A plug computer is a small-form-factor computer whose chassis contains the AC power plug, and thus plugs directly into the wall. Alternatively, the computer may resemble an AC adapter or a similarly small device. Plug computers are often configured for use in the home or office as compact computer. == Description == Plug computers consist of a high-performance, low-power system-on-a-chip processor, with several I/O hardware ports (USB ports, Ethernet connectors, etc.). Most versions do not have provisions for connecting a display and are best suited to running media servers, back-up services, or file sharing and remote access functions; thus acting as a bridge between in-home protocols (such as Digital Living Network Alliance (DLNA) and Server Message Block (SMB)) and cloud-based services. There are, however, plug computer offerings that have analog VGA monitor and/or HDMI connectors, which, along with multiple USB ports, permit the use of a display, keyboard, and mouse, thus making them full-fledged, low-power alternatives to desktop and laptop computers. They typically run any of a number of Linux distributions. Plug computers typically consume little power and are inexpensive. == History == A number of other devices of this type began to appear at the 2009 Consumer Electronics Show. On January 6, 2009 CTERA Networks launched a device called CloudPlug that provides online backup at local disk speeds and overlays a file sharing service. The device also transforms any external USB hard drive into a network-attached storage device. On January 7, 2009, Cloud Engines unveiled the Pogoplug network access server. On January 8, 2009, Axentra announced availability of their HipServ platform. On February 23, 2009, Marvell Technology Group announced its plans to build a mini-industry around plug computers. On August 19, 2009, CodeLathe announced availability of their TonidoPlug network access server. On November 13, 2009 QuadAxis launched its plug computing device product line and development platform, featuring the QuadPlug and QuadPC and running QuadMix, a modified Linux. On January 5, 2010, Iomega announced their iConnect network access server. On January 7, 2010 Pbxnsip launched its plug computing device the sipJack running pbxnsip: an IP Communications platform.
NCAA transfer portal
The NCAA transfer portal is a National Collegiate Athletic Association (NCAA) application, database, and compliance tool that facilitates student athletes' transfers between member institutions. It is intended to bring greater transparency to the transfer process and to enable student athletes to publicize their desire to transfer. The transfer portal is an NCAA-wide database covering all three NCAA divisions, although most media coverage of the transfer portal involves its use in the top-level Division I (D-I). The portal launched on October 15, 2018. Regulations adopted in 2021 allowed student-athletes in D-I football, men's and women's basketball, men's ice hockey, and baseball to transfer schools using the portal once without sitting out a year. In 2024, the NCAA authorized athletes unlimited transfers. == Process == For Divisions I and II, once an athlete desiring to transfer informs their school; the school must enter the athlete's name in the database within two business days. Then coaches and staff from other universities may contact the athlete about potentially transferring. Before the January 2026 NCAA convention, Division III schools were allowed, but not required, to enter such a student into the portal. A proposal to require use of the portal in that division was approved at the convention. The timeline for D-III members to enter athletes into the portal differs from that of the other divisions. Athletes wishing to enter the portal must first complete an educational module. Once completed, the school has seven calendar days to enter the athlete's transfer request into the portal. == Transfer windows == On August 31, 2022, the D-I board adopted a series of changes to transfer rules, introducing the concept of transfer windows, similar to those used in professional soccer worldwide. Student-athletes who wish to take advantage of the one-time transfer rule must, under normal circumstances, enter the portal within a designated window for their sport. These windows are slightly different for each NCAA sport, but are broadly grouped by the NCAA's three athletic "seasons". At that time, the windows were as follows: Fall sports – A 45-day winter window opening the day after championship selections are made in that sport, and a spring window from May 1–15. According to the NCAA, "reasonable accommodations" would be made for participants in football's FBS and FCS championship games (respectively the College Football Playoff National Championship and Division I Football Championship Game), both of which take place in early January. Participants in those games had a 14-day window opening on the day after the championship game, as well as the spring window. Winter sports – A 60-day window opening the day after championship selections are made in that sport. Spring sports – A winter window from December 1–15, and a 45-day spring window opening the day after championship selections are made in that sport. For sports included in the NCAA Emerging Sports for Women program, transfer windows are the same as those for fully recognized NCAA sports. As with fully recognized NCAA sports, transfer windows linked to championship events open on the day after selections are made for the generally recognized championship events in emerging sports. Student-athletes whose athletic aid is reduced, canceled, or not renewed by their school, as well as those