TransformedMessage.logpdf/pdf omit the transform Jacobian (base-space density at physical coordinates)

[Jammy2211]

Found while implementing the #1497 property sweep (priors/messages census follow-up; hubs #1330/#1331). Same failure shape as #1331-04: a generic-interface density path is silently wrong while the direct paths are right.

The finding

TransformedMessage inherits MessageInterface.logpdf (natural-parameter path). Its overrides forward natural_parameters, calc_log_base_measure and to_canonical_form to the base message — to_canonical_form is @transform-decorated, so the physical input is mapped to base coordinates — but no log_det change-of-variables term is ever added. The result: logpdf(x) at a physical x returns the base-space density at the mapped point, not the physical density. Meanwhile TransformedMessage.factor(x) does it correctly (base.logpdf(transform(x)) + log_det).

Reproduction on main @ 7d4d931 (numbers exact):

import numpy as np
from scipy.integrate import quad
import autofit as af

m = af.UniformPrior(0.0, 1.0).message      # TransformedMessage(UniformNormalMessage, LinearShift)

m.logpdf(np.asarray(0.7))                  # -1.05644  — physical density is 1.0, expect 0.0
m.factor(np.asarray(0.7))                  #  0.0      — correct

quad(lambda x: float(m.pdf(np.asarray(x))), 0.0, 1.0)   # 0.282095 = 1/(2*sqrt(pi)), not 1.0
quad(lambda x: float(np.exp(m.factor(np.asarray(x)))), 0.0, 1.0)  # 1.000000

For LogUniformPrior(0.01, 100) the generic-path pdf integrates to 15.83. Every TransformedMessage-wrapped prior (Uniform, LogUniform, LogGaussian) is affected; plain NormalMessage / TruncatedNormalMessage priors are not.

Doc contradiction

The composed_transform.py module docstring (added in #1334) states: "Evaluating logpdf(x) at a physical x runs the same three steps in reverse (physical → base) via _transform, accumulating the log-Jacobian log_det of each transform for the change of variables." The code does not accumulate it. Either the docstring describes the intended contract (then logpdf needs the log_det term) or logpdf is deliberately base-space for EP message arithmetic (then the docstring and pdf() are wrong/misleading).

What needs adjudication before a fix

• Which callers consume TransformedMessage.logpdf/pdf vs factor? The EP factor-graph path uses factor/exp_factor (correct). If any EP projection / log_norm path evaluates logpdf on transformed messages at physical coordinates, it inherits this error.
• logpdf_gradient returns the base log_likelihood with a Jacobian-corrected gradient — a third convention (gradient physical, value base-space) worth settling in the same pass.
• Note value_for, cdf, log_prior_from_value and factor are all verified correct — user-facing sampling and MCMC/MLE log-priors are unaffected. The exposure is anything treating transformed_message.pdf() as a physical density.

The #1497 property tests assert the physical-density path via factor for transformed messages and cite this issue at the site; when this is resolved they can be tightened to assert logpdf directly.

Sequencing suggestion: adjudicate alongside the parked single-source-density design (census C1/C4, prompts 12/13) — this is a fourth density-convention divergence of exactly the kind that design exists to eliminate.

[Jammy2211]

Caller analysis (adjudication point 1) — 2026-08-19, main @ 21288bb

Full inventory of logpdf/pdf/logpdf_gradient/factor call sites across the production autofit/ tree, numerically verified (UniformPrior(0,2): logpdf(1.3) = −0.993174 vs factor(1.3) = −0.693147 = log 0.5; ∫exp(logpdf) over the support = 0.564190 = 2/(2√π), ∫exp(factor) = 1.000000).

