Contribution
A technical review that turns the post hoc identity of Friston & Penny (2011) into a general toolkit. It states the reduced free energy and reduced posterior for any change of prior (Eqs. 9–10). It gives closed forms for the common conjugate priors (Table 1; the Dirichlet case is Eq. 12). It shows the method on three simulations with code, and recasts hierarchical (empirical Bayes) inversion as successive reductions (Eqs. 15–17).
Key insight
Comparing models usually means fitting each one. If the models differ only in what they assume before seeing the data, the fitted posterior of the most permissive model already contains everything needed: reweight it by how much more or less each alternative prior would have allowed, and the average weight is the change in evidence. In a variational scheme with conjugate forms, that average is a closed-form expression in sufficient statistics. Structure learning becomes the repeated question “would this model have been better off never having believed in that parameter?”
Assumptions
- Same likelihood for full and reduced models; the full model contains every parameter of every model to be compared (§3). Models of completely different form cannot be compared this way.
- A good approximate posterior and free energy for the full model. Eqs. 9–10 hold approximately, with F ≈ ln P(y) and Q ≈ P(θ | y). Any scheme that yields them will do, mean-field or not (§2).
- Known analytic forms for prior and posterior (Gaussian under the Laplace assumption, Dirichlet for categorical models, and so on); this is what makes the reduction closed-form.
- Switched-off parameters are given precise shrinkage priors, N(0, e⁻¹⁶) in the examples, against N(0, 1) or N(0, 1/16) for switched-on ones.
Key results
- Free energy decomposition (Eq. 4). F = ln P(y) − D_KL[Q ∥ P(θ|y)] = E_Q[ln P(y|θ)] − D_KL[Q ∥ P(θ)]: a bound, and accuracy minus complexity.
- Mean-field update (Eq. 5). The optimal factor Q(θ_i) is the softmax of E_{Q\i}[ln P(y, θ)], the expected log joint under the factor’s Markov blanket (citing Beal 2003).
- Reduced evidence and posterior (Eqs. 8–10). P̃(y)/P(y) = ∫ P(θ|y) P̃(θ)/P(θ) dθ, so F[P̃ : P] ≈ ln E_Q[P̃(θ)/P(θ)] + F[P] and ln Q̃ = ln Q + ln P̃/P − ln E_Q[P̃/P].
- Gaussian (Eq. 11) is Friston & Penny’s Eq. 9. A parameter whose prior variance is shrunk to zero has equal prior and posterior moments and drops out of the reduced free energy.
- Dirichlet (Eq. 12). Reduced posterior concentration = posterior + reduced prior − prior; ΔF = ln B(prior) − ln B(reduced prior) + ln B(reduced posterior) − ln B(posterior). “This equation returns the difference in free energy we would have observed, had we started observing outcomes with simpler prior beliefs.”
- §6.1 regression. 10 true and 10 spurious orthogonal regressors. A greedy search (spm_dcm_bmr_all) leaves 256 candidate models; the best has posterior probability 84% and the next 14.6%. The Bayesian model average prunes regressors 11–20 and also true regressor 6, whose effect was too small to detect; regressor 7 is present with probability 85.2%. “Around two seconds” on a desktop.
- §6.2 Gaussian mixture. Data from 5 clusters, a model started with 8. Clusters are removed when the reduced log evidence exceeds the full model’s. With an added rule merging clusters less than 3 nats apart (KL), the true number is recovered.
- §6.3 network discovery. An 8-node linear dynamical network (0.5 Hz feedforward, −1 Hz feedback, −0.25 Hz self-inhibition, noise precision 400, SNR 8.18 dB). Of three hypothesised models, the true structure wins with posterior “at ceiling”. An automatic search recovers the network except the connection from node 7 to node 8, which was not needed given node 6’s input.