affected by a university's elimination of a sports team, may enter the transfer portal at any time without penalty. A slightly different exception applies to those undergoing a head coaching change; student-athletes so affected in sports other than Division I football can enter the portal within 30 days of the change, starting on the day after the coach's departure is announced. The coaching change window also applied to Division I football before October 2025. Less than a month after transfer windows were adopted, the Division I Council adopted a change that affected only graduate transfers. Student-athletes who are set to graduate with remaining athletic eligibility, and plan to continue competition as postgraduate students, were exempt from transfer windows. They could enter the portal at any time during the academic year, and were not subject to the standard deadlines of May 1 for fall and winter sports and July 1 for spring sports. In April 2024, graduate transfers became subject to the same deadlines as all other transfer students. This change did not affect windows for student-athletes affected by a head coaching change, a loss of athletic aid, or the discontinuation of a team. Because the Ivy League allows neither redshirting nor athletic participation by graduate students, athletes at its member schools who are set to complete four years of attendance but still have remaining athletic eligibility may enter the portal at any time during their fourth academic year of attendance. In October 2024, the Division I Council reduced transfer windows in football and basketball to a total of 30 days. For FBS and FCS football, the fall window opened for 20 days, starting on the Monday after FBS conference championship games. Participants in postseason play had a 5-day window that opened on the day after each team's final game. A 10-day spring window opened in mid-April. In men's and women's basketball, a single 30-day window opens on the day after the second round of each Division I tournament concludes. The existing exceptions regarding head coaching changes, a loss of athletic aid, or the discontinuation of a team remained in place. Almost exactly a year later, Division I adopted more significant changes to the football transfer portal for both FBS and FCS. The previous two windows were abolished and replaced by a single window that opens from January 2–16. Participants in the College Football Playoff National Championship—the only game in FBS or FCS played after the closure of the new window—receive a 5-day window that opens on the day after that game. The window for players undergoing a head coaching change was also reduced. A new window of 15 days opens five calendar days after the hiring or public announcement of a new head coach. Should a school fail to hire or publicly announce a new head coach within 30 days after the previous coach's departure, the window will open on the 31st day after departure, provided that the 31st day is no earlier than January 3. This particular window, also open for 15 days, may open at any time before June 30. No change was announced to the exceptions for those affected by a loss of athletic aid or the discontinuation of a team. == Impact on high school recruiting == Effective July 1, 2025, the NCAA Division I Board of Directors implemented new DI roster limits following the court-approved House settlement. Additionally, according to the NCAA, "NCAA rules for Division I programs will no longer include sport-specific scholarship limits." As a result, many top Division I programs, especially those in power conferences, are relying heavily on the transfer portal to bring in conference- and national-level student-athletes. This shift in recruiting focus has already been exemplified across Division I men's and women's track and field especially, beginning in the recruitment cycle for 2025 college entries. Track and field coaches formerly managing rosters of 120-plus (60-plus men and 60-plus women) are now limited to 45 per side for a total of 90 roster spots across men's and women's track and field, meaning they are recruiting fewer student-athletes out of high school and more immediately impactful scholarship-worthy student-athletes via the transfer portal. Roster limits for track and field teams are even more stringent in the Southeastern Conference (SEC): 35 men and 35 women. For high school track and field athletes seeking opportunities with top DI programs, they no longer need to display potential to be point-scorers, but demonstrate the ability to contribute immediately, often by competing at a level aligned with conference scoring standards.
Amebis
Amebis from Kamnik is a company in Slovenia in the field of language technologies. The company has published several electronic dictionaries and encyclopedic dictionaries (e.g. ASP (32) dictionaries) and developed spell checkers, grammar checker Besana, hyphenators and lemmatizers for Slovene, Serbian and Albanian languages. The company maintains and edits the largest Slovenian dictionary portal Termania, which contains more than 135 dictionaries. The most used terminological dictionary on Termania is the Slovenian medical dictionary. In co-operation with company Alpineon and the Jožef Stefan Institute they have developed a speech synthesizer and screen reader Govorec (Speaker). They have also provided technical support for the largest text corpus of Slovene, called FidaPLUS, Fran and Franček. Amebis also developed the system of machine translation Amebis Presis, which incorporates the Slovenian language. On 11 October 2023 Amebis received award of the Father Stanislav Škrabec Foundation for special achievements in Slovene linguistics.
