AI Coding Wiki

AI Coding Wiki — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Artificial Linguistic Internet Computer Entity

    Artificial Linguistic Internet Computer Entity

    A.L.I.C.E. (Artificial Linguistic Internet Computer Entity), also referred to as Alicebot, or simply Alice, is a natural language processing chatbot—a program that engages in a conversation with a human by applying some heuristical pattern matching rules to the human's input. It was inspired by Joseph Weizenbaum's classical ELIZA program. It is one of the strongest programs of its type and has won the Loebner Prize, awarded to accomplished humanoid, talking robots, three times (in 2000, 2001, and 2004). The program is unable to pass the Turing test, as even the casual user will often expose its mechanistic aspects in short conversations. Alice was originally composed by Richard Wallace; it "came to life" on November 23, 1995. The program was rewritten in Java beginning in 1998. The current incarnation of the Java implementation is Program D. The program uses an XML Schema called AIML (Artificial Intelligence Markup Language) for specifying the heuristic conversation rules. Alice code has been reported to be available as open source. The AIML source is available from ALICE A.I. Foundation on Google Code and from the GitHub account of Richard Wallace. These AIML files can be run using an AIML interpreter like Program O or Program AB. == In popular culture == Spike Jonze has cited ALICE as the inspiration for his academy award-winning film Her, in which a human falls in love with a chatbot. In a New Yorker article titled “Can Humans Fall in Love with Bots?” Jonze said “that the idea originated from a program he tried about a decade ago called the ALICE bot, which engages in friendly conversation.” The Los Angeles Times reported:Though the film’s premise evokes comparisons to Siri, Jonze said he actually had the idea well before the Apple digital assistant came along, after using a program called Alicebot about ten years ago. As geek nostalgists will recall, that intriguing if at times crude software (it flunked the industry-standard Turing Test) would attempt to engage users in everyday chatter based on a database of prior conversations. Jonze liked it, and decided to apply a film genre to it. “I thought about that idea, and what if you had a real relationship with it?” Jonze told reporters. “And I used that as a way to write a relationship movie and a love story.”

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  • Margaret Mitchell (scientist)

    Margaret Mitchell (scientist)

    Margaret Mitchell is a computer scientist who works on algorithmic bias and fairness in machine learning. She is most well known for her work on automatically removing undesired biases concerning demographic groups from machine learning models, as well as more transparent reporting of their intended use. == Education == Mitchell obtained a bachelor's degree in linguistics from Reed College, Portland, Oregon, in 2005. After having worked as a research assistant at the OGI School of Science and Engineering for two years, she subsequently obtained a Master's in Computational Linguistics from the University of Washington in 2009. She enrolled in a PhD program at the University of Aberdeen, where she wrote a doctoral thesis on the topic of Generating Reference to Visible Objects, graduating in 2013. == Career and research == Mitchell is best known for her work on fairness in machine learning and methods for mitigating algorithmic bias. This includes her work on introducing the concept of 'Model Cards' for more transparent model reporting, and methods for debiasing machine learning models using adversarial learning. Margaret Mitchell created the framework for recognizing and avoiding biases by testing with a variable for the group of interest, predictor and an adversary. In 2012, Mitchell joined the Human Language Technology Center of Excellence at Johns Hopkins University as a postdoctoral researcher, before taking up a position at Microsoft Research in 2013. At Microsoft, Mitchell was the research lead of the Seeing AI project, an app that offers support for the visually impaired by narrating texts and images. In November 2016, she became a senior research scientist at Google Research and Machine intelligence. While at Google, she founded and co-led the Ethical Artificial Intelligence team together with Timnit Gebru. In May 2018, she represented Google in the Partnership on AI. In February 2018, she gave a TED talk on "How we can build AI to help humans, not hurt us". In January 2021, after Timnit Gebru's termination from Google, Mitchell reportedly used a script to search through her corporate account and download emails that allegedly documented discriminatory incidents involving Gebru. An automated system locked Mitchell's account in response. In response to media attention Google claimed that she "exfiltrated thousands of files and shared them with multiple external accounts". After a five-week investigation, Mitchell was fired. Prior to her dismissal, Mitchell had been a vocal advocate for diversity at Google, and had voiced concerns about research censorship at the company. In late 2021, she joined AI start-up Hugging Face. Mitchell is a co-founder of Widening NLP, a special interest group within the Association for Computational Linguistics (ACL) seeking to increase the proportion of women and minorities working in natural language processing; and Computational Linguistics and Clinical Psychology, an annual workshop within the ACL that brings together clinicians and computational linguists to advance the state of the art in clinical psychology.

