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Exposure Notification
The (Google/Apple) Exposure Notification System (GAEN) is a framework and protocol specification developed by Apple Inc. and Google to facilitate digital contact tracing during the COVID-19 pandemic. When used by health authorities, it augments more traditional contact tracing techniques by automatically logging close approaches among notification system users using Android or iOS smartphones. Exposure Notification is a decentralized reporting protocol built on a combination of Bluetooth Low Energy technology and privacy-preserving cryptography. It is an opt-in feature within COVID-19 apps developed and published by authorized health authorities. Unveiled on April 10, 2020, it was made available on iOS on May 20, 2020, as part of the iOS 13.5 update and on December 14, 2020, as part of the iOS 12.5 update for older iPhones. On Android, it was added to devices via a Google Play Services update, supporting all versions since Android Marshmallow. The Apple/Google protocol is similar to the Decentralized Privacy-Preserving Proximity Tracing (DP-3T) protocol created by the European DP-3T consortium and the Temporary Contact Number (TCN) protocol by Covid Watch, but is implemented at the operating system level, which allows for more efficient operation as a background process. Since May 2020, a variant of the DP-3T protocol is supported by the Exposure Notification Interface. Other protocols are constrained in operation because they are not privileged over normal apps. This leads to issues, particularly on iOS devices where digital contact tracing apps running in the background experience significantly degraded performance. The joint approach is also designed to maintain interoperability between Android and iOS devices, which constitute nearly all of the market. The ACLU stated the approach "appears to mitigate the worst privacy and centralization risks, but there is still room for improvement". In late April, Google and Apple shifted the emphasis of the naming of the system, describing it as an "exposure notification service", rather than "contact tracing" system. == Technical specification == Digital contact tracing protocols typically have two major responsibilities: encounter logging and infection reporting. Exposure Notification only involves encounter logging which is a decentralized architecture. The majority of infection reporting is centralized in individual app implementations. To handle encounter logging, the system uses Bluetooth Low Energy to send tracking messages to nearby devices running the protocol to discover encounters with other people. The tracking messages contain unique identifiers that are encrypted with a secret daily key held by the sending device. These identifiers change every 15–20 minutes as well as Bluetooth MAC address in order to prevent tracking of clients by malicious third parties through observing static identifiers over time. The sender's daily encryption keys are generated using a random number generator. Devices record received messages, retaining them locally for 14 days. If a user tests positive for infection, the last 14 days of their daily encryption keys can be uploaded to a central server, where it is then broadcast to all devices on the network. The method through which daily encryption keys are transmitted to the central server and broadcast is defined by individual app developers. The Google-developed reference implementation calls for a health official to request a one-time verification code (VC) from a verification server, which the user enters into the encounter logging app. This causes the app to obtain a cryptographically signed certificate, which is used to authorize the submission of keys to the central reporting server. The received keys are then provided to the protocol, where each client individually searches for matches in their local encounter history. If a match meeting certain risk parameters is found, the app notifies the user of potential exposure to the infection. Google and Apple intend to use the received signal strength (RSSI) of the beacon messages as a source to infer proximity. RSSI and other signal metadata will also be encrypted to resist deanonymization attacks. === Version 1.0 === To generate encounter identifiers, first a persistent 32-byte private Tracing Key ( t k {\displaystyle tk} ) is generated by a client. From this a 16 byte Daily Tracing Key is derived using the algorithm d t k i = H K D F ( t k , N U L L , 'CT-DTK' | | D i , 16 ) {\displaystyle dtk_{i}=HKDF(tk,NULL,{\text{'CT-DTK'}}||D_{i},16)} , where H K D F ( Key, Salt, Data, OutputLength ) {\displaystyle HKDF({\text{Key, Salt, Data, OutputLength}})} is a HKDF function using SHA-256, and D i {\displaystyle D_{i}} is the day number for the 