Optical braille recognition
Optical braille recognition is technology to capture and process images of braille characters into natural language characters. It is used to convert braille documents for people who cannot read them into text, and for preservation and reproduction of the documents. == History == In 1984, a group of researchers at the Delft University of Technology designed a braille reading tablet, in which a reading head with photosensitive cells was moved along set of rulers to capture braille text line-by-line. In 1988, a group of French researchers at the Lille University of Science and Technology developed an algorithm, called Lectobraille, which converted braille documents into plain text. The system photographed the braille text with a low-resolution CCD camera, and used spatial filtering techniques, median filtering, erosion, and dilation to extract the braille. The braille characters were then converted to natural language using adaptive recognition. The Lectobraille technique had an error rate of 1%, and took an average processing time of seven seconds per line. In 1993, a group of researchers from the Katholieke Universiteit Leuven developed a system to recognize braille that had been scanned with a commercially available scanner. The system, however, was unable to handle deformities in the braille grid, so well-formed braille documents were required. In 1999, a group at the Hong Kong Polytechnic University implemented an optical braille recognition technique using edge detection to translate braille into English or Chinese text. In 2001, Murray and Dais created a handheld recognition system, that scanned small sections of a document at once. Because of the small area scanned at once, grid deformation was less of an issue, and a simpler, more efficient algorithm was employed. In 2003, Morgavi and Morando designed a system to recognize braille characters using artificial neural networks. This system was noted for its ability to handle image degradation more successfully than other approaches. == Challenges == Many of the challenges to successfully processing braille text arise from the nature of braille documents. Braille is generally printed on solid-color paper, with no ink to produce contrast between the raised characters and the background paper. However, imperfections in the page can appear in a scan or image of the page. Many documents are printed inter-point, meaning they are double-sided. As such, the depressions of the braille of one side appear interlaid with the protruding braille of the other side. == Techniques == Some optical braille recognition techniques attempt to use oblique lighting and a camera to reveal the shadows of the depressions and protrusions of the braille. Others make use of commercially available document scanners.
Eric Xing
Eric Poe Xing (Chinese: 邢波) is an American computer scientist who has been serving as president of Mohamed bin Zayed University of Artificial Intelligence (MBZUAI) since January 2021. He is also a professor in the Carnegie Mellon University School of Computer Science where he founded the SAILING Lab in 2004, and is the co-founder of the AI companies Petuum and GenBio AI. Xing's research focuses on statistical machine learning, probabilistic graphical models, and systems for distributed machine learning. He was elected a Fellow of the Institute of Electrical and Electronics Engineers in 2019 for "contributions to machine learning algorithms and systems" and a Fellow of the Association for Computing Machinery in 2022 for "contributions to algorithms, architectures, and applications in machine learning." == Education == Xing earned a B.Sc. in physics from Tsinghua University in 1993, and an M.Sc. in computer science from Rutgers University in 1998. He earned a Ph.D. in molecular biology and biochemistry from Rutgers in 1999, supervised by molecular cancer researcher Chung S. Yang. His dissertation examined the inactivation of the Rb and p53 pathways in human esophageal squamous cell carcinoma. He earned a second Ph.D. in computer science from the University of California, Berkeley in 2004, supervised by Richard Karp, Michael