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  • JOONE

    JOONE

    JOONE (Java Object Oriented Neural Engine) is a component based neural network framework built in Java. == Features == Joone consists of a component-based architecture based on linkable components that can be extended to build new learning algorithms and neural networks architectures. Components are plug-in code modules that are linked to produce an information flow. New components can be added and reused. Beyond simulation, Joone also has to some extent multi-platform deployment capabilities. Joone has a GUI Editor to graphically create and test any neural network, and a distributed training environment that allows for neural networks to be trained on multiple remote machines. == Comparison == As of 2010, Joone, Encog and Neuroph are the major free component based neural network development environment available for the Java platform. Unlike the two other (commercial) systems that are in existence, Synapse and NeuroSolutions, it is written in Java and has direct cross-platform support. A limited number of components exist and the graphical development environment is rudimentary so it has significantly fewer features than its commercial counterparts. Joone can be considered to be more of a neural network framework than a full integrated development environment. Unlike its commercial counterparts, it has a strong focus on code-based development of neural networks rather than visual construction. While in theory Joone can be used to construct a wider array of adaptive systems (including those with non-adaptive elements), its focus is on backpropagation based neural networks.

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  • Philipp Koehn

    Philipp Koehn

    Philipp Koehn (born 1 August 1971 in Erlangen, West Germany) is a computer scientist and researcher in the field of machine translation. His primary research interest is statistical machine translation and he is one of the inventors of a method called phrase based machine translation. This is a sub-field of statistical translation methods that employs sequences of words (or so-called "phrases") as the basis of translation, expanding the previous word based approaches. A 2003 paper which he authored with Franz Josef Och and Daniel Marcu called Statistical phrase-based translation has attracted wide attention in Machine translation community and has been cited over a thousand times. Phrase based methods are widely used in machine translation applications in industry. Philipp Koehn received his PhD in computer science in 2003 from the University of Southern California, where he worked at the Information Sciences Institute advised by Kevin Knight. After a year as a postdoctoral fellow under Michael Collins at the Massachusetts Institute of Technology, he joined the University of Edinburgh as a lecturer in the School of Informatics in 2005. He was appointed reader in 2010 and professor in 2012. In 2014, he was appointed professor at the computer science department of The Johns Hopkins University, where he is affiliated with the Center for Language and Speech Processing. == Moses statistical machine translation decoder == The Moses machine translation decoder is an open source project that was created by and is maintained under the guidance of Philipp Koehn. The Moses decoder is a platform for developing Statistical machine translation systems given a parallel corpus for any language pair. The decoder was mainly developed by Hieu Hoang and Philipp Koehn at the University of Edinburgh and extended during a Johns Hopkins University Summer Workshop and further developed under Euromatrix and GALE project funding. The decoder (which is part of a complete statistical machine translation toolkit) is the de facto benchmark for research in the field. Although Koehn continues to play a major role in the development of Moses, the Moses decoder was supported by the European Framework 6 projects Euromatrix, TC-Star, the European Framework 7 projects EuroMatrixPlus, Let's MT, META-NET and MosesCore and the DARPA GALE project, as well as several universities such as the University of Edinburgh, the University of Maryland, ITC-irst, Massachusetts Institute of Technology, and others. Substantial additional contributors to the Moses decoder include Hieu Hoang, Chris Dyer, Josh Schroeder, Marcello Federico, Richard Zens, and Wade Shen. == Europarl corpus == The Europarl corpus is a set of documents that consists of the proceedings of the European Parliament from 1996 to the present. The corpus has been compiled and expanded by a group of researchers led by Philipp Koehn at University of Edinburgh. The data that makes up the corpus was extracted from the website of the European Parliament and then prepared for linguistic research. The latest release (2012) comprised up to 60 million words per language, with 21 European languages represented: Romanic (French, Italian, Spanish, Portuguese, Romanian), Germanic (English, Dutch, German, Danish, Swedish), Slavic (Bulgarian, Czech, Polish, Slovak, Slovene), Finno-Ugric (Finnish, Hungarian, Estonian), Baltic (Latvian, Lithuanian), and Greek. == Other interests and activities in chronological order == Koehn is a professor at Johns Hopkins University where he continues his research into machine translation through his affiliation with the Center for Language and Speech Processing Koehn is a professor and chair of machine translation at the University of Edinburgh School of Informatics and contributes to its statistical machine translation group which organises workshops, seminars and project related to the subject. Koehn has consulted to SYSTRAN periodically between 2006 and 2011. SYSTRAN was acquired by CLSI, a Korean machine translation company in April 2014. Koehn worked for Facebook/META AI Research from 2018 to 2022. Koehn is also chief scientist for Omniscien Technologies and a shareholder in Omniscien Technologies since 2007. Omniscien Technologies is a private company developing and commercialising machine translation technologies. Koehn authored a book titled "Statistical Machine Translation" in 2009 and a book titled "Neural Machine Translation" in 2020. == Awards and recognition == 2013: One of three finalists in the category of Research for the European Patent Office (EPO) 2013 European Inventor Award. Koehn was recognised for patent EP 1488338 B, Phrase-Based Joint Probability Model for Statistical Machine Translations, a translation model that uses mathematical probabilities to determine the most likely interpretation of chunks of text between foreign languages. 2015: Koehn received the Award of Honor of the International Association for Machine Translation. 2024: Koehn was named Fellow of the Association for Computational Linguistics (ACL).