Callers expecting a physical density (exposed to the bug)

• Prior.logpdf — the public prior API. UniformPrior.logpdf (uniform.py:124) calls message.logpdf explicitly, and its boundary-epsilon nudge operates in physical coordinates, so the method unambiguously intends physical semantics; every other prior reaches message.logpdf via Prior.__getattr__. Uniform / LogUniform / LogGaussian users are silently handed base-space values.
• MessageInterface.pdf (interface.py:45) — zero production callers; public-API surface only.
• The non-linear search stack never touches message logpdf — sampling and MCMC/MLE go through value_for / log_prior_from_value / factor, all verified correct. Exposure is API-level, not search-level.

EP-internal callers — all base-space, and mutually consistent

MeanField.logpdf/logpdf_gradient (mean_field.py:294,300), reached from FactorApproximation.__call__/func_gradient (mean_field.py:631-633,647) by the Laplace optimiser; MeanField is itself a Factor wrapping _logpdf, so projection(mode) in the from_mode_covariance log_norm calibration (mean_field.py:404) and AbstractMessage.__call__/factor_jacobian (abstract.py:392,398) route the same way.

This loop is coherent, not buggy: a likelihood factor is a density over data (measure-free in x), so fval(x) + Σ log q_base(T(x)) is exactly the base-space tilted log density at z = T(x). The Laplace mode it finds is the base-space tilted mode, and from_mode (transform_jac + jac.quad) projects mode and covariance into base space consistently. The missing log_det is the EP loop's working convention — and base-space Laplace is arguably the better-conditioned choice (unbounded support).

The one genuine cross-convention seam — PriorFactor

PriorFactor (declarative/factor/prior.py:23,62) wraps prior.factor — the physical density — as both its factor callable and its log_likelihood_function. Its tilted objective under the numerical optimiser is therefore

log π_base(T(x)) + log_det(x) + Σ log q_base(T(x))

— one x-dependent log_det(x) more than the coherent base-space tilted density (and one short of the coherent physical one). Because the #1337 seam strips the exact-update hooks (see the graphical/README.md roster row), declarative PriorFactors really do take this numerical path: prior-factor EP updates optimise a hybrid objective whose mode is neither the base-space nor the physical tilted mode.

Correction to adjudication point 2

logpdf_gradient is not a third convention. Its analytic gradient equals the central-difference derivative of logpdf exactly (−0.520142 both, at the probe point) — the jacobian multiplication is just the chain rule for the base-space composition base.logpdf(T(x)). So (logpdf, logpdf_gradient, numerical_logpdf_gradient_hessian) form one self-consistent base-space family and factor is the physical one: two conventions, not three.

New finding

Running the reproducer surfaced #1501: TransformedMessage.factor_gradient unpacks four values from _transform_det_jac, which returns three — crashes on first call, zero production callers.

Proposed resolution (adjudication point 3 — for ratification, nothing actioned)

A two-sided contract repair rather than a one-line logpdf patch:

1. EP-internal: declare base-space logpdf the message contract (fixing the composed_transform.py module-docstring claim that logpdf accumulates the Jacobian) and make PriorFactor consume the base-space density (message.logpdf) instead of prior.factor, making the EP loop fully base-space coherent. This changes prior-factor updates (removes the spurious log_det) but preserves the deliberate-looking base-space Laplace everywhere else. Belongs with the design: unify the Prior/Message hierarchy around a single-source density #1500 design (Q2/Q3 there).
2. Public API: Prior.logpdf and pdf promise a physical density and should route through factor (or be renamed/documented). Small, standalone, user-facing correctness fix.

The other coherent option — adding log_det to logpdf itself so everything is physical — touches every EP path at once and shifts EP Laplace modes to physical-space MAPs; that is a design change that should only be taken inside the #1500 single-source-density decision.

External verification requested

This analysis is AI-generated. Before acting on it we'd value a check from someone with a probabilistic-programming background on two claims in particular: (a) that the likelihood-factor tilted objective as implemented equals the base-space tilted density (the change-of-variables argument above), and (b) that PriorFactor's use of the physical density really does leave its tilted objective one log_det(x) off either coherent convention.

---

Generated by Claude Code