I. Jordan, and Stuart J. Russell. His thesis applied probabilistic graphical models to motif identification and haplotype inference in genomic data. == Career == Xing joined Carnegie Mellon University (CMU) as a faculty member in 2004, where he created the Statistical Artificial Intelligence and Integrative Genomics (SAILING) Lab. He held visiting appointments from 2010 to 2011, serving as a visiting research professor at Facebook Inc. and as a visiting associate professor in the Department of Statistics at Stanford University. He served as co-Program Chair of the International Conference on Machine Learning (ICML) in 2014 and General Chair in 2019. Xing served as the founding director of CMU’s Center for Machine Learning and Health, established in 2015 as part of the Pittsburgh Health Data Alliance, a collaboration between CMU, the University of Pittsburgh, and the University of Pittsburgh Medical Center. In 2016, Xing co-founded Petuum Inc., a US-based startup. In 2017, Petuum raised $93 million in a round of venture funding from SoftBank. In 2018 Petuum was named a World Economic Forum Technology Pioneer. In 2019, Xing received the Carnegie Science Award for Startup Entrepreneurs in recognition of his leadership of Petuum. On 29 November 2020, Xing was appointed president of the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), with the appointment taking effect in January 2021. In 2024, Xing co-founded GenBio AI where he is chief scientist. The US-based startup, which he co-founded with David Baker, Ziv Bar-Joseph, Emma Lundberg, Le Song and Fred Hu, aims to create AI-driven digital organisms (AIDO) for the purposes of modeling medical treatments. Xing has overseen the launch of the MBZUAI Institute of Foundation Models (IFM), which focuses on research and development of large-scale foundation models. In 2025–2026, IFM released the open-source reasoning model K2 Think, which was covered internationally as part of the UAE’s push to develop domestically controlled (“sovereign”) AI capabilities. IFM presented PAN as a “world model” research project and demonstrated related systems publicly. MBZUAI also collaborated with G42 and Cerebras Systems on the Jais language model, an open-source Arabic–English large language model released in 2023, according to Reuters. == Awards and honors == Xing is a recipient of the National Science Foundation (NSF) Career Award and the Alfred P. Sloan Research Fellowship. Xing is an elected Fellow of the following institutes and associations: Association for the Advancement of Artificial Intelligence (AAAI) 2016 Institute of Electrical and Electronics Engineers (IEEE) 2019 for "contributions to machine learning algorithms and systems" American Statistical Association (ASA) 2022 Association for Computing Machinery (ACM) 2022 for "contributions to algorithms, architectures, and applications in machine learning" Institute of Mathematical Statistics (IMS) 2023 International Society for Computational Biology (ISCB) 2026 == Selected publications == Eric P. Xing; Michael I. Jordan; Stuart J. Russell; Andrew Y. Ng (2003). "Distance Metric Learning with Application to Clustering with Side-Information" (PDF). Advances in Neural Information Processing Systems 15. Advances in Neural Information Processing Systems. Wikidata Q77691192. Edoardo M. Airoldi; David M. Blei; Stephen E Fienberg; Eric P Xing (1 September 2008). "Mixed Membership Stochastic Blockmodels". Journal of Machine Learning Research. 9: 1981–2014. ISSN 1533-7928. PMC 3119541. PMID 21701698. Wikidata Q35058357. Eric P. Xing; Michael I. Jordan; Richard M. Karp (28 June 2001), Feature selection for high-dimensional genomic microarray data, vol. 18, pp. 601–608, Wikidata Q138678867 Xing EP; Karp RM (1 January 2001). "CLIFF: clustering of high-dimensional microarray data via iterative feature filtering using normalized cuts". Bioinformatics. 17 Suppl 1: S306-15. doi:10.1093/BIOINFORMATICS/17.SUPPL_1.S306. ISSN 1367-4803. PMID 11473022. Wikidata Q30657299.
Apache Pig