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  • Taimi

    Taimi

    Taimi ( TAY-mee) is a dating app that caters to the LGBTQI+ community. The network matches its registered users based on their selected preferences and location. Originally an online dating service for gay men, by 2022 Taimi had become an app for all members of the LGBTQI+ community. It operates in more than 138 countries, including the US, UK, the Netherlands, Spain, Central and South America, Ukraine, and other European and Asian countries. Taimi runs on iOS and Android. The mobile app has a free and subscription-based premium version and offers a number of services for communication, including live streaming, chatting, and video calling. There is also an active blog that regularly posts articles and news about events of interest to the LGBTQ+ community. The application does not provide for non-Google e-mail log option, either phone number or Facebook account, during the registration process. The data controller for the non EU/UK users is based in a company, called Social Impact Inc., with its registered address at 1180 North Town Center Drive Suite 100, Las Vegas, Nevada, 89144, United States of America. == History == Taimi was launched in 2017 by Social Impact, Inc. in Las Vegas. Its founder, Alex Pasykov, originally called the app "Tame Me," a name that gradually morphed into Taimi. Over time, Taimi expanded into other countries, and expanding its reach to the LGBTQ+ community, so that, by 2022, it was fully inclusive of the entire queer community. In November 2020 the app was redesigned, with a new interface, branding, and logo. As of 2024, there are over 25 million registered users of Taimi worldwide. Pasykov states that he is an ally of the LGBTQ+ community and that he is focused on, among other things, partnering with NGOs to fight Homophobia and "regressive policies and laws" that negatively impact the community. == Features == Users register on the app and complete a profile, including personal information and preferences for compatibility, dating style, and relationship goals. An algorithm then finds and presents recommendations that a user accepts or rejects. Users are then free to chat via text or video with people they have connected with. Safety and security features include a two-step authentication process and an automated account verification along with a clear reporting system when breaches or policy violations occur. User responses to new features and policies drive changes and modifications that are made to all aspects of the site. == Partnerships and Collaborations == Taimi has a long history of collaborations and partnerships in Pride events, both in the US and abroad, including fund-raising efforts. Taimi has partnered with Rakuten Viber to create a bot focused on educating its members on key LGBTQ+ topics and to allow queer Viber users to connect. In 2023, Taimi collaborated with the Known Agency in an "America the Beautiful" campaign to shine a spotlight on current anti-LGBTQ+ policies and laws in a number of US states, and to counter these by highlighting the values and freedoms upon which America was founded. The campaign was nominated for The Drum Awards in the category "OOH For Good" and honored with the ANA Multicultural Excellence Award. Taimi also partnered with Goodparts, a queer-owned and operated retailer, in a "Body Beautiful" campaign focused on love and acceptance of all body types. In this campaign, well-known LGBTQ+ artists are providing artwork for Goodpart's product packaging. From October 31 to December 13, 2023, Taimi showed the "Taimi Moments" video, created in collaboration with Raygun Agency, on large screens between performances of LGBTQ+ artists Doja Cat, Ice Spice, and Doechii on their Scarlet Tour. In spring 2024, Taimi launched Queer Paradise, a series of live events in Southern California to celebrate diversity, sexual exploration, and dating fluidity. Each event in the series was curated to give the full spectrum of groups within the LGBTQ+ community a space to express their authentic selves. Taimi's partners for Queer Paradise include Hawtmess Productions, Eden Entertainment Group, Hump Events, Girls Gays & Theys, Damn Good Dyke Nights, and Gaybors Agency. In summer 2024, with support from GLAAD, Taimi has updated features and self-expression tools to better serve the LGBTQ+ people seeking connection in the app. Taimi allowed members to select multiple sexualities, unified the list of sexualities across all genders, added more pronoun options, and created a more inclusive and improved list of subcategories for non-binary users. Also, in summer 2024, Taimi has partnered with gender-affirming underwear brand Urbody to release a capsule collection. Focused on gender inclusivity and sexual fluidity, the capsule collection includes a range of underwear and compression tops intended to promote "joy, self-love and empowerment."

