[MEDIUM] `ClosedFormStrategy` throws `MethodError` for variational families with a non-constant base measure

[MEDIUM] ClosedFormStrategy throws MethodError for variational families with a non-constant base measure

Summary

ClosedFormStrategy (extension ClosedFormExpectationsExt) implements the exact-gradient (Williams-product) path, but its base-measure correction only has a method for ConstantBaseMeasure. Any variational family whose base measure is not constant (current ExponentialFamily.jl families: Poisson, Chisq, Binomial, Rayleigh, Weibull, NegativeBinomial) hits a MethodError during gradient computation.

Evidence

ext/ClosedFormExpectationsExt/ClosedFormExpectationsExt.jl:

• lines 18-25 define logbasemeasure_correction(::ClosedFormStrategy, ::ConstantBaseMeasure, ...) — the only method.
• compute_gradient! (lines 55-60) calls:
  logbasemeasure_correction(strategy, ExponentialFamily.isbasemeasureconstant(q_dist), q_dist, grad_target).

Verified (Julia 1.12.6, ClosedFormExpectations 0.4.1):

• methods(ClosedFormExpectationsExt.logbasemeasure_correction) returns exactly one method (ConstantBaseMeasure).
• Calling with ExponentialFamily.NonConstantBaseMeasure() throws MethodError.
• isbasemeasureconstant(convert(ExponentialFamilyDistribution, Poisson(3))) == NonConstantBaseMeasure() (also Chisq, Rayleigh, Binomial).

The docstring (src/strategies/closed_form.jl) states the strategy computes exact gradients for the target–variational pair with no caveat; this is only true for constant-base-measure families.

Root cause

The mathematical cancellation the comment relies on (E_q[(log p̃ - log h_q)(T-μ)], where the log h_q term vanishes because E_q[T] = μ) is only valid for a constant base measure. There is no implementation (and no explicit error) for the non-constant case.

Suggested direction

Either implement the -E_q[log h_q (T-μ)] correction for NonConstantBaseMeasure, or — until then — throw a clear, informative error from compute_gradient! when the variational q_dist has a non-constant base measure, rather than an opaque MethodError.

Start in ext/ClosedFormExpectationsExt/ClosedFormExpectationsExt.jl by reading logbasemeasure_correction and compute_gradient!, then compare their handling of ConstantBaseMeasure and NonConstantBaseMeasure. Decide which suggested behavior is appropriate, verify gradient computation no longer produces an opaque MethodError, and update the src/strategies/closed_form.jl docstring if the strategy remains limited.