24-hour window the broadcast is in starting from Unix Epoch Time. These generated keys are later sent to the central reporting server should a user become infected. From the daily tracing key a 16-byte temporary Rolling Proximity Identifier is generated every 10 minutes with the algorithm R P I i , j = Truncate ( H M A C ( d t k i , 'CT-RPI' | | T I N j ) , 16 ) {\displaystyle RPI_{i,j}={\text{Truncate}}(HMAC(dtk_{i},{\text{'CT-RPI'}}||TIN_{j}),16)} , where H M A C ( Key, Data ) {\displaystyle HMAC({\text{Key, Data}})} is a HMAC function using SHA-256, and T I N j {\displaystyle TIN_{j}} is the time interval number, representing a unique index for every 10 minute period in a 24-hour day. The Truncate function returns the first 16 bytes of the HMAC value. When two clients come within proximity of each other they exchange and locally store the current R P I i , j {\displaystyle RPI_{i,j}} as the encounter identifier. Once a registered health authority has confirmed the infection of a user, the user's Daily Tracing Key for the past 14 days is uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier used in the report period, matching it against the user's local encounter log. If a matching entry is found, then contact has been established and the app presents a notification to the user warning them of potential infection. === Version 1.1 === Unlike version 1.0 of the protocol, version 1.1 does not use a persistent tracing key, rather every day a new random 16-byte Temporary Exposure Key ( t e k i {\displaystyle tek_{i}} ) is generated. This is analogous to the daily tracing key from version 1.0. Here i {\displaystyle i} denotes the time is discretized in 10 minute intervals starting from Unix Epoch Time. From this two 128-bit keys are calculated, the Rolling Proximity Identifier Key ( R P I K i {\displaystyle RPIK_{i}} ) and the Associated Encrypted Metadata Key ( A E M K i {\displaystyle AEMK_{i}} ). R P I K i {\displaystyle RPIK_{i}} is calculated with the algorithm R P I K i = H K D F ( t e k i , N U L L , 'EN-RPIK' , 16 ) {\displaystyle RPIK_{i}=HKDF(tek_{i},NULL,{\text{'EN-RPIK'}},16)} , and A E M K i {\displaystyle AEMK_{i}} using the algorithm A E M K i = H K D F ( t e k i , N U L L , 'EN-AEMK' , 16 ) {\displaystyle AEMK_{i}=HKDF(tek_{i},NULL,{\text{'EN-AEMK'}},16)} . From these values a temporary Rolling Proximity Identifier ( R P I i , j {\displaystyle RPI_{i,j}} ) is generated every time the BLE MAC address changes, roughly every 15–20 minutes. The following algorithm is used: R P I i , j = A E S 128 ( R P I K i , 'EN-RPI' | | 0 x 000000000000 | | E N I N j ) {\displaystyle RPI_{i,j}=AES128(RPIK_{i},{\text{'EN-RPI'}}||{\mathtt {0x000000000000}}||ENIN_{j})} , where A E S 128 ( Key, Data ) {\displaystyle AES128({\text{Key, Data}})} is an AES cryptography function with a 128-bit key, the data is one 16-byte block, j {\displaystyle j} denotes the Unix Epoch Time at the moment the roll occurs, and E N I N j {\displaystyle ENIN_{j}} is the corresponding 10-minute interval number. Next, additional Associated Encrypted Metadata is encrypted. What the metadata represents is not specified, likely to allow the later expansion of the protocol. The following algorithm is used: Associated Encrypted Metadata i , j = A E S 128 _ C T R ( A E M K i , R P I i , j , Metadata ) {\displaystyle {\text{Associated Encrypted Metadata}}_{i,j}=AES128\_CTR(AEMK_{i},RPI_{i,j},{\text{Metadata}})} , where A E S 128 _ C T R ( Key, IV, Data ) {\displaystyle AES128\_CTR({\text{Key, IV, Data}})} denotes AES encryption with a 128-bit key in CTR mode. The Rolling Proximity Identifier and the Associated Encrypted Metadata are then combined and broadcast using BLE. Clients exchange and log these payloads. Once a registered health authority has confirmed the infection of a user, the user's Temporary Exposure Keys t e k i {\displaystyle tek_{i}} and their respective interval numbers i {\displaystyle i} for the past 14 days are uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier starting from interval number i {\displaystyle i} ,
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David Horn (Israeli physicist)