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  • AI Code Generators Reviews: What Actually Works in 2026

    AI Code Generators Reviews: What Actually Works in 2026

    Trying to pick the best AI code generator? An AI code generator 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 code generator 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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  • Daniel J. Hulme

    Daniel J. Hulme

    Daniel Hulme (born 21 February 1980) is a British businessman, investor, academic and commentator, working in the field of Artificial Intelligence (AI), applied technology and ethics. He is the CEO and founder of Satalia that exited to WPP plc in 2021 for a rumoured $100M where he is also Chief AI Officer. Hulme is also an angel investor in emerging technology companies. In 2024 Hulme co-founded Conscium, an AI Safety company which tests AI Agents and verifies that they do what they are supposed to do. It is also investigating whether AIs will soon become conscious, and how to test for that, and developing more efficient approaches to AI development using neuromorphic computing. Alongside building and scaling Satalia, Hulme was also a Co-Founding Director of Faculty (company) AI - previously ASI Data-Science. In 2026, Accenture announced it had agreed to acquire Faculty for $1bn. Hulme founded Satalia in 2008, a company that provides AI products and consultancy for governments and companies such as Tesco,DFS Furniture,PwC and the BBC. He received a masters and doctorate in AI from University College London (UCL), and is now their Computer Science Entrepreneur in residence, where he teaches how AI can be applied to solve business and social problems. After exiting Satalia to WPP plc Hulme took the dual role of Chief AI Officer at WPP where he is responsible for informing and coordinating AI across the group. In 2026 Hulme was elected as a Founding Fellow of the Academy for the Mathematical Sciences, in recognition of his contributions at the intersection of AI and applied mathematics. Hulme is an angel investor and also a frequent public speaker and writer on the topics of AI, ethics, technology, innovation, decentralization and organisational design. == Early life and education == Hulme was born in 1980. He grew up in the seaside town of Morecambe in north west England. After completing secondary school, Hulme moved to London to study at University College London. On completing his under graduate degree, Hulme stayed at UCL to complete a master's degree and then an EngD. All three degrees were in subjects related to AI. In 2009 Hulme was awarded a Kauffman Global Entrepreneur Scholarship, which saw him visit institutes in the United States to better understand their culture of innovation, and what UK business people could learn from it. This included a tour of Stanford, MIT, Berkeley and Harvard, along with a placement at Cisco Systems HQ in Silicon Valley. == Career == === Satalia === Hulme founded NPComplete Limited in 2007, and incorporated it in 2008, a few months before completing his PhD. NPComplete Limited trades as Satalia. The London-based company provides full-stack AI consultancy and products, helping organisations harness data science, machine learning and AI to solve complex problems, including real-time optimisation. NPComplete refers to mathematical NP-completeness, which describes a class of exponential problems in the field of computational complexity theory. The trading name of NPComplete, Satalia, is a portmanteau of SAT (Short for satisfiability, as in the Boolean satisfiability problem) and the Latin phrase Et alia. Satalia seeks to solve hard problems, in particular the class of exponentially hard problems found in academia and industry known as NP-hardness. In 2016, Satalia was the only UK company to appear in the Gartner Cool Vendors list for data science. In November 