TransportMaps.Densities.Decomposable.SequentialInferenceDensities

Module Contents

Classes

MarkovChainDensity	Distribution of a Markov process (optionally with hyper-parameters)
SequentialHiddenMarkovChainDensity	param pi_list: list of transition densities
MarkovComponentDensity	\(i\)-th Markov component of a HiddenMarkovChainDistribution

class TransportMaps.Densities.Decomposable.SequentialInferenceDensities.MarkovChainDensity(*args, **kwargs)

Bases: TransportMaps.Distributions.Decomposable.MarkovChainDistribution

Distribution of a Markov process (optionally with hyper-parameters)

For the index set \(A=[t_0,\ldots,t_k]\) with \(t_0<t_1<\ldots <t_k\), and the user defined distributions \(\pi({\bf Z}_{t_i} \vert {\bf Z}_{t_{i-1}}, \Theta)\), \(\pi({\bf Z}_{t_0} \vert \Theta)\) and \(\pi(\Theta)\) defines the distribution

\[\pi(\Theta, {\bf Z}_A) = \left( \prod_{i=1}^k \pi(t_i; {\bf Z}_{t_i} \vert {\bf Z}_{t_{i-1}}, \Theta) \right) \pi({\bf Z}_{t_0} \vert \Theta) \pi(\Theta)\]

associated to the process \({\bf Z}_A\).

Parameters:

• pi_list (list of ConditionalDistribution) – list of transition distributions \(\{\pi({\bf Z}_{t_0} \vert \Theta), \pi({\bf Z}_{t_1}\vert {\bf Z}_{t_{0}},\Theta), \ldots \}\)

• pi_hyper (Distribution) – prior on hyper-parameters \(h(\Theta)\)

class TransportMaps.Densities.Decomposable.SequentialInferenceDensities.SequentialHiddenMarkovChainDensity(*args, **kwargs)

Bases: TransportMaps.Distributions.Decomposable.SequentialHiddenMarkovChainDistribution

Parameters:

• pi_list (list of ConditionalDistribution) – list of transition densities \([\pi({\bf Z}_{t_0}\vert\Theta), \pi({\bf Z}_{t_1}\vert{\bf Z}_{t_{0}},\Theta), \ldots ]\)

• ll_list (list of LogLikelihood) – list of log-likelihoods \(\{\log\mathcal{L}({\bf y}_t \vert {\bf Z}_t,\Theta)\}_{t\in B}\)

• pi_hyper (Distribution) – prior on hyper-parameters \(h(\Theta)\)

class TransportMaps.Densities.Decomposable.SequentialInferenceDensities.MarkovComponentDensity(*args, **kwargs)

Bases: TransportMaps.Distributions.Decomposable.MarkovComponentDistribution

\(i\)-th Markov component of a HiddenMarkovChainDistribution

If \(i=-1\) the Markov component is given by

\[\pi^{0:n}(\Theta, {\bf Z}_{t_0}, \ldots, {\bf Z}_{t_n}) := \left( \prod_{t \in \{t_0,\ldots,t_n\} \cap B} \mathcal{L}({\bf y}_t \vert \Theta, {\bf Z}_t) \right) \left( \prod_{i=1}^n \pi({\bf Z}_{t_i}\vert \Theta, {\bf Z}_{t_{i-1}}) \right) \pi({\bf Z}_{t_0}\vert\Theta) \pi(\Theta) \;.\]

If \(i>=0\) then the Markov component is

\[\pi^{i:i+n}\left(\Theta, {\bf Z}_{t_i}, \ldots, {\bf Z}_{t_{i+n}}\right) := \eta(\Theta, {\bf Z}_{t_i}) \left( \prod_{t \in \left\{t_{i+1},\ldots,t_{i+n}\right\} \cap B} \mathcal{L}\left({\bf y}_t \vert \mathfrak{T}_{i-1}^{\Theta}(\Theta), {\bf Z}_t\right) \right) \left( \prod_{k=i+1}^{i+n-1} \pi\left({\bf Z}_{t_k+1}\vert {\bf Z}_{t_{k}}, \mathfrak{T}_{i-1}^{\Theta}(\Theta) \right) \right) \pi\left({\bf Z}_{t_{i+1}} \vert \mathfrak{M}_{i-1}^{1}(\Theta, {\bf Z}_{t_i}), \mathfrak{T}_{i-1}^{\Theta}(\Theta) \right) \;,\]

where \(\mathfrak{T}_{i-1}^{\Theta}\) and \(\mathfrak{M}_{i-1}^{1}\) are the hyper-parameter and forecast components of the map computed at step \(i-1\), using the sequential algorithm described in [TM3].

Parameters:

• idx0 (int) – index \(i\) of the Markov component

• pi_list (list of Distribution) – list of \(n\) transition densities

• ll_list (list of LogLikelihood) – list of \(n\) log-likelihoods (None for missing data) \(\{\log\mathcal{L}({\bf y}_t \vert \Theta, {\bf Z}_t)\}_{t\in B}\)

• state_dim (int) – dimension of the state-space

• hyper_dim (int) – dimension of the parameter-space

• pi_hyper (Distribution) – prior on hyper-parameters \(h(\Theta)\)

• state_map (TransportMap) – forecast map \(\mathfrak{M}_{i-1}^{1}\) from step \(i-1\).

• hyper_map (TransportMap) – hyper-parameter map \(\mathfrak{T}_{i-1}^{\Theta}\) from step \(i-1\).