Table 1: A Generative Model of Perception.

Prior Beliefs (Generative Model) (P)	Approximate Posterior Beliefs (Q)
$P\left(s\right)=\underset{\text{state}\text{prior}}{\underbrace{Cat\left(\mathbf{D}\right)}}$	$Q\left(s\right)=\underset{\text{state}\text{posterior}}{\underbrace{Cat\left(\overline{\mathbf{s}}\right)}}$
$P\left(o\right|s)=\underset{\text{likelihood}}{\underbrace{Cat\left(\mathbf{A}\right)}}$	$\overline{\mathbf{s}}=\sigma (\underset{\genfrac{}{}{0pt}{}{\mathrm{prior}}{\mathrm{beliefs}}}{\underbrace{ln\mathbf{D}}}+\underset{\genfrac{}{}{0pt}{}{\mathrm{sensory}}{\mathrm{evidence}}}{\underbrace{ln\mathbf{A}·o}})$
$\underset{\genfrac{}{}{0pt}{}{\mathrm{state}}{\mathrm{expectations}}}{\underbrace{\mathbf{s}}}=\mathbf{D}$
$\underset{\genfrac{}{}{0pt}{}{\mathrm{outcome}}{\mathrm{expectations}}}{\underbrace{\mathbf{o}}}=\mathbf{As}$

Notes: The generative model is defined in terms of prior beliefs about hidden states $P\left(s\right)=Cat\left(\mathbf{D}\right)$ (where $\mathbf{D}$ is a vector encoding the prior probability of each state) and a likelihood mapping $P\left(o\right|s)=Cat(\mathbf{A})$ (where $\mathbf{A}$ is a matrix encoding the probability of each outcome given a particular state). $Cat\left(\mathbf{X}\right)$ denotes a categorical probability distribution (see also the supplementary information A3). Through variational inference, the beliefs about hidden states $s$ are updated given an observed sensory outcome $o$, thus arriving at an approximate posterior $Q\left(s\right)=Cat\left(\overline{\mathbf{s}}\right)$ (see also supplementary information in appendix A1), where $\overline{\mathbf{s}}=\sigma (ln\mathbf{D}+ln\mathbf{A}·o)$. Here, the dot notation indicates backward matrix multiplication (in the case of a normalized set of probabilities and a likelihood mapping): for a given outcome, $\mathbf{A}·o$ returns the (renormalized) probability or likelihood of each hidden state s (see also the supplementary information in appendix A2).