Result · CLASSIFIER-MIXTURE-SCORE-FORMULA

Class probabilities determine the mixture score

Derived from the literaturepublication: derived from published

Meaning

Classifier posteriors yield explicit mixture score coordinates under the stated model and calibration assumptions.

Statement

For multiclass posteriors eta_alpha with training priors pi_alpha, s_alpha(x)=[eta_alpha/pi_alpha]/Σ_beta theta0_beta[eta_beta/pi_beta] in the stated mixture parameterization.

Closest prior work

Nearest prior work

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

Relationships

Rests on

The full chain beneath it

Local score can be obtained from a local density ratioRATIO-LOCAL-SCOREdepth 2

Enables

Posterior error to score error for mixture-fraction scores: one Lipschitz inequality, and the excess Brier score as the classifier requirementCLASSIFIER-CALIBRATION-SCORE-LIPSCHITZ

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.

From Score, ratio and classifier access · chapter contents