- §7.3 hierarchy (Eqs. 15–17). The free energy of an n-level model is expressible recursively by reduced free energies of the levels below, so higher levels are optimised from lower-level priors and posteriors alone. In Friston et al. (2016), 16 simulated subjects with 158 neuronal parameters each: the between-subject level and its reduction took “a few seconds” against about a minute per subject-level inversion.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The free energy and posterior of a model with any new prior follow from the full model’s prior, posterior and free energy | strong (derivation), approximate to the extent F and Q are | Eqs. 6–10 |
| C2 | This generalises the Savage–Dickey density ratio to arbitrary reduced priors | strong as a formal statement; the paper asserts it, the derivation is Friston & Penny’s Eq. 6 | §3 |
| C3 | Closed forms exist for Gaussian, Dirichlet, beta, gamma, categorical and multinomial priors | strong (derivations; gamma worked in the appendix) | Eqs. 11–12, Table 1, Eqs. 18–20 |
| C4 | Greedy BMR recovers sparse structure in regression, mixtures and linear dynamical networks | weak to moderate: one simulation each, from the model class | §§6.1–6.3, Figs. 1–5 |
| C5 | Hierarchical inversion can be done as a feedforward cascade of reductions with no loss under Laplace or Dirichlet forms | moderate: derivation plus one cited application | Eqs. 15–17; Friston et al. 2016 |
| C6 | BMR is a model of offline synaptic pruning in the brain, as in sleep | weak: a metaphor supported by simulations in cited work, not tested here | §7.2 |
Concepts
- reduced free energy F[P̃ : P]: the free energy of the model with prior P̃, computed from the full model’s quantities (Eq. 9).
- full (parent) model: the model with the least informative priors, containing every parameter of every model compared.
- structure learning: here, finding which parameters a model needs, by scoring reductions of a parent model.
- parametric empirical Bayes (PEB): hierarchical inversion of subject-level DCMs with a between-subject general linear model, done by reduction (§7.3).
Connections
- Friston & Penny (LIT-620). The source of the identity and of the Gaussian form; cited for the derivation of Eq. 11.
- Savage–Dickey. The paper cites Savage (1972) and Verdinelli & Wasserman (1995) for the ratio, not Dickey. The record’s statement of the ratio is Wagenmakers et al. (LIT-609).
- Smith et al. (LIT-615) cite this paper (as Friston et al. 2018) for post hoc model optimisation, and apply the Dirichlet form to an agent’s prior over hidden states.
- Active Inference, Curiosity and Insight (Friston et al. 2017a) is the source of the Dirichlet reduction and the rule-learning simulation of §7.2. The record does not hold it.
Bearing on the record
- Markov blankets. Eq. 5 is the statistical home of the term: the blanket of a factor in a variational posterior, relative to whatever factorisation was chosen. It supports the formal point of THEORY-066 (a blanket is defined relative to a chosen set) by example, and claims nothing about individuation, which is what THEORY-073 disputes in the free-energy literature on life.
- With Friston & Penny it would source a THEORY that structure learning by pruning a parent model is evidence maximisation over priors, with Savage–Dickey as its limiting case.
- No ML instruction. Nothing here belongs in the anthology.
Limitations
- Only reductions. BMR cannot add structure the parent model lacks; the parent must already contain every candidate parameter.
- The approximation is inherited. If the full model’s Q is poor (strong nonlinearity, local optima), so are all reduced free energies. The paper does not revisit Friston & Penny’s open question of how good the reduced free energy is as a proxy.
- Search. Large model spaces still need a greedy search, and the pruning outcome can depend on it.
- Evidence. The demonstrations are simulations from the assumed model class, and the empirical review is, by the authors’ own description, mostly their lab’s and collaborators’ work.
Open questions
- How does BMR behave when the parent model is misspecified, so that no reduction contains the truth?
- Can model expansion, adding parameters, be scored with the same economy? Smith et al. (LIT-615) approach it by building spare capacity into the parent model.
Corrections
- Citation. The brief’s “Friston, Parr & Zeidman et al.” has a spurious “et al.”: there are exactly three authors.