachinko allocation model (PAM) is a topic model. Topic models are a suite of algorithms to uncover the hidden thematic structure of a collection of documents. The algorithm improves upon earlier topic models such as latent Dirichlet allocation (LDA) by modeling correlations between topics in addition to the word correlations which constitute topics. PAM provides more flexibility and greater expressive power than latent Dirichlet allocation. While first described and implemented in the context of natural language processing, the algorithm may have applications in other fields such as bioinformatics. The model is named for pachinko machines—a game popular in Japan, in which metal balls bounce down around a complex collection of pins until they land in various bins at the bottom. == History == Pachinko allocation was first described by Wei Li and Andrew McCallum in 2006. The idea was extended with hierarchical Pachinko allocation by Li, McCallum, and David Mimno in 2007. In 2007, McCallum and his colleagues proposed a nonparametric Bayesian prior for PAM based on a variant of the hierarchical Dirichlet process (HDP). The algorithm has been implemented in the MALLET software package published by McCallum's group at the University of Massachusetts Amherst. == Model == PAM connects words in V and topics in T with an arbitrary directed acyclic graph (DAG), where topic nodes occupy the interior levels and the leaves are words. The probability of generating a whole corpus is the product of the probabilities for every document: P ( D | α ) = ∏ d P ( d | α ) {\displaystyle P(\mathbf {D} |\alpha )=\prod _{d}P(d|\alpha )}
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AI Background Removers Reviews: What Actually Works in 2026

Trying to pick the best AI background remover? An AI background remover is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI background remover slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
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Best AI Sales Assistants in 2026

Shopping for the best AI sales assistant? An AI sales assistant is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI sales assistant slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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BBC Own It

The BBC Own It app was a British information site designed to protect and support children using the Internet. The app was launched in 2017 and retired in 2022, though the website retired in 2024 and has since moved to BBC Teach. As part of the BBC's partnership with Internet Matters, the not-for-profit contributed to content on the BBC Own It website. == History == In 2016, The Royal Foundation of The Duke and Duchess of Cambridge established The Royal Foundation Taskforce on the Prevention of Cyberbullying. Work began in 2017 by the BBC to create an app about cyberbullying and online safety (later titled Own It) in response to a call for action from the Taskforce. In December 2017, the BBC launched Own It. In November 2018, work on the BBC Own It App was announced by Prince William. In September 2019, the BBC Own It App was launched into the AppStore and Google Play. In 2022, the BBC discontinued the app, although the website was still active, however in 2024, the website was discontinued, and now any links to the website now redirect to a BBC Teach page. == Awards == UXUK award for Best Education or Learning Experience (2019) Banff World Media Festival Rockies Award for Children & Youth Interactive Content (2020) CogX Award for Best Innovation In Natural Language Processing (2020)
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Is an AI Code-review Tool Worth It in 2026?

Looking for the best AI code-review tool? An AI code-review tool is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI code-review tool slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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The Best Free AI Blog Writer for Beginners

Looking for the best AI blog writer? An AI blog writer is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI blog writer slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Bernhard Schölkopf

Bernhard Schölkopf (born 20 February 1968) is a German computer scientist known for his work in machine learning, especially on kernel methods and causality. He is a director at the Max Planck Institute for Intelligent Systems in Tübingen, Germany, where he heads the Department of Empirical Inference. He is also an affiliated professor at ETH Zürich, honorary professor at the University of Tübingen and Technische Universität Berlin, and chairman of the European Laboratory for Learning and Intelligent Systems (ELLIS). == Research == === Kernel methods === Schölkopf developed SVM methods achieving world record performance on the MNIST pattern recognition benchmark at the time. With the introduction of kernel PCA, Schölkopf and coauthors argued that SVMs are a special case of a much larger class of methods, and all algorithms that can be expressed in terms of dot products can be generalized to a nonlinear setting by means of what is known as reproducing kernels. Another significant observation was that the data on which the kernel is defined need not be vectorial, as long as the kernel Gram matrix is positive definite. Both insights together led to the foundation of the field of kernel methods, encompassing SVMs and many other algorithms. Kernel methods are now textbook knowledge and one of the major machine learning paradigms in research and applications. Developing kernel PCA, Schölkopf extended it to extract invariant features and to design invariant kernels and showed how to view other major dimensionality reduction methods such as LLE and Isomap as special cases. In further work with Alex Smola and others, he extended the SVM method to regression and classification with pre-specified sparsity and quantile/support estimation. He proved a representer theorem implying that SVMs, kernel PCA, and most other kernel algorithms, regularized by a norm in a reproducing kernel Hilbert space, have solutions taking the form of kernel expansions on the training data, thus reducing an infinite dimensional optimization problem to a