Karl Friston, Thomas Parr and Peter Zeidman (2018; revised 2019), arXiv preprint — ARXIV-1805.07092
Citation verified. The brief gave “Friston, Parr & Zeidman et al. (2018)”. arXiv and the PDF list three authors and no others, so “et al.” is wrong. 2018 is right for the first version. The version read is v2 of 14 October 2019, which the authors describe as thoroughly updated, with more derivations and three worked examples.
Key takeaways
- The identity, in variational form. For a reduced prior P̃(θ) on the same likelihood, ln P̃(y) = ln ∫ Q(θ) P̃(θ)/P(θ) dθ + ln P(y), so F[P̃ : P] ≈ ln E_Q[P̃/P] + F[P] (Eq. 9), and ln Q̃ = ln Q + ln(P̃/P) − ln E_Q[P̃/P] (Eq. 10). The authors: “Some readers will recognise this as a generalisation of the Savage-Dickey density ratio … to any new prior.”
- Closed forms beyond Gaussians. The Gaussian case is Friston & Penny’s. The Dirichlet case is ã = a + ã₀ − a₀ and ΔF = ln B(a₀) − ln B(ã₀) + ln B(ã) − ln B(a), with B the multivariate beta function (Eq. 12; ₀ marks priors, no subscript posteriors). Table 1 adds beta, categorical, gamma and multinomial.
- Structure learning by pruning. Switching parameters off with precise shrinkage priors (N(0, e⁻¹⁶)) and keeping reductions that raise the free energy recovers sparse structure in simulation: a 20-regressor regression, an 8-cluster mixture pruned to the true 5, and an 8-node network.
- Hierarchical inversion as a cascade of reductions. The reduced free energy of level i, under the empirical prior from level i + 1, acts as that level’s likelihood (Eqs. 16–17). This is how SPM’s parametric empirical Bayes inverts group studies without re-fitting each subject.
Standing in the record
Filed on 2026-10-03 at the owner’s request, in the model-comparison batch. No anthology topic holds it. It is a statistical technique reviewed for neuroimaging and theoretical neurobiology, and it carries no instruction for machine-learning practice, though it names the ELBO and cites variational-inference work.
It extends Friston & Penny (LIT-620). It cites that paper for the Gaussian case and for the derivation of its Eq. 11, restates its identity (Eqs. 6–8 here against Eqs. 3–4 there), and adds the conjugate exponential-family forms that Friston & Penny did not give. The generalisation-of-Savage–Dickey claim is the same one Friston & Penny make with their Eq. 6. The record reads the ratio itself in Wagenmakers et al. (LIT-609).
Where it touches the record’s free-energy holdings:
- Markov blankets, in their original statistical sense. Eq. 5 states the mean-field update: the optimal approximate posterior for a subset of parameters is the softmax of the expected log joint “under its Markov blanket”. This is the graphical (Pearl) blanket in a factorised posterior. It is a computational convenience, defined relative to whichever subset is being updated, and claims to individuate nothing. That is the reading of blankets that THEORY-066 says is all the formalism supports, and that THEORY-073 says the free-energy literature on life, Friston’s Life as we know it (LIT-526), goes beyond. This paper does not take a side; it shows the device in the setting it came from.
- Active inference. §7.2 reviews BMR in an agent, after Friston et al. (2017a): offline “thinking about things”, likened to sleep, prunes Dirichlet likelihood parameters and improves later learning. Smith et al. (LIT-615) build their concept-learning model on this. Sophisticated Inference (LIT-577) uses the same discrete generative models (A, B, C, D matrices) but is about planning, not structure.