Result · CLASSIFIER-RATIO-ORACLE

Calibrated posterior odds recover density ratios

Establishedpublication: published

Meaning

Calibrated posterior odds recover component density ratios after correcting for class priors.

Scientific context

With known class priors, calibrated classifier posterior odds recover component density ratios, which can then be transformed into the local mixture score. This continues the same simulation-based-inference line as the local-score identity (RATIO-LOCAL-SCORE), using a classifier's posterior odds rather than an explicit density model.

Statement

With known class priors, posterior odds recover component density ratios, which can then be transformed into the local mixture score.

Assumptions

true full-observation posterior probabilities P(Y=a|X=x), not merely reliability-calibrated probabilities
known strictly positive training priors; the same class-conditional component laws

classifier-based score oracle

Where it stops

Reliability calibration alone identifies density ratios of the forecast representation, not necessarily the full observation; see O8.4 and AUDIT-SCORE-ERROR-BUDGET-001.

Closest prior work

Cranmer, Pavez, Louppe (2015)Approximating Likelihood Ratios with Calibrated Discriminative Classifiers

Relationships

Enables

Raises

Classifier calibration error to Fisher-information lossOPEN-CLASSIFIER-CALIBRATION-FI

Open next door

Classifier calibration error to Fisher-information lossOPEN-CLASSIFIER-CALIBRATION-FI

In the library

Carried by ratios_from_posteriors. Reference (opens in a new tab).

Every object and refusal

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

O3. Calibrated classifier posteriors provide ratios

With class priors $\pi_\alpha$ , posterior odds recover component density ratios. In the mixture parameterization,

\boxed{ s_\alpha(x)= \frac{\eta_\alpha(x)/\pi_\alpha} {\sum_\beta\theta_{0\beta}\eta_\beta(x)/\pi_\beta}. }

Here the posterior means the true conditional probability given the full observation, not merely a reliability-calibrated forecast. For normalized mixture fractions, apply the tangent chart (for example, component-minus-reference differences) to these ratios to obtain a mean-zero score. Reliability calibration alone recovers ratios of the compressed representation, not necessarily the full observation; see O8.4.

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