finite dimensional one. He co-developed kernel embeddings of distributions methods to represent probability distributions in Hilbert Spaces, with links to Fraunhofer diffraction as well as applications to independence testing. === Causality === Starting in 2005, Schölkopf turned his attention to causal inference. Causal mechanisms in the world give rise to statistical dependencies as epiphenomena, but only the latter are exploited by popular machine learning algorithms. Knowledge about causal structures and mechanisms is useful by letting us predict not only future data coming from the same source, but also the effect of interventions in a system, and by facilitating transfer of detected regularities to new situations. Schölkopf and co-workers addressed (and in certain settings solved) the problem of causal discovery for the two-variable setting and connected causality to Kolmogorov complexity. Around 2010, Schölkopf began to explore how to use causality for machine learning, exploiting assumptions of independence of mechanisms and invariance. His early work on causal learning was exposed to a wider machine learning audience during his Posner lecture at NeurIPS 2011, as well as in a keynote talk at ICML 2017. He assayed how to exploit underlying causal structures in order to make machine learning methods more robust with respect to distribution shifts and systematic errors, the latter leading to the discovery of a number of new exoplanets including K2-18b, which was subsequently found to contain water vapour in its atmosphere, a first for an exoplanet in the habitable zone. == Education and employment == Schölkopf studied mathematics, physics, and philosophy in Tübingen and London. He was supported by the Studienstiftung and won the Lionel Cooper Memorial Prize for the best M.Sc. in Mathematics at the University of London. He completed a Diplom in Physics, and then moved to Bell Labs in New Jersey, where he worked with Vladimir Vapnik, who became co-adviser of his PhD thesis at TU Berlin (with Stefan Jähnichen). His thesis, defended in 1997, won the annual award of the German Informatics Association. In 2001, following positions in Berlin, Cambridge and New York, he founded the Department for Empirical Inference at the Max Planck Institute for Biological Cybernetics, which grew into a leading center for research in machine learning. In 2011, he became founding director at the Max Planck Institute for Intelligent Systems. With Alex Smola, Schölkopf co-founded the series of Machine Learning Summer Schools. He also co-founded a Cambridge-Tübingen PhD Programme and the Max Planck-ETH Center for Learning Systems. In 2016, he co-founded the Cyber Valley research consortium. He participated in the IEEE Global Initiative on "Ethically Aligned Design". Schölkopf is co-editor-in-Chief of the Journal of Machine Learning Research, a journal he helped found, being part of a mass resignation of the editorial board of Machine Learning (journal). He is among the world’s most cited computer scientists. Alumni of his lab include Ulrike von Luxburg, Carl Rasmussen, Matthias Hein, Arthur Gretton, Gunnar Rätsch, Matthias Bethge, Stefanie Jegelka, Jason Weston, Olivier Bousquet, Olivier Chapelle, Joaquin Quinonero-Candela, and Sebastian Nowozin. As of late 2023, Schölkopf is also a scientific advisor to French research group Kyutai which is being funded by Xavier Niel, Rodolphe Saadé, Eric Schmidt, and others. == Awards and recognition == Schölkopf’s awards include the Royal Society Milner Award and, shared with Isabelle Guyon and Vladimir Vapnik, the BBVA Foundation Frontiers of Knowledge Award in the Information and Communication Technologies category. He was the first scientist working in Europe to receive this award. He was elected a Fellow of the Royal Society in 2026.
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Feature hashing

In machine learning, feature hashing, also known as the hashing trick (by analogy to the kernel trick), is a fast and space-efficient way of vectorizing features, i.e. turning arbitrary features into indices in a vector or matrix. It works by applying a hash function to the features and using their hash values as indices directly (after a modulo operation), rather than looking the indices up in an associative array. In addition to its use for encoding non-numeric values, feature hashing can also be used for dimensionality reduction. This trick is often attributed to Weinberger et al. (2009), but there exists a much earlier description of this method published by John Moody in 1989. == Motivation == === Motivating example === In a typical document classification task, the input to the machine learning algorithm (both during learning and classification) is free text. From this, a bag of words (BOW) representation is constructed: the individual tokens are extracted and counted, and each distinct token in the training set defines a feature (independent variable) of each of the documents in both the training and test sets. Machine learning algorithms, however, are typically defined in terms of numerical vectors. Therefore, the bags of words for a set of documents is regarded as a term-document matrix where each row is a single document, and each column is a single feature/word; the entry i, j in such a matrix captures the frequency (or weight) of the j'th term of the vocabulary in document i. (An alternative convention swaps the rows and columns of the matrix, but this difference is immaterial.) Typically, these vectors are extremely sparse—according to Zipf's law. The common approach is to construct, at learning time or prior to that, a dictionary representation of the vocabulary of the training set, and use that to map words to indices. Hash tables and tries are common candidates for dictionary implementation. E.g., the three documents John likes to watch movies. Mary likes movies too. John also likes football. can be converted, using the dictionary to the term-document matrix ( John likes to watch movies Mary too also football 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 ) {\displaystyle {\begin{pmatrix}{\textrm {John}}&{\textrm {likes}}&{\textrm {to}}&{\textrm {watch}}&{\textrm {movies}}&{\textrm {Mary}}&{\textrm {too}}&{\textrm {also}}&{\textrm {football}}\\1&1&1&1&1&0&0&0&0\\0&1&0&0&1&1&1&0&0\\1&1&0&0&0&0&0&1&1\end{pmatrix}}} (Punctuation was removed, as is usual in document classification and clustering.) The problem with this process is that such dictionaries take up a large amount of storage space and grow in size as the training set grows. On the contrary, if the vocabulary is kept fixed and not increased with a growing training set, an adversary may try to invent new words or misspellings that are not in the stored vocabulary so as to circumvent a machine learned filter. To address this challenge, Yahoo! Research attempted to use feature hashing for their spam filters. Note that the hashing trick isn't limited to text classification and similar tasks at the document level, but can be applied to any problem that involves large (perhaps unbounded) numbers of features. === Mathematical motivation === Mathematically, a token is an element t {\displaystyle t} in a finite (or countably infinite) set T {\displaystyle T} . Suppose we only need to process a finite corpus, then we can put all tokens appearing in the corpus into T {\displaystyle T} , meaning that T {\displaystyle T} is finite. However, suppose we want to process all possible words made of the English letters, then T {\displaystyle T} is countably infinite. Most neural networks can only operate on real vector inputs, so we must construct a "dictionary" function ϕ : T → R n {\displaystyle \phi :T\to \mathbb {R} ^{n}} . When T {\displaystyle T} is finite, of size | T | = m ≤ n {\displaystyle |T|=m\leq n} , then we can use one-hot encoding to map it into R n {\displaystyle \mathbb {R} ^{n}} . First, arbitrarily enumerate T = { t 1 , t 2 , . . , t m } {\displaystyle T=\{t_{1},t_{2},..,t_{m}\}} , then define ϕ ( t i ) = e i {\displaystyle \phi (t_{i})=e_{i}} . In other words, we assign a unique index i {\displaystyle i} to each token, then map the token with index i {\displaystyle i} to the unit basis vector e i {\displaystyle e_{i}} . One-hot encoding is easy to interpret, but it requires one to maintain the arbitrary enumeration of T {\displaystyle T} . Given a token t ∈ T {\displaystyle t\in T} , to compute ϕ ( t ) {\displaystyle \phi (t)} , we must find out the index i {\displaystyle i} of the token t {\displaystyle t} . Thus, to implement ϕ {\displaystyle \phi } efficiently, we need a fast-to-compute bijection h : T → { 1 , . . . , m } {\displaystyle h:T\to \{1,...,m\}} , then we have ϕ ( t ) = e h ( t ) {\displaystyle \phi (t)=e_{h(t)}} . In fact, we can relax the requirement slightly: It suffices to have a fast-to-compute injection h : T → { 1 , . . . , n } {\displaystyle h:T\to \{1,...,n\}} , then use ϕ ( t ) = e h ( t ) {\displaystyle \phi (t)=e_{h(t)}} . In practice, there is no simple way to construct an efficient injection h : T → { 1 , . . . , n } {\displaystyle h:T\to \{1,...,n\}} . However, we do not need a strict injection, but only an approximate injection. That is, when t ≠ t ′ {\displaystyle t\neq t'} , we should probably have h ( t ) ≠ h ( t ′ ) {\displaystyle h(t)\neq h(t')} , so that probably ϕ ( t ) ≠ ϕ ( t ′ ) {\displaystyle \phi (t)\neq \phi (t')} . At this point, we have just specified that h {\displaystyle h} should be a hashing function. Thus we reach the idea of feature hashing. == Algorithms == === Feature hashing (Weinberger et al. 2009) === The basic feature hashing algorithm presented in (Weinberger et al. 2009) is defined as follows. First, one specifies two hash functions: the kernel hash h : T → { 1 , 2 , . . . , n } {\displaystyle h:T\to \{1,2,...,n\}} , and the sign hash ζ : T → { − 1 , + 1 } {\displaystyle \zeta :T\to \{-1,+1\}} . Next, one defines the feature hashing function: ϕ : T → R n , ϕ ( t ) = ζ ( t ) e h ( t ) {\displaystyle \phi :T\to \mathbb {R} ^{n},\quad \phi (t)=\zeta (t)e_{h(t)}} Finally, extend this feature hashing function to strings of tokens by ϕ : T ∗ → R n , ϕ ( t 1 , . . . , t k ) = ∑ j = 1 k ϕ ( t j ) {\displaystyle \phi :T^{}\to \mathbb {R} ^{n},\quad \phi (t_{1},...,t_{k})=\sum _{j=1}^{k}\phi (t_{j})} where T ∗ {\displaystyle T^{}} is the set of all finite strings consisting of tokens in T {\displaystyle T} . Equivalently, ϕ ( t 1 , . . . , t k ) = ∑ j = 1 k ζ ( t j ) e h ( t j ) = ∑ i = 1 n ( ∑ j : h ( t j ) = i ζ ( t j ) ) e i {\displaystyle \phi (t_{1},...,t_{k})=\sum _{j=1}^{k}\zeta (t_{j})e_{h(t_{j})}=\sum _{i=1}^{n}\left(\sum _{j:h(t_{j})=i}\zeta (t_{j})\right)e_{i}} ==== Geometric properties ==== We want to say something about the geometric property of ϕ {\displaystyle \phi } , but T {\displaystyle T} , by itself, is just a set of tokens, we cannot impose a geometric structure on it except the discrete topology, which is generated by the discrete metric. To make it nicer, we lift it to T → R T {\displaystyle T\to \mathbb {R} ^{T}} , and lift ϕ {\displaystyle \phi } from ϕ : T → R n {\displaystyle \phi :T\to \mathbb {R} ^{n}} to ϕ : R T → R n {\displaystyle \phi :\mathbb {R} ^{T}\to \mathbb {R} ^{n}} by linear extension: ϕ ( ( x t ) t ∈ T ) = ∑ t ∈ T x t ζ ( t ) e h ( t ) = ∑ i = 1 n ( ∑ t : h ( t ) = i x t ζ ( t ) ) e i {\displaystyle \phi ((x_{t})_{t\in T})=\sum _{t\in T}x_{t}\zeta (t)e_{h(t)}=\sum _{i=1}^{n}\left(\sum _{t:h(t)=i}x_{t}\zeta (t)\right)e_{i}} There is an infinite sum there, which must be handled at once. There are essentially only two ways to handle infinities. One may impose a metric, then take its completion, to allow well-behaved infinite sums, or one may demand that nothing is actually infinite, only potentially so. Here, we go for the potential-infinity way, by restricting R T {\displaystyle \mathbb {R} ^{T}} to contain only vectors with finite support: ∀ ( x t ) t ∈ T ∈ R T {\displaystyle \forall (x_{t})_{t\in T}\in \mathbb {R} ^{T}} , only finitely many entries of ( x t ) t ∈ T {\displaystyle (x_{t})_{t\in T}} are nonzero. Define an inner product on R T {\displaystyle \mathbb {R} ^{T}} in the obvious way: ⟨ e t , e t ′ ⟩ = { 1 , if t = t ′ , 0 , else. ⟨ x , x ′ ⟩ = ∑ t , t ′ ∈ T x t x t ′ ⟨ e t , e t ′ ⟩ {\displaystyle \langle e_{t},e_{t'}\rangle ={\begin{cases}1,{\text{ if }}t=t',\\0,{\text{ else.}}\end{cases}}\quad \langle x,x'\rangle =\sum _{t,t'\in T}x_{t}x_{t'}\langle e_{t},e_{t'}\rangle } As a side note, if T {\displaystyle T} is infinite, then the inner product space R T {\displaystyle \mathbb {R} ^{T}} is not complete. Taking its completion would get us to a Hilbert space, which allows well-behaved infinite sums. Now we have an inner product space, with enough structure to describe the geometry of the feature hashing function ϕ : R T → R n {\displaystyle \phi :\ma
