Table 1:
Glossary of Variables.
NotationVariable
$P(·)$ Probability distribution
$Q(·)$ Variational posterior or empirical prior distribution
$F$ Variational free energy
$G$ Expected free energy
$uτ$ Action at time $τ$
$o=(o1,o2,…,oτ,…)$ Observation
$s=(s1,s2,…,sτ,…)$ Hidden (latent) states
$π$ Policy (sequence of actions)
$sτπ$ Expectation of state at time $τ$ under $Q(sτ|π)$
$sτu$ Expectation of state at time $τ$ under $Q(sτ|uτ)$
$vτπ$ Log expectation of state at time $τ$ under $Q(sτ|π)$
$oτu$ Expectation of observation at time $τ$ under $Q(oτ|u<τ)$
$uτo$ Expectation of action at time $τ$ under $Q(uτ|oτ)$
A Parameters of categorical likelihood distribution
B Parameters of categorical transition probabilities
C Parameters of categorical prior preferences
D Parameters of categorical initial state probabilities
H Conditional entropy of likelihood distribution
$a,a$ Prior and posterior Dirichlet parameters for A
$b,b$ Prior and posterior Dirichlet parameters for B
$d,d$ Prior and posterior Dirichlet parameters for D
$Cat(·)$ Categorical probability distribution
$Dir(·)$ Dirichlet probability distribution
$EP[·]$ Expectation under the subscripted probability distribution
$H[·]$ Shannon entropy of a probability distribution
$DKL[·∥·]$ Kullback-Leibler divergence between probability distributions
$ψ(·)$ Digamma function
$σ(·)$ Softmax (normalized exponential) function
NotationVariable
$P(·)$ Probability distribution
$Q(·)$ Variational posterior or empirical prior distribution
$F$ Variational free energy
$G$ Expected free energy
$uτ$ Action at time $τ$
$o=(o1,o2,…,oτ,…)$ Observation
$s=(s1,s2,…,sτ,…)$ Hidden (latent) states
$π$ Policy (sequence of actions)
$sτπ$ Expectation of state at time $τ$ under $Q(sτ|π)$
$sτu$ Expectation of state at time $τ$ under $Q(sτ|uτ)$
$vτπ$ Log expectation of state at time $τ$ under $Q(sτ|π)$
$oτu$ Expectation of observation at time $τ$ under $Q(oτ|u<τ)$
$uτo$ Expectation of action at time $τ$ under $Q(uτ|oτ)$
A Parameters of categorical likelihood distribution
B Parameters of categorical transition probabilities
C Parameters of categorical prior preferences
D Parameters of categorical initial state probabilities
H Conditional entropy of likelihood distribution
$a,a$ Prior and posterior Dirichlet parameters for A
$b,b$ Prior and posterior Dirichlet parameters for B
$d,d$ Prior and posterior Dirichlet parameters for D
$Cat(·)$ Categorical probability distribution
$Dir(·)$ Dirichlet probability distribution
$EP[·]$ Expectation under the subscripted probability distribution
$H[·]$ Shannon entropy of a probability distribution
$DKL[·∥·]$ Kullback-Leibler divergence between probability distributions
$ψ(·)$ Digamma function
$σ(·)$ Softmax (normalized exponential) function
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