Apache Pig is a high-level platform for creating programs that run on Apache Hadoop. The language for this platform is called Pig Latin. Pig can execute its Hadoop jobs in MapReduce, Apache Tez, or Apache Spark. Pig Latin abstracts the programming from the Java MapReduce idiom into a notation which makes MapReduce programming high level, similar to that of SQL for relational database management systems. Pig Latin can be extended using user-defined functions (UDFs) which the user can write in Java, Python, JavaScript, Ruby or Groovy and then call directly from the language. == History == Apache Pig was originally developed at Yahoo Research around 2006 for researchers to have an ad hoc way of creating and executing MapReduce jobs on very large data sets. In 2007, it was moved into the Apache Software Foundation. === Naming === Regarding the naming of the Pig programming language, the name was chosen arbitrarily and stuck because it was memorable, easy to spell, and for novelty. The story goes that the researchers working on the project initially referred to it simply as 'the language'. Eventually they needed to call it something. Off the top of his head, one researcher suggested Pig, and the name stuck. It is quirky yet memorable and easy to spell. While some have hinted that the name sounds coy or silly, it has provided us with an entertaining nomenclature, such as Pig Latin for the language, Grunt for the shell, and PiggyBank for the CPAN-like shared repository. == Example == Below is an example of a "Word Count" program in Pig Latin: The above program will generate parallel executable tasks which can be distributed across multiple machines in a Hadoop cluster to count the number of words in a dataset such as all the webpages on the internet. == Pig vs SQL == In comparison to SQL, Pig has a nested relational model, uses lazy evaluation, uses extract, transform, load (ETL), is able to store data at any point during a pipeline, declares execution plans, supports pipeline splits, thus allowing workflows to proceed along DAGs instead of strictly sequential pipelines. On the other hand, it has been argued DBMSs are substantially faster than the MapReduce system once the data is loaded, but that loading the data takes considerably longer in the database systems. It has also been argued RDBMSs offer out of the box support for column-storage, working with compressed data, indexes for efficient random data access, and transaction-level fault tolerance. Pig Latin is procedural and fits very naturally in the pipeline paradigm while SQL is instead declarative. In SQL users can specify that data from two tables must be joined, but not what join implementation to use (You can specify the implementation of JOIN in SQL, thus "... for many SQL applications the query writer may not have enough knowledge of the data or enough expertise to specify an appropriate join algorithm."). Pig Latin allows users to specify an implementation or aspects of an implementation to be used in executing a script in several ways. In effect, Pig Latin programming is similar to specifying a query execution plan, making it easier for programmers to explicitly control the flow of their data processing task. SQL is oriented around queries that produce a single result. SQL handles trees naturally, but has no built in mechanism for splitting a data processing stream and applying different operators to each sub-stream. Pig Latin script describes a directed acyclic graph (DAG) rather than a pipeline. Pig Latin's ability to include user code at any point in the pipeline is useful for pipeline development. If SQL is used, data must first be imported into the database, and then the cleansing and transformation process can begin.
Generalized filtering