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Internettolken

Internettolken (or InternetPreter) is a web-based machine translating tool. As the first Swedish online translating service, it was started in 2002 and included the English and Swedish languages. Today, there are 14 languages with more than 120 possible combinations. The service is free up to 150 words per day, and as a 2,000-word free testing account. It is available both on its website, and as a gadget on iGoogle. The interface is either English or Swedish. Being a dictionary-based tool, with its own translation software, it can sometimes offer a more accurate translation than Google Translate and others, although the grammar will be incorrect. == Languages currently available ==
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Cognitive computer

A cognitive computer is a computer that hardwires artificial intelligence and machine learning algorithms into an integrated circuit that closely reproduces the behavior of the human brain. It generally adopts a neuromorphic engineering approach. Synonyms include neuromorphic chip and cognitive chip. In 2023, IBM's proof-of-concept NorthPole chip (optimized for 2-, 4- and 8-bit precision) achieved remarkable performance in image recognition. In 2013, IBM developed Watson, a cognitive computer that uses neural networks and deep learning techniques. The following year, it developed the 2014 TrueNorth microchip architecture which is designed to be closer in structure to the human brain than the von Neumann architecture used in conventional computers. In 2017, Intel also announced its version of a cognitive chip in "Loihi, which it intended to be available to university and research labs in 2018. Intel (most notably with its Pohoiki Beach and Springs systems), Qualcomm, and others are improving neuromorphic processors steadily. == IBM TrueNorth chip == TrueNorth was a neuromorphic CMOS integrated circuit produced by IBM in 2014. It is a manycore processor network on a chip design, with 4096 cores, each one having 256 programmable simulated neurons for a total of just over a million neurons. In turn, each neuron has 256 programmable "synapses" that convey the signals between them. Hence, the total number of programmable synapses is just over 268 million (228). Its basic transistor count is 5.4 billion. In 2023 Zhejiang University and Alibaba developed Darwin a neuromorphic chip The darwin3 chip was designed around 2023 so it is fairly modern compared to IBM's TrueNorth or Intel's LoihI. === Details === Memory, computation, and communication are handled in each of the 4096 neurosynaptic cores, TrueNorth circumvents the von Neumann-architecture bottleneck and is very energy-efficient, with IBM claiming a power consumption of 70 milliwatts and a power density that is 1/10,000th of conventional microprocessors. The SyNAPSE chip operates at lower temperatures and power because it only draws power necessary for computation. Skyrmions have been proposed as models of the synapse on a chip. The neurons are emulated using a Linear-Leak Integrate-and-Fire (LLIF) model, a simplification of the leaky integrate-and-fire model. According to IBM, it does not have a clock, operates on unary numbers, and computes by counting to a maximum of 19 bits. The cores are event-driven by using both synchronous and asynchronous logic, and are interconnected through an asynchronous packet-switched mesh network on chip (NOC). IBM developed a new network to program and use TrueNorth. It included a simulator, a new programming language, an integrated programming environment, and libraries. This lack of backward compatibility with any previous technology (e.g., C++ compilers) poses serious vendor lock-in risks and other adverse consequences that may prevent it from commercialization in the future. === Research === In 2018, a cluster of TrueNorth network-linked to a master computer was used in stereo vision research that attempted to extract the depth of rapidly moving objects in a scene. == IBM NorthPole chip == In 2023, IBM released its NorthPole chip, which is a proof-of-concept for dramatically improving performance by intertwining compute with memory on-chip, thus eliminating the Von Neumann bottleneck. It blends approaches from IBM's 2014 TrueNorth system with modern hardware designs to achieve speeds about 4,000 times faster than TrueNorth. It can run ResNet-50 or Yolo-v4 image recognition tasks about 22 times faster, with 25 times less energy and 5 times less space, when compared to GPUs which use the same 12-nm node process that it was fabricated with. It includes 224 MB of RAM and 256 processor cores and can perform 2,048 operations per core per cycle at 8-bit precision, and 8,192 operations at 2-bit precision. It runs at between 25 and 425 MHz. This is an inferencing chip, but it cannot yet handle GPT-4 because of memory and accuracy limitations == Intel Loihi chip == === Pohoiki Springs === Pohoiki Springs is a system that incorporates Intel's self-learning neuromorphic chip, named Loihi, introduced in 2017, perhaps named after the Hawaiian seamount Lōʻihi. Intel claims Loihi is about 1000 times more energy efficient than general-purpose computing systems used to train neural networks. In theory, Loihi supports both machine learning training and inference on the same silicon independently of a cloud connection, and more efficiently than convolutional neural networks or deep learning neural networks. Intel points to a system for monitoring a person's heartbeat, taking readings after events such as exercise or eating, and using the chip to normalize the data and work out the ‘normal’ heartbeat. It can then spot abnormalities and deal with new events or conditions. The first iteration of the chip was made using Intel's 14 nm fabrication process