David Horn (Hebrew: דוד הורן; born 10 September 1937) is a Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University (TAU), Israel. He has served as Vice-Rector of TAU, Chairman of the School of Physics and Astronomy and as Dean of the Faculty of Exact Sciences in TAU. He is a fellow of the American Physical Society, nominated for "contributions to theoretical particle physics, including the seminal work on finite energy sum rules, research of the phenomenology of hadronic processes, and investigation of Hamiltonian lattice theories". == Early life and education == David Horn was born and educated in Haifa. He graduated from the Reali School in 1955. He began his academic studies in Physics at the Technion in Haifa in 1957, and received his B.Sc. (Summa Cum Laude) in 1961, and M.Sc. in 1962. He continued his Ph.D. studies at the Hebrew University of Jerusalem until 1965. His thesis on "Some Aspects of the Structure of Weak Interactions" was supervised by Prof. Yuval Ne'eman. == Career == Horn joined the newly founded Tel Aviv University as an assistant in 1962. He became a lecturer in 1965, a senior lecturer in 1967 and an associate professor in 1968. He was promoted to full professor of Physics in 1972. In 1974 he became the incumbent of the Edouard and Francoise Jaupart Chair of Theoretical Physics of Particles and Fields, a position he held until 2007. Horn has supervised 43 graduate students at TAU and authored over 240 scientific publications. He retired as a professor emeritus in 2005, and continues to be an active researcher. Horn spent a significant part of his career holding visiting academic positions at other universities and research institutes, including: Postdoctoral Fellow at Argonne National Lab, ILL, Research Fellow and three times Visiting Associate at California Institute of Technology, Pasadena, CA, Visitor at CERN in Geneva, Visiting Professor at Cornell University, NY, Member of the Institute for Advanced Study, Princeton, NJ, Visiting Professor at SLAC in Stanford University, CA, and Visiting Professor at Kyoto University, Japan. Beginning from 1980, Horn held official positions at Tel Aviv University, starting with tenure as Vice-Rector (1980-1983), a position he left for research at SLAC. After returning he was nominated Chairman of the Department of High Energy Physics (1984-1986), followed by tenures as Chairman of the School of Physics and Astronomy (1986-9), Dean of the Raymond and Beverly Sackler Faculty of Exact Sciences (1990-1995), and first Director of the Adams Super Center for Brain Studies (1993-2000). Horn has also held national and international professional positions. He was Chairman of the Israel Commission for High Energy Physics (1983-2003), and, in this capacity, served as an Israeli observer of the council of CERN (1991-2003). He served as member of the Israel Council for Higher Education (1987-1991), member of the Executive Committee of the European Physical Society (1989-1992) and member of the European Strategy Forum on Research Infrastructures (2005-2017). He chaired the Israeli Committee of Research Infrastructures (2012-2016), issuing roadmaps for scientific RI in 2013 and 2016. == Research == Horn's research work focused on theory and phenomenology of High Energy Physics until 1990. He then shifted his interests to Neural Computation and Machine Learning and, since 2005, he has also published in Bioinformatics. Together with Richard Dolen and Christoph Schmid he discovered the Finite Energy Sum Rules in 1967. It was a realization of the bootstrap approach to hadronic structure, and became known as the Dolen-Horn-Schmid Duality. Together with Richard Silver he investigated a model of coherent production of pions at high energy hadron collisions in 1971, and together with Jeffrey Mandula he undertook the investigation of mesons with constituent gluons in 1978. Moving to lattice gauge theories in 1979, he discovered, together with Shimon Yankielowic and Marvin Weinstein, a non-confining phase in Z(N) theories for large N. In 1981 he demonstrated the existence of finite matrix models with link gauge fields, nowadays known as quantum link models. In 1984 Horn and Weinstein developed the t-expansion methodology. Horn's contributions to neural modeling include a novel mechanism for memory maintenance via neuronal regulation in 1998, developed with Nir Levy and Eytan Ruppin and unsupervised learning of natural languages in 2005, a joint work with Zach Solan, Eytan Ruppin and Shimon Edelman, introducing novel algorithms for motif and grammar extraction from text. Horn has contributed to algorithms of clustering, an important topic in Machine Learning, by developing Support Vector Clustering (SVC) in 2001, together with Asa Ben Hur, Hava Siegelmann and Vladimir Vapnik. This was followed shortly thereafter by a joint work with Assaf Gottlieb on Quantum Clustering (QC). His contributions to Bioinformatics include motif descriptions of function and structure of proteins, as well as motif studies of genomic structures. Together with Erez Persi he studied compositional order of proteomes, and repeat instability of genomes, as evolution markers of organisms and of cancer (a joint work with Persi and others). == Honors == Horn is a Fellow of the American Physical Society (1985) and a Fellow of the Israel Physical Society (2018). == Publications == === Selected articles === R. Dolen, D. Horn and C. Schmid; Prediction of Regge-parameters of rho poles from low-energy pi-N