Why ScoreQuant

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.

What binning costs, and where the cost lives

For a regular model the local sensitivity of one event is its score, the gradient of the log-likelihood at a reference point, and the Fisher information of the unbinned sample is the score's second moment. A hard rule that maps each event to one of \(K\) labels keeps only the between-cell part of that second moment; the difference is exactly the within-cell scatter of the score. The method overview states the identity and the book derives it. Two consequences matter here.

Binning can only lose information, and refining a partition can only help. And the loss is governed by how the score varies inside each cell, not by how the observation does: a cell that is narrow in the measurement variable but flat in the score costs almost nothing, while a cell that looks reasonable on a histogram but straddles a region where the score swings costs a great deal. Equal-width and equal-population rules know nothing about the score, so their loss is arbitrary with respect to the quantity anyone actually cares about.

Binning by a single discriminant is the near miss. For a scalar parameter the optimal cells really are intervals of the score, and ScoreQuant solves that case exactly. 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 discards the multivariate structure before the binning ever sees it.

Why score space

Score space is the natural coordinate system because the loss identity is written in it. It has one coordinate per parameter, however many measurement variables the events have, so a forty-channel measurement feeding a two-parameter fit becomes a two-dimensional problem. An analytic likelihood, a linear component model and an estimated density ratio all produce score vectors, so one optimizer serves all of them. And its origin means something: the retained information is a second moment about zero, because a score of zero is the direction of no sensitivity. ScoreQuant therefore never centers scores, and it projects out numerically singular directions rather than repairing them with a ridge, because a ridge would invent information the sample does not contain.

Score space also draws the honesty boundary. When the supplied vectors 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 model. Every result carries score provenance, and information_kind reads exact_fisher only when the provenance permits it. Three doors describes the input routes and Chapter 13 what a surrogate does and does not mean.

Two tasks

Labelling the rows of a fixed table and fitting a rule for rows not yet seen are different questions, and one fit method cannot answer both honestly. optimize_partition solves the first: a combinatorial assignment whose answer is a label vector, and PartitionResult has no predict method because many rules reproduce the same labels on a sample and disagree everywhere else. fit_quantizer solves the second: a geometric rule on score space, defined everywhere, with prediction through predict_scores. One theorem connects them, and compile_quantizer() exposes it only where its hypotheses are verified. Choosing your workflow decides between the tasks; Chapter 6 is the argument.

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 Chapter 13 explains how to check it.