and houses 128 clusters of 1,024 artificial neurons each for a total of 131,072 simulated neurons. This offers around 130 million synapses, far less than the human brain's 800 trillion synapses, and behind IBM's TrueNorth. Loihi is available for research purposes among more than 40 academic research groups as a USB form factor. In October 2019, researchers from Rutgers University published a research paper to demonstrate the energy efficiency of Intel's Loihi in solving simultaneous localization and mapping. In March 2020, Intel and Cornell University published a research paper to demonstrate the ability of Intel's Loihi to recognize different hazardous materials, which could eventually aid to "diagnose diseases, detect weapons and explosives, find narcotics, and spot signs of smoke and carbon monoxide". === Pohoiki Beach === Intel's Loihi 2, named Pohoiki Beach, was released in September 2021 with 64 cores. It boasts faster speeds, higher-bandwidth inter-chip communications for enhanced scalability, increased capacity per chip, a more compact size due to process scaling, and improved programmability. === Hala Point === Hala Point packages 1,152 Loihi 2 processors produced on Intel 3 process node in a six-rack-unit chassis. The system supports up to 1.15 billion neurons and 128 billion synapses distributed over 140,544 neuromorphic processing cores, consuming 2,600 watts of power. It includes over 2,300 embedded x86 processors for ancillary computations. Intel claimed in 2024 that Hala Point was the world’s largest neuromorphic system. It uses Loihi 2 chips. It is claimed to offer 10x more neuron capacity and up to 12x higher performance. The Darwin3 chip exceeds these specs. Hala Point provides up to 20 quadrillion operations per second, (20 petaops), with efficiency exceeding 15 trillion (8-bit) operations s−1 W−1 on conventional deep neural networks. Hala Point integrates processing, memory and communication channels in a massively parallelized fabric, providing 16 PB s−1 of memory bandwidth, 3.5 PB s−1 of inter-core communication bandwidth, and 5 TB s−1 of inter-chip bandwidth. The system can process its 1.15 billion neurons 20 times faster than a human brain. Its neuron capacity is roughly equivalent to that of an owl brain or the cortex of a capuchin monkey. Loihi-based systems can perform inference and optimization using 100 times less energy at speeds as much as 50 times faster than CPU/GPU architectures. Intel claims that Hala Point can create LLMs. Much further research is needed == SpiNNaker == SpiNNaker (Spiking Neural Network Architecture) is a massively parallel, manycore supercomputer architecture designed by the Advanced Processor Technologies Research Group at the Department of Computer Science, University of Manchester. == Criticism == Critics argue that a room-sized computer – as in the case of IBM's Watson – is not a viable alternative to a three-pound human brain. Some also cite the difficulty for a single system to bring so many elements together, such as the disparate sources of information as well as computing resources. In 2021, The New York Times released Steve Lohr's article "What Ever Happened to IBM’s Watson?". He wrote about some costly failures of IBM Watson. One of them, a cancer-related project called the Oncology Expert Advisor, was abandoned in 2016 as a costly failure. During the collaboration, Watson could not use patient data. Watson struggled to decipher doctors’ notes and patient histories. The development of LLMs has placed a new emphasis on cognitive computers, because the Transformer technology that underpins LLMs demands huge energy for GPUs and PCs. Cognitive computers use significantly less energy, but the details of STDPs and neuron models cannot yet match the accuracy of backprop, and so ANN to SNN weight translations such as QAT and PQT or progressive quantization are becoming popular, with their own limitations.
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Collostructional analysis

Collostructional analysis is a family of methods developed by (in alphabetical order) Stefan Th. Gries (University of California, Santa Barbara) and Anatol Stefanowitsch (Free University of Berlin). Collostructional analysis aims at measuring the degree of attraction or repulsion that words exhibit to constructions, where the notion of construction has so far been that of Goldberg's construction grammar. == Collostructional methods == Collostructional analysis so far comprises three different methods: collexeme analysis, to measure the degree of attraction/repulsion of a lemma to a slot in one particular construction; distinctive collexeme analysis, to measure the preference of a lemma to one particular construction over another, functionally similar construction; multiple distinctive collexeme analysis extends this approach to more than two alternative constructions; covarying collexeme analysis, to measure the degree of attraction of lemmas in one slot of a construction to lemmas in another slot of the same construction. == Input frequencies == Collostructional analysis requires frequencies of words and constructions and is similar to a wide variety of collocation statistics. It differs from raw frequency counts by providing not only observed co-occurrence frequencies of words and constructions, but also (i) a comparison of the observed frequency to the one expected by chance; thus, collostructional analysis can distinguish attraction and repulsion of words and constructions; (ii) a measure of the strength of the attraction or repulsion; this is usually the log-transformed p-value of a Fisher-Yates exact test. == Versus other collocation statistics == Collostructional analysis differs from most collocation statistics such that (i) it measures not the association of words to words, but of words to syntactic patterns or constructions; thus, it takes syntactic structure more seriously than most collocation-based analyses; (ii) it has so far only used the most precise statistics, namely the Fisher-Yates exact test based on the hypergeometric distribution; thus, unlike t-scores, z-scores, chi-square tests etc., the analysis is not based on, and does not violate, any distributional assumptions.
