Glossary¶

Bin¶

One category of a hard partition. After binning, events are represented only by integer labels or aggregate counts.

Calibration¶

The property of a probability estimate that its numerical values are quantitatively meaningful, not merely correctly ordered. Score construction from a classifier requires calibrated posteriors (or a ratio-estimation loss), because a ranking score or an arbitrary monotone transform of a likelihood ratio does not determine the density ratios the score is built from.

Compile bridge¶

The one theorem that turns a sample partitioning result into a reusable quantizer without inventing a rule: when a D-optimal finite partition is exchange-stable and its information matrix is nonsingular, it is provably identical to the nearest-cell rule in its own \(I_q^{-1}\)-Mahalanobis metric, so PartitionResult.compile_quantizer() returns exactly that rule and verifies label reproduction before returning it — at the gain_tolerance the partition was optimized at, since that is the precision the solver actually delivers. The bridge exists for the log determinant only; a profiled-\(D_s\) partition has an analogous population geometry but no exact finite implication, so it refuses to compile rather than approximate one.

D-efficiency¶

Geometric mean of retained-information eigenvalues, also called the geometric-mean retention. It summarizes balanced local information retention across informative parameter directions and equals a single retained/unbinned ratio when there is only one informative direction.

Density ratio¶

A model density divided by another model density: \(p(x\mid\theta)/p(x\mid\theta_0)\), or a component ratio \(\phi_k/\phi_{\rm ref}\). The score is the gradient of a log density ratio, so ratios are the minimal statistical representation a score provider needs — absolute normalization cancels, and any common event-wise factor is a free gauge. Distinct from an importance ratio.

Efficient score¶

In a model with interest parameters \(\psi\) and nuisance parameters \(\lambda\), the part of the interest score \(s_\psi\) that is left after regressing away the nuisance score \(s_\lambda\): \(e(s) = s_\psi - BC^{-1}s_\lambda\), where \(B\) and \(C\) are blocks of the unbinned information matrix. Profiled \(D_s\)-optimality's population stationarity condition is a nearest-cell rule in this projection alone; the directions it annihilates matter only through the regression coefficient, not directly.

Exchange stability¶

A property of a finite labeling: no single row can move to another cell and improve the objective by more than a stated gain_tolerance, evaluated by one exact scan over every admissible relocation. PartitionResult.exchange_stable records whether the solver's own output has this property at the tolerance it ran at, and exchange_stability_report checks it for a labeling from any source, recording the tolerance on the report. Exchange stability is necessary for the compile bridge but is checked independently of it.

Finite assignment¶

See sample partitioning.

Fisher information¶

Expected outer product of the score. It measures local model sensitivity, not estimator bias.

Hard quantizer¶

A deterministic score-space mapping that assigns every score to exactly one bin. Contrast with a randomized quantizer, which assigns a probability distribution over bins.

Importance ratio¶

The factor \(p_{\theta_0}(x)/g(x)\) that reweights a sample drawn from a proposal distribution \(g\) so that weighted averages estimate expectations under the reference law. It is a property of the measure and enters ScoreQuant as source weights — never through a score provider, which is where density ratios live.

Intensity¶

An unnormalized event-rate model. Unlike a probability density, its integral may encode expected yield rather than one.

Likelihood¶

The probability density or mass of observed data, viewed as a function of model parameters.

Likelihood ratio¶

Ratio of two likelihoods or component densities; see density ratio. A classifier posterior divided by its class prior can estimate component ratios up to a common event-wise factor, and direct estimators (KLIEP, uLSIF, neural ratio estimation) target the same object without a classifier.

Oracle¶

A calculation using the exact data-generating likelihood or exact score. A learned classifier ratio is not an oracle merely because it is used without binning.

Population design¶

Optimization of a measurable rule under a specified score law itself, rather than only its finite realization. It is the inductive half of space quantization; fit_quantizer performs it exactly when given an IntegrationSource and approximately, as empirical inductive fitting, when given a finite sample.

Ratio closure¶

The identity that exact density ratios relative to a reference measure integrate to one under it. ratio_closure_report measures the residual; a large value flags estimator bias, a misdeclared training prior, or a measure mismatch. The check is necessary but not sufficient, so closure never upgrades estimated provenance to exact.

Reference point¶

Parameter value \(\theta_0\) at which scores and Fisher information are evaluated.

Retention¶

The fraction of unbinned Fisher information a labeling keeps, reported per informative direction as an eigenvalue ratio of \(I_{\text{full}}^{-1}I_q\) and summarized as geometric_mean_retention (equivalently, D-efficiency) or arithmetic_mean_retention. A retention of \(1\) loses nothing; a retention of \(0\) means a direction carries no information after binning. \(1/\sqrt{\text{retention}}\) is the resulting inflation of a Gaussian standard error in that direction.

Sample partitioning¶

The transductive task: given one fixed weighted table of scores, choose the labels that maximize the retained information of those rows. optimize_partition performs it and returns a PartitionResult, which deliberately has no predict method — a labeling of one table does not by itself determine what happens to a score that was not in it. Also called finite assignment.

Score¶

Gradient of log likelihood with respect to parameters. It describes an event's local parameter sensitivity.

Score law¶

The distribution or intensity measure induced on score space by a reference source and an observation-to-score provider.

Score provider¶

A map from observations to score vectors. It does not supply a reference measure; a source must supply that separately.

Simplex¶

Set of nonnegative fractions that sum to one. A \(K\)-component mixture has \(K-1\) free directions.

Source¶

An empirical table or integration rule that supplies the reference measure used by an objective.

Space quantization¶

The inductive task: given a score law — an empirical sample or a density over a bounded box — choose a reusable rule that assigns any future score to a bin. fit_quantizer performs it and returns a QuantizerResult, whose answer is a geometric object (a transform, centers, sometimes a metric) with a well-defined predict_scores method, because the rule is defined everywhere rather than only on the rows it was fit from.

Surrogate information¶

Between-cell information computed from estimated scores. It is exact for those supplied vectors but not automatically Fisher information of the original statistical model.

Template¶

Conditional bin probabilities \(P(B_j\mid k)\) for a model component \(k\).

Three doors¶

The three ways a weighted table of score rows can arise, differing in which statistical representation you already possess rather than in what the optimizer does with the result: door 1, precomputed (event, score) rows supplied directly as a ScoreSample; door 2, component densities or an analytic score model, reached through an ObservationSample or IntegrationSource together with LinearComponentScore or ScoreFunction; door 3, density ratios — analytic, classifier-derived, or from a direct ratio estimator — reached through an ObservationSample with DensityRatioScore or CentralLogRatioScore. All three doors open onto the same object — a weighted score table — and are validated together with a source, never supplied alone.

Whitening¶

Linear scaling of informative score directions so that their unbinned Fisher matrix is the identity. ScoreQuant does not mean-center scores while whitening.