Generalized filtering is a generic Bayesian filtering scheme for nonlinear state-space models. It is based on a variational principle of least action, formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates as used in (multibody) dynamical systems analysis. Generalized filtering furnishes posterior densities over hidden states (and parameters) generating observed data using a generalized gradient descent on variational free energy, under the Laplace assumption. Unlike classical (e.g. Kalman-Bucy or particle) filtering, generalized filtering eschews Markovian assumptions about random fluctuations. Furthermore, it operates online, assimilating data to approximate the posterior density over unknown quantities, without the need for a backward pass. Special cases include variational filtering, dynamic expectation maximization and generalized predictive coding. == Definition == Definition: Generalized filtering rests on the tuple ( Ω , U , X , S , p , q ) {\displaystyle (\Omega ,U,X,S,p,q)} : A sample space Ω {\displaystyle \Omega } from which random fluctuations ω ∈ Ω {\displaystyle \omega \in \Omega } are drawn Control states U ∈ R {\displaystyle U\in \mathbb {R} } – that act as external causes, input or forcing terms Hidden states X : X × U × Ω → R {\displaystyle X:X\times U\times \Omega \to \mathbb {R} } – that cause sensory states and depend on control states Sensor states S : X × U × Ω → R {\displaystyle S:X\times U\times \Omega \to \mathbb {R} } – a probabilistic mapping from hidden and control states Generative density p ( s ~ , x ~ , u ~ ∣ m ) {\displaystyle p({\tilde {s}},{\tilde {x}},{\tilde {u}}\mid m)} – over sensory, hidden and control states under a generative model m {\displaystyle m} Variational density q ( x ~ , u ~ ∣ μ ~ ) {\displaystyle q({\tilde {x}},{\tilde {u}}\mid {\tilde {\mu }})} – over hidden and control states with mean μ ~ ∈ R {\displaystyle {\tilde {\mu }}\in \mathbb {R} } Here ~ denotes a variable in generalized coordinates of motion: u ~ = [ u , u ′ , u ″ , … ] T {\displaystyle {\tilde {u}}=[u,u',u'',\ldots ]^{T}} === Generalized filtering === The objective is to approximate the posterior density over hidden and control states, given sensor states and a generative model – and estimate the (path integral of) model evidence p ( s ~ ( t ) | m ) {\displaystyle p({\tilde {s}}(t)\vert m)} to compare different models. This generally involves an intractable marginalization over hidden states, so model evidence (or marginal likelihood) is replaced with a variational free energy bound. Given the following definitions: μ ~ ( t ) = a r g m i n μ ~ { F ( s ~ ( t ) , μ ~ ) } {\displaystyle {\tilde {\mu }}(t)={\underset {\tilde {\mu }}{\operatorname {arg\,min} }}\{F({\tilde {s}}(t),{\tilde {\mu }})\}} G ( s ~ , x ~ , u ~ ) = − ln p ( s ~ , x ~ , u ~ | m ) {\displaystyle G({\tilde {s}},{\tilde {x}},{\tilde {u}})=-\ln p({\tilde {s}},{\tilde {x}},{\tilde {u}}\vert m)} Denote the Shannon entropy of the density q {\displaystyle q} by H [ q ] = E q [ − log ( q ) ] {\displaystyle H[q]=E_{q}[-\log(q)]} . We can then write the variational free energy in two ways: F ( s ~ , μ ~ ) = E q [ G ( s ~ , x ~ , u ~ ) ] − H [ q ( x ~ , u ~ | μ ~ ) ] = − ln p ( s ~ | m ) + D K L [ q ( x ~ , u ~ | μ ~ ) | | p ( x ~ , u ~ | s ~ , m ) ] {\displaystyle F({\tilde {s}},{\tilde {\mu }})=E_{q}[G({\tilde {s}},{\tilde {x}},{\tilde {u}})]-H[q({\tilde {x}},{\tilde {u}}\vert {\tilde {\mu }})]=-\ln p({\tilde {s}}\vert m)+D_{KL}[q({\tilde {x}},{\tilde {u}}\vert {\tilde {\mu }})\vert \vert p({\tilde {x}},{\tilde {u}}\vert {\tilde {s}},m)]} The second equality shows that minimizing variational free energy (i) minimizes the Kullback-Leibler divergence between the variational and true posterior density and (ii) renders the variational free energy (a bound approximation to) the negative log evidence (because the divergence can never be less than zero). Under the Laplace assumption q ( x ~ , u ~ ∣ μ ~ ) = N ( μ ~ , C ) {\displaystyle q({\tilde {x}},{\tilde {u}}\mid {\tilde {\mu }})={\mathcal {N}}({\tilde {\mu }},C)} the variational density is Gaussian and the precision that minimizes free energy is C − 1 = Π = ∂ μ ~ μ ~ G ( μ ~ ) {\displaystyle C^{-1}=\Pi =\partial _{{\tilde {\mu }}{\tilde {\mu }}}G({\tilde {\mu }})} . This means that free-energy can be expressed in terms of the variational mean (omitting constants): F = G ( μ ~ ) + 1 2 ln | ∂ μ ~ μ ~ G ( μ ~ ) | {\displaystyle F=G({\tilde {\mu }})+\textstyle {1 \over 2}\ln \vert \partial _{{\tilde {\mu }}{\tilde {\mu }}}G({\tilde {\mu }})\vert } The variational means that minimize the (path integral) of free energy can now be recovered by solving the generalized filter: μ ~ ˙ = D μ ~ − ∂ μ ~ F ( s ~ , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}-\partial _{\tilde {\mu }}F({\tilde {s}},{\tilde {\mu }})} where D {\displaystyle D} is a block matrix derivative operator of identify matrices such that D u ~ = [ u ′ , u ″ , … ] T {\displaystyle D{\tilde {u}}=[u',u'',\ldots ]^{T}} === Variational basis === Generalized filtering is based on the following lemma: The self-consistent solution to μ ~ ˙ = D μ ~ − ∂ μ ~ F ( s , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}-\partial _{\tilde {\mu }}F(s,{\tilde {\mu }})} satisfies the variational principle of stationary action, where action is the path integral of variational free energy S = ∫ d t F ( s ~ ( t ) , μ ~ ( t ) ) {\displaystyle S=\int dt\,F({\tilde {s}}(t),{\tilde {\mu }}(t))} Proof: self-consistency requires the motion of the mean to be the mean of the motion and (by the fundamental lemma of variational calculus) μ ~ ˙ = D μ ~ ⇔ ∂ μ ~ F ( s ~ , μ ~ ) = 0 ⇔ δ μ ~ S = 0 {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}\Leftrightarrow \partial _{\tilde {\mu }}F({\tilde {s}},{\tilde {\mu }})=0\Leftrightarrow \delta _{\tilde {\mu }}S=0} Put simply, small perturbations to the path of the mean do not change variational free energy and it has the least action of all possible (local) paths. Remarks: Heuristically, generalized filtering performs a gradient descent on variational free energy in a moving frame of reference: μ ~ ˙ − D μ ~ = − ∂ μ ~ F ( s , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}-D{\tilde {\mu }}=-\partial _{\tilde {\mu }}F(s,{\tilde {\mu }})} , where the frame itself minimizes variational free energy. For a related example in statistical physics, see Kerr and Graham who use ensemble dynamics in generalized coordinates to provide a generalized phase-space version of Langevin and associated Fokker-Planck equations. In practice, generalized filtering uses local linearization over intervals Δ t {\displaystyle \Delta t} to recover discrete updates Δ μ ~ = ( exp ( Δ t ⋅ J ) − I ) J − 1 μ ~ ˙ J = ∂ μ ~ μ ~ ˙ = D − ∂ μ ~ μ ~ F ( s ~ , μ ~ ) {\displaystyle {\begin{aligned}\Delta {\tilde {\mu }}&=(\exp(\Delta t\cdot J)-I)J^{-1}{\dot {\tilde {\mu }}}\\J&=\partial _{\tilde {\mu }}{\dot {\tilde {\mu }}}=D-\partial _{{\tilde {\mu }}{\tilde {\mu }}}F({\tilde {s}},{\tilde {\mu }})\end{aligned}}} This updates the means of hidden variables at each interval (usually the interval between observations). == Generative (state-space) models in generalized coordinates == Usually, the generative density or model is specified in terms of a nonlinear input-state-output model with continuous nonlinear functions: s = g ( x , u ) + ω s x ˙ = f ( x , u ) + ω x {\displaystyle {\begin{aligned}s&=g(x,u)+\omega _{s}\\{\dot {x}}&=f(x,u)+\omega _{x}\end{aligned}}} The corresponding generalized model (under local linearity assumptions) obtains the from the chain rule s ~ = g ~ ( x ~ , u ~ ) + ω ~ s s = g ( x , u ) + ω s s ′ = ∂ x g ⋅ x ′ + ∂ u g ⋅ u ′ + ω s ′ s ″ = ∂ x g ⋅ x ″ + ∂ u g ⋅ u ″ + ω s ″ ⋮ x ~ ˙ = f ~ ( x ~ , u ~ ) + ω ~ x x ˙ = f ( x , u ) + ω x x ˙ ′ = ∂ x f ⋅ x ′ + ∂ u f ⋅ u ′ + ω x ′ x ˙ ″ = ∂ x f ⋅ x ″ + ∂ u f ⋅ u ″ + ω x ″ ⋮ {\displaystyle {\begin{aligned}{\tilde {s}}&={\tilde {g}}({\tilde {x}},{\tilde {u}})+{\tilde {\omega }}_{s}\\\\s&=g(x,u)+\omega _{s}\\s'&=\partial _{x}g\cdot x'+\partial _{u}g\cdot u'+\omega '_{s}\\s''&=\partial _{x}g\cdot x''+\partial _{u}g\cdot u''+\omega ''_{s}\\&\vdots \\\end{aligned}}\qquad {\begin{aligned}{\dot {\tilde {x}}}&={\tilde {f}}({\tilde {x}},{\tilde {u}})+{\tilde {\omega }}_{x}\\\\{\dot {x}}&=f(x,u)+\omega _{x}\\{\dot {x}}'&=\partial _{x}f\cdot x'+\partial _{u}f\cdot u'+\omega '_{x}\\{\dot {x}}''&=\partial _{x}f\cdot x''+\partial _{u}f\cdot u''+\omega ''_{x}\\&\vdots \end{aligned}}} Gaussian assumptions about the random fluctuations ω {\displaystyle \omega } then prescribe the likelihood and empirical priors on the motion of hidden states p ( s ~ , x ~ , u ~ | m ) = p ( s ~ | x ~ , u ~ , m ) p ( D x ~ | x , u ~ , m ) p ( x | m ) p ( u ~ | m ) p ( s ~ | x ~ , u ~ , m ) = N ( g ~ ( x ~ , u ~ ) , Σ ~ ( x ~ , u ~ ) s ) p ( D x ~ | x , u ~ , m ) = N ( f ~ ( x ~ , u ~ ) , Σ ~ ( x ~ , u ~ ) x ) {\displayst