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Color histogram

In image processing and photography, a color histogram is a representation of the distribution of colors in an image. For digital images, a color histogram represents the number of pixels that have colors in each of a fixed list of color ranges that span the image's color space (the set of all possible colors). A color histogram can be built for any kind of color space, although the term is more often used for three-dimensional spaces such as RGB or HSV. For monochromatic images, the term intensity histogram may be used instead. For multi-spectral images, where each pixel is represented by an arbitrary number of measurements (for example, beyond the three measurements in RGB), a color histogram is N-dimensional, with N being the number of measurements taken. Each measurement has its own wavelength range of the light spectrum, some of which may be outside the visible spectrum. If the set of possible color values is sufficiently small, each of those colors may be placed on a range by itself; then the histogram is merely the count of pixels that have each possible color. Most often, the space is divided into an appropriate number of ranges, often arranged as a regular grid, each containing many similar color values. A color histogram may also be represented and displayed as a smooth function defined over the color space that approximates the pixel counts. Like other kinds of histograms, a color histogram is a statistic that can be viewed as an approximation of an underlying continuous distribution of color values. == Overview == Color histograms are flexible constructs that can be built from images in various color spaces, whether RGB, rg chromaticity or any other color space of any dimension. A histogram of an image is produced first by discretization of the colors in the image into a number of bins, and counting the number of image pixels in each bin. For example, a red–blue chromaticity histogram can be formed by first normalizing color pixel values by dividing RGB values by R+G+B, then quantizing the normalized R and B coordinates into N bins each. A two-dimensional histogram of red–blue chromaticity divided into four bins (N=4) may yield a histogram similar to this table: A histogram can be N-dimensional. Although harder to display, a three-dimensional color histogram for the above example could be thought of as four separate red–blue histograms, where each of the four histograms contains the red–blue values for a bin of green (0–63, 64–127, 128–191, and 192–255). The histogram provides a compact summarization of the distribution of data in an image. A color histogram of an image is relatively invariant with translation and rotation about the viewing axis, and varies only slowly with the angle of view. By comparing histogram signatures of two images and matching the color content of one image with the other, a color histogram is particularly well suited for the problem of recognizing an object of unknown position and rotation within a scene. Importantly, translation of an RGB image into the illumination invariant rg-chromaticity space allows the histogram to operate well in varying light levels. 1. What is a histogram? A histogram is a graphical representation of the number of pixels in an image. In a more simple way to explain, a histogram is a bar graph, whose X-axis represents the tonal scale (black at the left and white at the right), and Y-axis represents the number of pixels in an image in a certain area of the tonal scale. For example, the graph of a luminance histogram shows the number of pixels for each brightness level (from black to white), and when there are more pixels, the peak at the certain luminance level is higher. 2. What is a color histogram? A color histogram of an image represents the distribution of the composition of colors in the image. It shows different types of colors appeared and the number of pixels in each type of the colors appeared. The relation between a color histogram and a luminance histogram is that a color histogram can be also expressed as “three luminance histograms”, each of which shows the brightness distribution of each individual red/green/blue color channel. == Characteristics of a color histogram == A color histogram focuses only on the proportion of the number of different types of colors, regardless of the spatial location of the colors. The values of a color histogram are from statistics. They show the statistical distribution of colors and the essential tone of an image. In general, as the color distributions of the foreground and background in an image are different, there might be a bimodal distribution in the histogram. For the luminance histogram alone, there is no perfect histogram and in general, the histogram can tell whether it is over-exposure or not, but there are times when you might think the image is over exposed by viewing the histogram; however, in reality it is not. == Principles of the formation of a color histogram == The formation of a color histogram is rather simple. From the definition above, we can simply count the number of pixels for each 256 scales in each of the 3 RGB channel, and plot them on 3 individual bar graphs. In general, a color histogram is based on a certain color space, such as RGB or HSV. When we compute the pixels of different colors in an image, if the color space is large, then we can first divide the color space into certain numbers of small intervals. Each of the intervals is called a bin. This process is called color quantization. Then, by counting the number of pixels in each of the bins, we get a color histogram of the image. The concrete steps of the principles can be viewed in Example 1. == Examples == === Example 1 === Given the following image of a cat (an original version and a version that has been reduced to 256 colors for easy histogram purposes), the following data represents a color histogram in the RGB color space, using four bins. Bin 0 corresponds to intensities 0–63 Bin 1 is 64–127 Bin 2 is 128–191 and Bin 3 is 192–255. === Example 2 === Application in camera: Nowadays, some cameras have the ability to show the 3 color histograms when we take photos. We can examine clips (spikes on either the black or white side of the scale) in each of the 3 RGB color histograms. If we find one or more clipping on a channel of the 3 RGB channels, then this would result in a loss of detail for that color. To illustrate this, consider this example: We know that each of the three R, G, B channels has a range of values from 0 to 255 (8 bit). So consider a photo that has a luminance range of 0–255. Assume the photo we take is made of 4 blocks that are adjacent to each other and we set the luminance scale for each of the 4 blocks of original photo to be 10, 100, 205, 245. Thus, the image looks like the topmost figure on the right. Then, we overexpose the photo a little, say, the luminance scale of each block is increased by 10. Thus, the luminance scale for each of the 4 blocks of new photo is 20, 110, 215, 255. Then, the image looks like the second figure on the right. There is not much difference between both figures, all we can see is that the whole image becomes brighter (the contrast for each of the blocks remain the same). Now, we overexpose the original photo again, this time the luminance scale of each block is increased by 50. Thus, the luminance scale for each of the 4 blocks of the new photo is 60, 150, 255, 255. The new image now looks like the third figure on the right. Note that the scale for the last block is 255 instead of 295, for 255 is the top scale and thus the last block has clipped. When this happens, we lose the contrast of the last 2 blocks, and thus we cannot recover the image no matter how we adjust it. To conclude, when taking photos with a camera that displays histograms, always keep the brightest tone in the image below the largest scale 255 on the histogram in order to avoid losing details. == Drawbacks and other approaches == The main drawback of histograms for classification is that the representation is dependent on the color of the object being studied, ignoring its shape and texture. Color histograms can potentially be identical for two images with different object content which happens to share color information. Conversely, without spatial or shape information, similar objects of different color may be indistinguishable based solely on color histogram comparisons. There is no way to distinguish a red and white cup from a red and white plate. Put it another way: histogram-based algorithms have no concept of a generic 'cup', and a model of a red and white cup is no use when given an otherwise identical blue and white cup. Another problem is that color histograms have high sensitivity to noisy interference such as lighting intensity changes and quantization errors. High dimensionality (bins) color histograms are also another issue. Some color histogram feature spaces often occupy more than one hundred di
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Interlingual machine translation

Interlingual machine translation is one of the classic approaches to machine translation. In this approach, the source language, i.e. the text to be translated is transformed into an interlingua, i.e., an abstract language-independent representation. The target language is then generated from the interlingua. Within the rule-based machine translation paradigm, the interlingual approach is an alternative to the direct approach and the transfer approach. In the direct approach, words are translated directly without passing through an additional representation. In the transfer approach the source language is transformed into an abstract, less language-specific representation. Linguistic rules which are specific to the language pair then transform the source language representation into an abstract target language representation and from this the target sentence is generated. The interlingual approach to machine translation has advantages and disadvantages. The advantages are that it requires fewer components in order to relate each source language to each target language, it takes fewer components to add a new language, it supports paraphrases of the input in the original language, it allows both the analysers and generators to be written by monolingual system developers, and it handles languages that are very different from each other (e.g. English and Arabic). The obvious disadvantage is that the definition of an interlingua is difficult and maybe even impossible for a wider domain. The ideal context for interlingual machine translation is thus multilingual machine translation in a very specific domain. For example, Interlingua has been used as a pivot language in international conferences and has been proposed as a pivot language for the European Union. == History == The first ideas about interlingual machine translation appeared in the 17th century with Descartes and Leibniz, who came up with theories of how to create dictionaries using universal numerical codes, not unlike numerical tokens used by large language models nowadays. Others, such as Cave Beck, Athanasius Kircher and Johann Joachim Becher worked on developing an unambiguous universal language based on the principles of logic and iconographs. In 1668, John Wilkins described his interlingua in his "Essay towards a Real Character and a Philosophical Language". In the 18th and 19th centuries many proposals for "universal" international languages were developed, the most well known being Esperanto. That said, applying the idea of a universal language to machine translation did not appear in any of the first significant approaches. Instead, work started on pairs of languages. However, during the 1950s and 60s, researchers in Cambridge headed by Margaret Masterman, in Leningrad headed by Nikolai Andreev and in Milan by Silvio Ceccato started work in this area. The idea was discussed extensively by the Israeli philosopher Yehoshua Bar-Hillel in 1969. During the 1970s, noteworthy research was done in Grenoble by researchers attempting to translate physics and mathematical texts from Russian to French, and in Texas a similar project (METAL) was ongoing for Russian to English. Early interlingual MT systems were also built at Stanford in the 1970s by Roger Schank and Yorick Wilks; the former became the basis of a commercial system for the transfer of funds, and the latter's code is preserved at The Computer Museum at Boston as the first interlingual machine translation system. In the 1980s, renewed relevance was given to interlingua-based, and knowledge-based approaches to machine translation in general, with much research going on in the field. The uniting factor in this research was that high-quality translation required abandoning the idea of requiring total comprehension of the text. Instead, the translation should be based on linguistic knowledge and the specific domain in which the system would be used. The most important research of this era was done in distributed language translation (DLT) in Utrecht, which worked with a modified version of Esperanto, and the Fujitsu system in Japan. In 2016, Google Neural Machine Translation achieved "zero-shot translation", that is it directly translates one language into another. For example, it might be trained just for Japanese-English and Korean-English translation, but can perform Japanese-Korean translation. The system appears to have learned to produce a language-independent intermediate representation of language (an "interlingua"), which allows it to perform zero-shot translation by converting from and to the interlingua. == Outline == In this method of translation, the interlingua can be thought of as a way of describing the analysis of a text written in a source language such that it is possible to convert its morphological, syntactic, semantic (and even pragmatic) characteristics, that is "meaning" into a target language. This interlingua is able to describe all of the characteristics of all of the languages which are to be translated, instead of simply translating from one language to another. Sometimes two interlinguas are used in translation. It is possible that one of the two covers more of the characteristics of the source language, and the other possess more of the characteristics of the target language. The translation then proceeds by converting sentences from the