A package implementing distributions, Markov kernels, and likelihoods that all play nice with eachother. The main motivation is to simplify the implementation of Bayesian filtering and smoothing algorithms. Let $\pi(x)$ be a probability distribution and $k(y\mid x)$ a Markov kernel then only the following operations are required for Bayesian state estimation
$$ k(y) = \int k(y\mid x) \pi(x) \mathrm{d} x, $$
which gives the prediction step in Bayesian filtering.
$$
k(y\mid x)\pi(x) = \pi(x \mid y) k(y),
$$
where evaluation of $\pi(x \mid y)$ at $y$ gives Bayes' rule and $k(y)$ is the marginal distribution of $y$ (used for prediction error decomposition of the marginal likelihood). In fact, the prediction step may be implemented with the inverse factorization operation as well, in which case $\pi(x\mid y)$ is the backwards kernel used to compute smoothing distributions in the Rauch-Tung-Striebel recursion. Please see the tutorials in the documentation.
] add MarkovKernels
Types for representing marginal distributions, Markov kernels, and likelihoods:
abstract type AbstractAffineMap end # used to represent affine conditional means
abstract type AbstractDistribution end
abstract type AbstractMarkovKernel end
abstract type AbstractLikelihood end
Currently, the following concrete types are defined:
Normal # Vector valued Normal distributons
Dirac # Vector valued Dirac distributions
NormalKernel # Vector valued Normal kernels
DiracKernel # Vector valued Dirac kernels
Likelihood # AbstractMarkovKernel paired with a measurement
The following type union is used to represent the (conditional) covariance:
const CovarianceParameter{T} = Union{HermOrSym{T},Factorization{T}}
Additionally, the following aliases are defined:
const AffineNormalKernel{T} = NormalKernel{T,<:AbstractAffineMap,<:CovarianceParameter}
const AffineDiracKernel{T} = DiracKernel{T,<:AbstractAffineMap}
For the purpose of Bayesian state estimation, ideally the following functions are defined:
marginalize(D::AbstractDistribution, K::AbstractMarkovKernel)
invert(D::AbstractDistribution, K::AbstractMarkovKernel)
posterior(D::AbstractDistribution, L::AbstractLikelihood)
posterior_and_loglike(D::AbstractDistribution, L::AbstractLikelihood)
These are currently implemented for Normal, AffineNormalKernel, AffineDiracKernel. Additionally, marginalize is implemented for Dirac with respect to the aforementioned kernels.
In practice, these functions can not be implemented exactly for a given distribution / Markov kernel pair. Therefore, it is up to the user to define, when required, appropriate approximations, i.e.:
predict(D::AbstractDistribution, K::AbstractMarkovKernel)
update(D::AbstractDistribution, L::AbstractLikelihood)