2019, City A.M. reported that Satalia was the 39th fastest growing tech firm in the UK, with three year growth at 886%. Satalia was acquired by WPP plc in August 2021 for a rumored $100,000,000, where Hulme was the majority shareholder. === Conscium === Conscium is the World's first commercial organisation dedicated to the understanding, verification and validation of conscious AI and its implications for developing safe, efficient neuromorphic models. Conscium is an AI safety company with three workstreams: AI agent verification. Verification of AI agents developed by third parties to ensure they are beneficial and not harmful. Development of neuromorphic systems. Neuromorphic computing refers to technologies that can process information more like a biological brain compared to existing approaches, making them far more adaptive, scalable and efficient than current AI. Research into artificial consciousness. This workstream is led by Mark Solms, Chair of Neuropsychology at the University of Cape Town. This research aims to better understand what consciousness in AI systems and machines would look like, and, if and when machines do reach consciousness, what the moral and ethical implications would be. Conscium was founded in 2024 in London by a team including Hulme, Ed Charvet, Calum Chace, Ted Lappas, and Panagiotis Repoussis. Conscium has recruited some of the world’s leading neuroscientists and computer scientists to its advisory board, including Anil Seth, Mark Solms, Karl J. Friston, Anthony Finkelstein, Benjamin Rosman, David Wood, Jonathan Shock, Megan Peters, Moran Cerf, Nicholas Humphrey, Nicky Clayton, Nikola Kasabov, Steve Furber, and Suzanne Livingston. Supported by these world-leading experts, Conscium is creating a neuromorphic computing lab to research and validate the capacity of machines to acquire consciousness, making them safer for humanity. Conscium has published an open letter warning of the risks of AI suffering if care is not taken to mitigate against that possibility when and if AI becomes conscious. Signatories of the letter included Stephen Fry, Karl Friston and Anthony Finkelstein. === The Partnership for Research Into Sentient Machines (PRISM) === Hulme is one of the founding partners of PRISM - The Partnership for Research Into Sentient Machines, a non-profit set up to help prepare society for a future with conscious, or seemingly conscious, artificial intelligence. === Academia === Hulme's master's degree topic was on simulating artificial life, where he used Evolutionary algorithm's to generate emergent intelligence in AI agent's with Artificial Neural Network brains. His PhD spanned modelling bumblebee brains and mathematical optimization. Hulme maintained his connection with UCL after completing his doctorate, staying on in various teaching positions. From 2014 to Oct 2019 he was the Director of UCL's Business analytics MSc, which dealt with the application of AI to government, social, and business problems. As of 2020, Hulme is UCL's (University College London) Entrepreneur-in-Residence. He is also a faculty member and lecturer at Singularity University, and a visiting lecturer at London School of Economics's Marshall Institute. === Public engagement === Hulme frequently speaks for TEDx, Google and at various other events. He specialises in Artificial Intelligence, Decentralization, Organisational Design, and Innovation. He has written numerous articles and contributed to several books, largely concerning AI, as well as applied technology and related ethical issues. In 2017, along with Elon Musk, Stuart J. Russell, Geoffrey Hinton and Demis Hassabis, Hulme was one of the 116 founders of robotics and AI companies to sign an open letter to the United Nations, warning against the use of AI in autonomous weapons. Hulme also consults with various companies, governments and other organisations, independently of Satalia.