first language into sentences closer to the target language through two stages. The system may also be set up such that the second interlingua uses a more specific vocabulary that is closer, or more aligned with the target language, and this could improve the translation quality. The above-mentioned system is based on the idea of using linguistic proximity to improve the translation quality from a text in one original language to many other structurally similar languages from only one original analysis. This principle is also used in pivot machine translation, where a natural language is used as a "bridge" between two more distant languages. For example, in the case of translating to English from Ukrainian using Russian as an intermediate language. == Translation process == In interlingual machine translation systems, there are two monolingual components: the analysis of the source language and the interlingual, and the generation of the interlingua and the target language. It is however necessary to distinguish between interlingual systems using only syntactic methods (for example the systems developed in the 1970s at the universities of Grenoble and Texas) and those based on artificial intelligence (from 1987 in Japan and the research at the universities of Southern California and Carnegie Mellon). The first type of system corresponds to that outlined in Figure 1. while the other types would be approximated by the diagram in Figure 4. The following resources are necessary to an interlingual machine translation system: Dictionaries (or lexicons) for analysis and generation (specific to the domain and the languages involved). A conceptual lexicon (specific to the domain), which is the knowledge base about events and entities known in the domain. A set of projection rules (specific to the domain and the languages). Grammars for the analysis and generation of the languages involved. One of the problems of knowledge-based machine translation systems is that it becomes impossible to create databases for domains larger than very specific areas. Another is that processing these databases is very computationally expensive. == Efficacy == One of the main advantages of this strategy is that it provides an economical way to make multilingual translation systems. With an interlingua it becomes unnecessary to make a translation pair between each pair of languages in the system. So instead of creating n ( n − 1 ) {\displaystyle n(n-1)} language pairs, where n {\displaystyle n} is the number of languages in the system, it is only necessary to make 2 n {\displaystyle 2n} pairs between the n {\displaystyle n} languages and the interlingua. The main disadvantage of this strategy is the difficulty of creating an adequate interlingua. It should be both abstract and independent of the source and target languages. The more languages added to the translation system, and the more different they are, the more potent the interlingua must be to express all possible translation directions. Another problem is that it is difficult to extract meaning from texts in the original languages to create the intermediate representation. == Existing interlingual machine translation systems == Calliope-Aero Carabao Linguistic Virtual Machine Grammatical Framework Number Translator Google Translate use English internally as a pivot language for some language pairs such as Chinese and Japanese, and more generally those with "higher quality" neural-network translators with English but not between each other.
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Markov chain central limit theorem

In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity. == Statement == Suppose that: the sequence X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 {\textstyle X_{1}} , is the stationary distribution, so that X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g {\textstyle g} is some (measurable) real-valued function for which var ⁡ ( g ( X 1 ) ) < + ∞ . {\textstyle \operatorname {var} (g(X_{1}))<+\infty .} Now let μ = E ⁡ ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ⁡ ( n μ ^ n ) = lim n → ∞ n var ⁡ ( μ ^ n ) = var ⁡ ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ⁡ ( g ( X 1 ) , g ( X 1 + k ) ) . {\displaystyle {\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k})\\\sigma ^{2}&:=\lim _{n\to \infty }\operatorname {var} ({\sqrt {n}}{\widehat {\mu }}_{n})=\lim _{n\to \infty }n\operatorname {var} ({\widehat {\mu }}_{n})=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (g(X_{1}),g(X_{1+k})).\end{aligned}}} Then as n → ∞ , {\textstyle n\to \infty ,} we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , {\displaystyle {\sqrt {n}}({\hat {\mu }}_{n}-\mu )\ {\xrightarrow {\mathcal {D}}}\ {\text{Normal}}(0,\sigma ^{2}),} where the decorated arrow indicates convergence in distribution. == Monte Carlo Setting == The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = { 1 , … , n 1 } × { 1 , … , n 2 } ⊆ Z 2 {\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}} . A proper configuration on X {\displaystyle X} consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ {\displaystyle \chi } denote the set of all proper configurations on X {\displaystyle X} , N χ ( n 1 , n 2 ) {\displaystyle N_{\chi }(n_{1},n_{2})} be the total number of proper configurations and π be the uniform distribution on χ {\displaystyle \chi } so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) {\displaystyle W(x)} is the number of white points in x ∈ χ {\displaystyle x\in \chi } then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) {\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}} If n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are even moderately large then we will have to resort to an approximation to E π W {\displaystyle E_{\pi }W} . Consider the following Markov chain on χ {\displaystyle \chi } . Fix p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and set X 1 = x 1 {\displaystyle X_{1}=x_{1}} where x 1 ∈ χ {\displaystyle x_{1}\in \chi } is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X {\displaystyle (x,y)\in X} and independently draw U ∼ U n i f o r m ( 0 , 1 ) {\displaystyle U\sim \mathrm {Uniform} (0,1)} . If u ≤ p {\displaystyle u\leq p} and all of the adjacent points are black then color ( x , y ) {\displaystyle (x,y)} white leaving all other points alone. Otherwise, color ( x , y ) {\displaystyle (x,y)} black and leave all other points alone. Call the resulting configuration X 1 {\displaystyle X_{1}} . Continuing in this fashion yields a Harris ergodic Markov chain { X 1 , X 2 , X 3 , … } {\displaystyle \{X_{1},X_{2},X_{3},\ldots \}} having π {\displaystyle \pi } as its invariant distribution. It is now a simple matter to estimate E π W {\displaystyle E_{\pi }W} with w n ¯ = ∑ i = 1 n W ( X i ) / n {\displaystyle {\overline {w_{n}}}=\sum _{i=1}^{n}W(X_{i})/n} . Also, since χ {\displaystyle \chi } is finite (albeit potentially large) it is well known that X {\displaystyle X} will converge exponentially fast to π {\displaystyle \pi } which implies that a CLT holds for w n ¯ {\displaystyle {\overline {w_{n}}}} . == Implications == Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
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