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Notation Table: Active Inference 401: Advanced PhD Seminar

Canonical symbols and notation used throughout the curriculum. All modules must use these symbols consistently.

Variational and Free Energy Quantities

Symbol	Meaning	LaTeX	Definition
$F$	Variational Free Energy	$F$	$F = D_{KL}[q(s) \| p(s)] - \mathbb{E}_{q(s)}[\ln p(o \mid s)]$; upper bound on surprisal
$G$	Expected Free Energy	$G$	$G(\pi) = \mathbb{E}_{q(o,s \mid \pi)}[\ln q(s \mid \pi) - \ln p(o, s)]$; decomposes into Risk + Ambiguity
$\tilde{F}$	Generalized Free Energy	$\tilde{F}$	Combines VFE and EFE under a single variational functional
$\mathbb{F}$	Free energy functional (path integral)	$\mathbb{F}$	Path integral formulation over state trajectories
$D_{KL}$	Kullback-Leibler Divergence	$D_{KL}$	$D_{KL}[q \| p] = \mathbb{E}_q[\ln q - \ln p]$; always $\geq 0$ (Gibbs' inequality)
$\mathcal{L}$	ELBO / Negative VFE	$\mathcal{L}$	$\mathcal{L} = -F = \ln p(o) - D_{KL}[q(s) \| p(s \mid o)]$

Generative Model Components

Symbol	Meaning	LaTeX	Definition
$p(o,s)$	Generative model (joint)	$p(o,s)$	Joint distribution over observations $o$ and hidden states $s$
$p(o \mid s)$	Likelihood	$p(o \mid s)$	Maps hidden states to predicted observations
$p(s' \mid s, a)$	Transition model	$p(s' \mid s, a)$	State dynamics conditioned on action
$q(s)$	Approximate posterior	$q(s)$	Variational approximation to $p(s \mid o)$ (recognition density)
$\pi$	Policy	$\pi$	Sequence of actions $\pi = (a_1, \ldots, a_T)$
$\gamma$	Policy precision	$\gamma$	Inverse temperature for policy selection: $p(\pi) = \sigma(\gamma \cdot G(\pi))$

POMDP Matrices (Discrete State-Space)

Symbol	Meaning	LaTeX	Definition
A	Likelihood matrix	$\mathbf{A}$	$A_{ij} = p(o_i \mid s_j)$; categorical observation model
B	Transition matrix	$\mathbf{B}$	$B_{ij}^{(a)} = p(s_i' \mid s_j, a)$; action-conditioned state transitions
C	Preference vector	$\mathbf{C}$	$C_i = \ln p(o_i)$; log prior preferences over observations
D	Prior over initial states	$\mathbf{D}$	$D_i = p(s_i^{(0)})$; initial state distribution
E	Habit vector	$\mathbf{E}$	$E_\pi = p(\pi)$; prior over policies before EFE evaluation

Markov Blanket and Partition States

Symbol	Meaning	LaTeX	Definition
$\eta$	External states	$\eta$	States outside the Markov blanket; hidden from internal states
$\mu$	Internal states	$\mu$	States inside the blanket; parameterize beliefs about $\eta$
$s$	Sensory states	$s$	Blanket states influenced by $\eta$ but not $\mu$
$a$	Active states	$a$	Blanket states influenced by $\mu$ but not $\eta$
$b = (s, a)$	Blanket states	$b$	Union of sensory and active states
$p = (\mu, b)$	Particular states	$p$	Internal + blanket states; constitutes the "agent"

Dynamical and Geometric Quantities

Symbol	Meaning	LaTeX	Definition
$\Gamma$	Fisher information metric	$\Gamma$	Riemannian metric on $\mathcal{M}$; $\Gamma_{ij} = \mathbb{E}[\partial_i \ln p \cdot \partial_j \ln p]$
$Q$	Solenoidal flow matrix	$Q$	Antisymmetric ($Q = -Q^T$); generates non-dissipative flow
$f_\mu$	Flow of internal states	$f_\mu$	$f_\mu = (Q_{\mu\mu} - \Gamma_{\mu\mu}) \nabla_\mu F$
$p^*(\cdot)$	NESS density	$p^*(\cdot)$	Non-equilibrium steady-state distribution; $\partial_t p^* = 0$
$\mathcal{M}$	Statistical manifold	$\mathcal{M}$	Space of probability distributions with Fisher metric

Information-Theoretic Quantities

Symbol	Meaning	LaTeX	Definition
$H[p]$	Shannon entropy	$H[p]$	$H[p] = -\mathbb{E}_p[\ln p]$
$I(X;Y)$	Mutual information	$I(X;Y)$	$I(X;Y) = D_{KL}[p(x,y) \| p(x)p(y)]$
$\mathfrak{s}(o)$	Surprisal	$\mathfrak{s}(o)$	$\mathfrak{s}(o) = -\ln p(o)$; self-information
$\ln p(o \mid m)$	Log model evidence	$\ln p(o \mid m)$	Marginal likelihood under model $m$

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