The Bayesian Learning Rule Beyond KL Geometry

Conjugate Bayes plays a key role in machine learning, ensuring computationally simple posterior updates. We generalize the variational principle behind this computation by using the Fenchel-Young (FY) inequality, which lets us define a generalized evidence lower bound beyond the KL geometry, and at equality, characterizes new generalized posteriors. A generalized notion of conjugacy follows, defined as closure of prior-potential families that are affine under translation by the loss. We show that such families not only preserve additive canonical updating, but also posterior inference with parameter-dependent supports and tuneably heavy tails. This combination is not feasible with standard conjugate Bayes. We derive a new FY generalization of expectation maximization (EM), which recovers classical and Hard EM, and amortized versions. We show examples, including generalized Bayesian linear regression with closed-form bounded support and heavy-tailed posteriors, and a Gaussian mixture with sparse cluster assignments.

Exponential families encompass the distributions central to modern machine learning -- softmax, Gaussians, and Boltzmann distributions -- and underlie the theory of variational inference, entropy-regularized reinforcement learning, and RLHF. We isolate a simple identity for exponential families that expresses the KL difference $\mathrm{KL}(q \| p_{λ_2}) - \mathrm{KL}(q \| p_{λ_1})$ in terms of the log-partition function $A(λ)$ and the moment $μ_q$. Remarkably, this identity together with the single fact that $\mathrm{KL} \geq 0$ (with equality iff $p = q$) suffices, by direct substitution and rearrangement, to derive a cluster of results that are classically obtained by separate, heavier arguments: a generalized three-point identity for arbitrary reference distributions, Pythagorean theorems for I-projections and reverse I-projections, convexity of the log-partition function, identification of its Legendre dual in KL terms, the Gibbs variational principle, and the explicit optimizer in KL-regularized reward maximization, including the exponential tilting formula underlying entropy-regularized control and RLHF. Beyond these purely algebraic consequences, standard analytic arguments recover the gradient formula for the log-partition function, the Bregman representation of within-family KL divergence, and the surjectivity of the moment map. The note is self-contained.

The forward and reverse Kullback-Leibler (KL) divergences arise as limiting objectives in learning and inference yet induce markedly different inductive biases that cannot be explained at the level of expectations alone. In this work, we introduce the Surprisal-Rényi Free Energy (SRFE), a log-moment-based functional of the likelihood ratio that lies outside the class of $f$-divergences. We show that SRFE recovers forward and reverse KL divergences as singular endpoint limits and derive local expansions around both limits in which the variance of the log-likelihood ratio appears as a first-order correction. This reveals an explicit mean-variance tradeoff governing departures from KL-dominated regimes. We further establish a Gibbs-type variational characterization of SRFE as the unique minimizer of a weighted sum of KL divergences and prove that SRFE directly controls large deviations of excess code-length via Chernoff-type bounds, yielding a precise Minimum Description Length interpretation. Together, these results identify SRFE as a variance- and tail-sensitive free-energy functional that clarifies the geometric and large-deviation structure underlying forward and reverse KL limits, without unifying or subsuming distinct learning frameworks.