scattering data Phys. Rev. Lett. 19 (1967) 402–407. Finite-Energy Sum Rules and Their Application to pi-N Charge Exchange Phys. Rev. 166 (1968) 1768–1781. D. Horn and R. Silver: Coherent production of pions, Annals Phys. 66 (1971) 509-541 T. Banks, D. Horn and H. Neuberger: Bosonization of the SU(N) Thirring Models, Nucl. Phys. B108, 119 (1976). D. Horn and J. Mandula: Model of Mesons with Constituent Gluons, Phys. Rev. D17, 898 (1978). D. Horn, M. Weinstein and S. Yankielowicz: Hamiltonian Approach to Z(N) Lattice Gauge Theories, Phys. Rev. D19, 3715 (1979). D. Horn: Finite Matrix Models with Continuous Local Gauge Invariance, Phys. Lett. 100B, 149-151 (1981). T. Banks, Y. Dothan and D. Horn: Geometric Fermions, Phys. Lett. 117B, 413 (1982). D. Horn and M. Weinstein: The t expansion: A nonperturbative analytic tool for Hamiltonian systems. Phys. Rev. D 30, 1256-1270 (1984). Ury Naftaly, Nathan Intrator and David Horn: Optimal Ensemble Averaging of Neural Networks. Network, Computation in Neural Systems, 8, 283-296 (1997). David Horn, Nir Levy, Eytan Ruppin: Memory Maintenance via Neuronal Regulation, Neural Computation, 10, 1-18 (1998). Asa Ben-Hur, David Horn, Hava Siegelmann and Vladimir Vapnik: Support Vector Clustering. Journal of Machine Learning Research 2, 125-137 (2001). David Horn and Assaf Gottlieb: Algorithm for data clustering in pattern recognition problems based on quantum mechanics, Phys. Rev. Lett. 88 (2002) 18702 Zach Solan, David Horn, Eytan Ruppin and Shimon Edelman: Unsupervised learning of natural languages, Proc. Natl. Acad. Sc. 102 (2005) 11629–11634. Vered Kunik, Yasmine Meroz, Zach Solan, Ben Sandbank, Uri Weingart, Eytan Ruppin and David Horn: Functional representation of enzymes by specific peptides. PLOS Computational Biology 2007, 3(8):e167. Benny Chor, David Horn, Yaron Levy, Nick Goldman and Tim Massingham: Genomic DNA k-mer spectra: models and modalities. Genome Biology 2009, 10(10):R108 Erez Persi and David Horn. Systematic Analysis of Compositional Order of Proteins Reveals New Characteristics of Biological Functions and a Universal Correlate of Macroevolution. PLoS Comput Biol 9 (2013): e1003346. David Horn. Taxa counting using Specific Peptides of Aminoacyl tRNA Synthetases Encyclopedia of Metagenomics, Springer, 2013. Sagi Shporer, Benny Chor, Saharon Rosset, David Horn. Inversion symmetry of DNA k-mer counts: validity and deviations. BMC Genomics 2016, 17:696 Erez Persi, Davide Prandi, Yuri I. Wolf, Yair Pozniak, Christopher Barbieri, Paola Gasperini, Himisha Beltran, Bishoy M. Faltas, Mark A. Rubin, Tamar Geiger, Eugene V. Koonin, Francesca Demichelis, David Horn. Proteomic and Genomic Signatures of Repeat Instability in Cancer and Adjacent Normal Tissues. PNAS 116, 34, 2019 - 08790 === Book === David Horn and Fredrick Zachariasen: Hadron Physics at Very High Energies. Benjamin 1973. === Patents === Method and Apparatus for Quantum Clustering. USA Patent No. 7,653,646 B2. Method for discovering relationships in data by dynamic quantum clustering USA Patent No 8874412 and USA Patent No. 9,646,074. == Personal life == Horn was married to Nira Fuss since 1963 until her death in 2019. He is a father of three, Yuval, Tamar, and Oded, and grandfather of nine. He lives in Tel Aviv, Israel.
Phrase structure grammar
The term phrase structure grammar was originally introduced by Noam Chomsky as the term for grammar studied previously by Emil Post and Axel Thue (Post canonical systems). Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense, phrase structure grammars are also known as constituency grammars. The defining character of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation of dependency grammars. == History == In 1956, Chomsky wrote, "A phrase-structure grammar is defined by a finite vocabulary (alphabet) Vp, and a finite set Σ of initial strings in Vp, and a finite set F of rules of the form: X → Y, where X and Y are strings in Vp." == Constituency relation == In linguistics, phrase structure grammars are all those grammars that are based on the constituency relation, as opposed to the dependency relation associated with dependency grammars; hence, phrase structure grammars are also known as constituency grammars. Any of several related theories for the parsing of natural language qualify as constituency grammars, and most of them have been developed from Chomsky's work, including Government and binding theory Generalized phrase structure grammar Head-driven phrase structure grammar Lexical functional grammar The minimalist program Nanosyntax Further grammar frameworks and formalisms also qualify as constituency-based, although they may not think of themselves as having spawned from Chomsky's work, e.g. Arc pair grammar, and Categorial grammar.
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