Why ScoreQuant

The problem: some analyses have to bin

A great deal of statistical practice ends in counts. A template fit needs the expected number of events per category for each model component. A trigger has to route an event into one of a few tiers. A cytometry protocol reports the fraction of cells in named gates. A binned likelihood needs bins. In every one of these the continuous measurement is replaced by an integer label, and that step is irreversible.

The bins are usually chosen for readability: equal width, equal population, a threshold on one discriminant, or whatever the previous analysis used. Those are choices about presentation. They are also, silently, choices about how much parameter sensitivity survives — and nothing in the usual workflow reports the size of what was given up.

Why naive binning loses information

Write the local sensitivity of one event as the score at a reference point \(\theta_0\),

\[ s(x)=\nabla_\theta\log p(x\mid\theta)\big|_{\theta_0}, \]

with the corresponding event score \(\nabla_\theta\log\lambda(x;\theta)\) for an intensity model. The Fisher information of the unbinned sample is the weighted second moment \(I_\infty=E[ss^\top]\). A hard rule \(q\) that maps each event to one of \(K\) labels retains only the between-cell part,

\[ I_q=\sum_b W_b\,\mu_b\mu_b^\top, \qquad W_b=E[\mathbf 1_{q=b}],\qquad \mu_b=E[s\mid q=b], \]

and the difference is exactly the within-cell scatter of the score,

\[ I_\infty-I_q=\sum_b E\!\left[\mathbf 1_{q=b}\,(s-\mu_b)(s-\mu_b)^\top\right]\succeq 0 . \]

Two consequences follow immediately. First, binning can only lose information, never create it, and refining a partition can only help. Second — and this is the operative point — the loss is governed entirely by how the score varies inside each cell, not by how the observation does. A bin that is narrow in the measurement variable but flat in \(s\) costs almost nothing. A bin that looks perfectly reasonable on a histogram but straddles a region where \(s\) swings costs a great deal. Equal-width and equal-population rules know nothing about \(s\), so their loss is arbitrary with respect to the quantity anyone actually cares about.

Binning by a single discriminant is the near miss. It is the right idea in one dimension: for a scalar parameter the optimal cells really are intervals of the score, and ScoreQuant solves that case exactly. But with several parameters the score is a vector, and a single ranking cannot separate directions that matter for different parameters. Compressing to one axis first, then binning, discards the multivariate structure before the binning ever gets to see it.

Why score space

The score is the natural coordinate system for this problem because the loss identity above is written entirely in it. Three practical consequences:

• It is the right dimension. Score space has one coordinate per parameter, however many measurement variables the events have. A 40-channel measurement feeding a two-parameter fit becomes a two-dimensional quantization problem.
• It is comparable across sources. An analytic likelihood, a linear component model, and an estimated density ratio all produce score vectors, so one optimizer serves all of them.
• Its origin means something. \(I_q\) is a second moment about zero, not a variance about the sample mean, because \(s=0\) is the direction of no sensitivity. ScoreQuant therefore never centers scores. It projects out numerically singular directions rather than repairing them with a ridge, because a ridge would invent information that the sample does not contain.

Score space also draws the honesty boundary. When the supplied vectors really are the model score, their second moment is Fisher information. When they are estimates — most often built from estimated density ratios — the same algebra is exact for the vectors you supplied and only a surrogate for the original model:

\[ \widehat I_q=\operatorname{Var}\!\big(E[\hat s\mid q(\hat s)]\big), \qquad I_q^{\mathrm{true}}=\operatorname{Var}\!\big(E[s\mid q(\hat s)]\big). \]

Every result therefore carries score provenance — including how any density ratios were obtained — and information_kind reads exact_fisher only when the provenance permits it. Ratio estimation, calibration, and cross-fitting belong to the application; their error must never be reported as quantization loss.

Two kinds of ratios

The word "ratio" appears in two roles with different semantics, and ScoreQuant keeps them in different places.

Model density ratios are a statistical representation. The score is the gradient of a log density ratio, \(s(x)=\nabla_\theta\log\big(p(x\mid\theta)/p(x\mid\theta_0)\big)\big|_{\theta_0}\), and for component models the relative densities \(\phi_k/\phi_{\rm ref}\) determine it completely — absolute normalization cancels. They enter through a score provider, and ScoreQuant names the estimand: it ships the exact ratio-to-score algebra and a closure diagnostic (ratio_closure_report) that bounds visible estimator bias, while estimating and certifying the ratios themselves stays with the application.

Importance ratios reweight expectations. When a sample is drawn from a proposal \(g\) instead of the reference law, \(E_{p_{\theta_0}}[f]=E_g[(p_{\theta_0}/g)\,f]\), and the factor \(p_{\theta_0}(x_i)/g(x_i)\) belongs in the source weights. It is a property of the measure, not of the score map, and it never passes through a provider.

Why two tasks and not one

There are two genuinely different questions here, and one fit method cannot answer both honestly.

Sample partitioning is transductive. You have a finite weighted table of scores and you want the labels that maximize the retained information of those rows. This is a combinatorial assignment problem, and its answer is a vector of labels. It does not by itself say anything about a score you have not seen: many different rules reproduce the same labels on the sample and disagree everywhere else. That is why PartitionResult has no predict method. Adding one would force the library to pick a rule the mathematics did not determine, and users would apply it to new data believing the sample optimality transferred.

Space quantization is inductive. You have a score law — an empirical sample, or a density over a bounded box — and you want a reusable rule that assigns any future score to a bin. Its answer is a geometric object: a transform, a set of centers, sometimes a metric. Prediction is well defined because the rule is defined everywhere.

The two tasks are connected by exactly one theorem, and ScoreQuant exposes that connection explicitly rather than assuming it. If a finite D-optimal partition is stable against every admissible single-row relocation, and its between-cell information is nonsingular, then it is already a strict self-consistent Voronoi partition in the \(I_B^{-1}\)-Mahalanobis metric. Only then is there a canonical rule to hand back, and compile_quantizer() returns it after verifying that the rule reproduces every positive-weight training label. An unstable or degenerate partition is refused. The implication also does not generalize: for profiled \(D_s\), exact fixtures show globally optimal finite assignments that violate the corresponding nearest-cell geometry, so a profiled partition has no compilation method that could succeed by accident.

When to use it, and when not

ScoreQuant is worth reaching for when downstream inference needs hard gates, categories, or template counts, when several parameters matter at once, and when local parameter sensitivity matters more than proximity in measurement space.

It is not a general-purpose compressor, a ratio estimator, or a complete likelihood framework, and it cannot certify that an upstream simulator or a learned density ratio is unbiased — the closure check bounds visible bias from below, never from above. It optimizes what the supplied scores say; the quality of the representation behind them is your responsibility, and the book chapter on estimated ratios and scores explains how to check it.