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
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
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
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
) 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
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
:
* ''A sample space''
from which random fluctuations
are drawn
* ''Control states''
– that act as external causes, input or forcing terms
* ''Hidden states''
– that cause sensory states and depend on control states
* ''Sensor states''
– a probabilistic mapping from hidden and control states
* ''Generative density''
– over sensory, hidden and control states under a generative model
* ''Variational density''
– over hidden and control states with mean
Here ~ denotes a variable in generalized coordinates of motion:
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
A marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample from a prior and is therefore often referred to as model evi ...
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:
:
:
Denote the
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum W ...
of the density
by