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  • AI Writing Assistants: Free vs Paid (2026)

    AI Writing Assistants: Free vs Paid (2026)

    Curious about the best AI writing assistant? An AI writing assistant is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI writing assistant 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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  • SPL notation

    SPL notation

    SPL (Sentence Plan Language) is an abstract notation representing the semantics of a sentence in natural language. In a classical Natural Language Generation (NLG) workflow, an initial text plan (hierarchically or sequentially organized factoids, often modelled in accordance with Rhetorical Structure Theory) is transformed by a sentence planner (generator) component to a sequence of sentence plans modelled in a Sentence Plan Language. A surface generator can be used to transform the SPL notation into natural language sentences. Probably the most widely used SPL language used today (2022) is AMR (Abstract Meaning Representation, see there for further references), but is owes parts of its popularity to its application to NLP problems other than NLG, e.g., machine translation and semantic parsing.

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  • AI Text-to-image Tools: Free vs Paid (2026)

    AI Text-to-image Tools: Free vs Paid (2026)

    Shopping for the best AI text-to-image tool? An AI text-to-image tool 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 text-to-image tool 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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  • Baum–Welch algorithm

    Baum–Welch algorithm

    In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. The Baum–Welch algorithm, the primary method for inference in hidden Markov models, is numerically unstable due to its recursive calculation of joint probabilities. As the number of variables grows, these joint probabilities become increasingly small, leading to the forward recursions rapidly approaching values below machine precision. == History == The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. == Description == A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state. The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. We assume there are N {\displaystyle N} states in total). We assume the P ( X t ∣ X t − 1 ) {\displaystyle P(X_{t}\mid X_{t-1})} is independent of time t {\displaystyle t} , which leads to the definition of the time-independent stochastic transition matrix A = { a i j } = P ( X t = j ∣ X t − 1 = i ) . {\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).} The initial state distribution (i.e. when t = 1 {\displaystyle t=1} ) is given by π i = P ( X 1 = i ) . {\displaystyle \pi _{i}=P(X_{1}=i).} The observation variables Y t {\displaystyle Y_{t}} can take one of K {\displaystyle K} possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation y i {\displaystyle y_{i}} at time t {\displaystyle t} for state X t = j {\displaystyle X_{t}=j} is given by b j ( y i ) = P ( Y t = y i ∣ X t = j ) . {\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).} Taking into account all the possible values of Y t {\displaystyle Y_{t}} and X t {\displaystyle X_{t}} , we obtain the N × K {\displaystyle N\times K} matrix B = { b j ( y i ) } {\displaystyle B=\{b_{j}(y_{i})\}} where b j {\displaystyle b_{j}} belongs to all the possible states and y i {\displaystyle y_{i}} belongs to all the observations. An observation sequence is given by Y = ( Y 1 = y 1 , Y 2 = y 2 , … , Y T = y T ) {\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})} . Thus we can describe a hidden Markov chain by θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} . The Baum–Welch algorithm finds a local maximum for θ ∗ = a r g m a x θ ⁡ P ( Y ∣ θ ) {\displaystyle \theta ^{}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )} (i.e. the HMM parameters θ {\displaystyle \theta } that maximize the probability of the observation). === Algorithm === Set θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum. ==== Forward procedure ==== Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t} . This is found recursively: α i ( 1 ) = π i b i ( y 1 ) , {\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),} α i ( t + 1 ) = b i ( y t + 1 ) ∑ j = 1 N α j ( t ) a j i . {\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.} Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling α {\displaystyle \alpha } in the forward and β {\displaystyle \beta } in the backward procedure below. ==== Backward procedure ==== Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t {\displaystyle t} . We calculate β i ( t ) {\displaystyle \beta _{i}(t)} as, β i ( T ) = 1 , {\displaystyle \beta _{i}(T)=1,} β i ( t ) = ∑ j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) . {\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).} ==== Update ==== We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) β i ( t ) ∑ j = 1 N α j ( t ) β j ( t ) , {\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},} which is the probability of being in state i {\displaystyle i} at time t {\displaystyle t} given the observed sequence Y {\displaystyle Y} and the parameters θ {\displaystyle \theta } ξ i j ( t ) = P ( X t = i , X t + 1 = j ∣ Y , θ ) = P ( X t = i , X t + 1 = j , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) ∑ k = 1 N ∑ w = 1 N α k ( t ) a k w β w ( t + 1 ) b w ( y t + 1 ) , {\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},} which is the probability of being in state i {\displaystyle i} and j {\displaystyle j} at times t {\displaystyle t} and t + 1 {\displaystyle t+1} respectively given the observed sequence Y {\displaystyle Y} and parameters θ {\displaystyle \theta } . The denominators of γ i ( t ) {\displaystyle \gamma _{i}(t)} and ξ i j ( t ) {\displaystyle \xi _{ij}(t)} are the same ; they represent the probability of making the observation Y {\displaystyle Y} given the parameters θ {\displaystyle \theta } . The parameters of the hidden Markov model θ {\displaystyle \theta } can now be updated: π i ∗ = γ i ( 1 ) , {\displaystyle \pi _{i}^{}=\gamma _{i}(1),} which is the expected frequency spent in state i {\displaystyle i} at time 1 {\displaystyle 1} . a i j ∗ = ∑ t = 1 T − 1 ξ i j ( t ) ∑ t = 1 T − 1 γ i ( t ) , {\displaystyle a_{ij}^{}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},} which is the expected number of transitions from state i to state j compared to the expected total number of transitions starting in state i, including from state i to itself. The number of transitions starting in state i is equivalent to the number of times state i is observed in the sequence from t = 1 to t = T − 1. b i ∗ ( v k ) = ∑ t = 1 T 1 y t = v k γ i ( t ) ∑ t = 1 T γ i ( t ) , {\displaystyle b_{i}^{}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},} where 1 y t = v k = { 1 if y t = v k , 0 otherwise {\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}} is an indicator function, and b i ∗ ( v k ) {\displaystyle b_{i}^{}(v_{k})} is the expected number of times the output observations have been equal to v k {\displaystyle v_{k}} while in state i {\displaystyle i} over the expected total number of times in state i {\displaystyle i} . These steps are now repeated iteratively until a desired level of convergence. Note: It is possible to over-fit a particular data set. 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