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Conjugate update for an exponential-family likelihood

Statement

Let $η : Θ \to Ω$ be measurable, let $n \geq 1$ , and let $μ$ be a sigma-finite data measure. Suppose, given $θ$ , the observations $Y_{1}, \dots, Y_{n}$ are IID with density $h (y) \exp {η (θ)^{T} T (y) - A (η (θ))}$ relative to $μ$ . On the product data space with measure $μ^{\otimes n}$ , assume the resulting joint density is specified and jointly measurable and that the sampling family is a parameter-measurable probability kernel. Let the prior be $Π_{χ, κ}$ from Conjugate-form prior for an exponential family with $0 < Z (χ, κ) < \infty$ . For an observed $y_{1 : n}$ , put $s = \sum_{i} T (y_{i})$ . If $h (y_{i}) > 0$ for each $i$ and $0 < Z (χ + s, κ + n) < \infty$ , then the evidence is $m (y_{1 : n}) = (\prod_{i = 1}^{n} h (y_{i})) \frac{Z (χ + s, κ + n)}{Z (χ, κ)} \in (0, \infty),$ and the posterior density relative to the parameter measure is $\frac{\exp {η (θ)^{T} (χ + s) - (κ + n) A (η (θ))}}{Z (χ + s, κ + n)} .$ Thus the update adds $\sum_{i} T (y_{i})$ to $χ$ and $n$ to $κ$ only on finite-positive evidence fibers.

Facts & Assumptions

Given: The measurable exponential-family sampling model, proper conjugate prior, finite IID sample, and observed data satisfying the stated normalizer conditions. The joint density on the product data space is the IID product $\prod_{i = 1}^{n} h (y_{i}) \exp {η (θ)^{T} T (y_{i}) - A (η (θ))} .$

[F1]

The exponential-family sampling density has kernel $h (y) \exp {η^{T} T (y) - A (η)}$ . Exponential families and full rank.

[F2]

The conjugate-form prior density is proportional to $\exp {η (θ)^{T} χ - κ A (η (θ))}$ , normalized only for finite positive $Z (χ, κ)$ . Conjugate-form prior for an exponential family.

[F3]

The specified proper prior and parameter-measurable sampling kernel constitute a Bayesian model. Bayesian model as a prior and sampling kernel.

[F4]

The evidence is the prior integral of the specified nonnegative likelihood density, and the quotient update is defined only for finite positive evidence. Evidence normalizer and fiberwise posterior update.

[F5]

A likelihood-weighted prior measure admits probability normalization exactly when its evidence is finite and positive. Fiberwise posterior propriety is equivalent to finite positive evidence.

[F6]

For a proper prior and a jointly measurable dominated sampling density, the normalized likelihood quotient on positive finite fibers is a posterior conditional kernel. Bayes formula for dominated kernels.

Proof

Given: The assumptions in the Statement and the data $y_{1 : n}$ .

1.1F1F3F6given

Multiplying the stated IID densities from [F1] gives the specified joint likelihood $L_{θ} (y_{1 : n}) = (\prod_{i} h (y_{i})) \exp {η (θ)^{T} s - n A (η (θ))} .$ The stated joint measurability and kernel hypotheses ensure this is the dominated likelihood to which [F3] and [F6] apply; no separate pointwise Radon–Nikodym representatives are being promoted to a jointly measurable density.

2.1F2F4F5step 1.1algebra

Multiply this likelihood by the prior density [F2]. The parameter-dependent factor becomes $\exp {η (θ)^{T} (χ + s) - (κ + n) A (η (θ))},$ so integrating it against the parameter measure gives $m (y_{1 : n}) = (\prod_{i} h (y_{i})) \frac{Z (χ + s, κ + n)}{Z (χ, κ)} .$ All factors are finite positive by the hypotheses; [F4] defines this evidence, and [F5] confirms that it is exactly the valid normalization regime.

3.1F2F4F6step 2.1algebra∎

Divide the product of likelihood and prior density by the evidence in step 2.1. The data-only carrier and the prior normalizer cancel, leaving the displayed density with updated parameters $(χ + s, κ + n)$ . Since the evidence is finite positive, [F6] identifies this normalized kernel as the posterior on this fiber; no quotient is formed when any stated positivity or finiteness condition fails.

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Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §2.4 (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families, 2020 (standard reference, not scraped)