Table 1.

Glossary of expressions.

Expression	Description
$o_{\tau }=(o_{\tau }^1,\dots ,o_{\tau }^M)\in \{0,1\},o_{\tau }\in [0,1],{\stackrel{⌢}{o}}_{\tau }=\ln o_{\tau }$	Outcomes in M modalities, their posterior expectations and logarithms
õ = (o1, …, ot)	Sequences of outcomes up until the current time
$s_{\tau }=(s_{\tau }^1,\dots ,s_{\tau }^N)\in \{0,1\},s_{\tau }\in [0,1],{\stackrel{⌢}{s}}_{\tau }=\ln s_{\tau }$	Hidden states in N factors, their posterior expectations and logarithms
$\tilde{s}=(s_1,\dots ,s_T)$	Sequences of hidden states up until the end of the current trial
$\pi =({\pi }_1,\dots ,{\pi }_K),\pi \in [0,1],\stackrel{⌢}{\pi }=\ln \pi$	Policies specifying action sequences, their posterior expectations and logarithms
a = π(t) = (a1, …, aN)	Action or control variables for each factor of hidden states
$\gamma ,\gamma =\frac{1}{\beta }$	The precision (inverse temperature) of beliefs about policies and its posterior expectation
β	Prior expectation of temperature (inverse precision) of beliefs about policies
Am	Likelihood array mapping from hidden states to the m-th modality
${\mathbf{\text{B}}}^n(a),{\mathbf{\text{B}}}_{\tau }^{n,\pi }={\mathbf{\text{B}}}^n(a=\pi (\tau ))$	Transition probability for the n-th hidden state under each action
${\mathbf{\text{C}}}_{\tau }^m$	Logarithm of the prior probability of the m-th outcome; i.e., preferences or utility
Dn	Prior expectation of the n-th hidden state at the beginning of each trial
F:F${}_{\pi }=F(\pi )=\sum _{\tau }F(\pi ,\tau )=\sum _{n,\tau }F(\pi ,\tau ,n)$	Variational free energy for each policy
G:G${}_{\pi }=G(\pi )=\sum _{\tau }G(\pi ,\tau )=\sum _{m,\tau }G(\pi ,\tau ,m)$	Expected free energy for each policy
Hm	Entropy of the m-th outcome
A· s${}^{\bullet }=\sum _{i,j,k\dots }{\mathbf{\text{A}}}_{i,j,k\dots }{\mathbf{\text{s}}}_i^1{\mathbf{\text{s}}}_j^2{\mathbf{\text{s}}}_k^3\dots$	Dot product (or sum of products), returning a scalar
A· s∕n	A dot product over all but the n-th vector