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arxiv: 2606.29516 · v1 · pith:YJXCR726new · submitted 2026-06-28 · 💻 cs.LG · stat.ML

A Mathematical Optimization Approach for Expert-Informed Bayesian Best Subset Selection

Nolan Alexander , Henning Mortveit This is my paper

Pith reviewed 2026-06-30 07:35 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords best subset selectionmixed-integer optimizationBayesian feature selectionexpert priorsMAP estimationPoisson binomialfeature relevance
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The pith

Expert probability estimates of feature relevance are incorporated into the MIO best-subsets problem as a log-odds penalty in a MAP framework.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that adds domain-expert views on which features matter to the classical best subset selection problem. Expert opinions from multiple sources are combined into a prior probability for each feature using Poisson binomial distribution, win rates, or normalized ranks. These priors enter the mixed-integer optimization objective as a log-odds term that encourages or discourages feature selection. The approach reverts exactly to standard best subsets when experts provide no information. Readers would care because many modeling tasks have useful expert knowledge that current data-only methods ignore.

Core claim

The EBBS model formulates best subset selection as a maximum a posteriori problem in which aggregated expert probabilities appear as additive log-odds penalty terms in the MIO objective, thereby allowing expert knowledge to influence the globally optimal sparse solution without altering the underlying optimization structure.

What carries the argument

The MAP objective augmented with a log-odds penalty derived from expert prior probabilities, which is optimized via mixed-integer optimization.

Load-bearing premise

Expert assessments, once aggregated via Poisson binomial, win rates, or normalized ranks, constitute a valid prior that improves the MAP objective without introducing systematic bias that the data cannot correct.

What would settle it

Generate synthetic data where the true relevant features directly contradict the supplied expert priors; if EBBS returns subsets with higher validation error than standard best subsets, the benefit of the expert term is refuted.

read the original abstract

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes Expert-Implied Bayesian Best Subsets (EBBS), a MAP formulation that augments the MIO encoding of best-subset selection with a linear log-odds penalty term derived from per-feature inclusion probabilities p_i. These probabilities are obtained by aggregating expert assessments via the Poisson binomial distribution (for marginals), pairwise win rates, or normalized mean ranks. The resulting objective is claimed to be equivalent to -log p(data|subset) - log p(subset|expert p), reducing exactly to ordinary best subsets when all p_i = 1/2. Analytic derivations of the MAP equivalence and theoretical properties are stated to be supplied, with empirical results on synthetic and real data noted as forthcoming.

Significance. If the claimed MAP equivalence and theoretical properties can be verified, the construction supplies a direct, MIO-compatible mechanism for injecting external expert probabilities into sparse regression. This could be useful in applied domains where domain knowledge is available but currently ignored by best-subsets solvers. The reduction to the classical problem when experts are uninformative is a clean special case. However, because no derivations, closed-form checks, or empirical results appear in the manuscript, the practical significance and improvement over standard best subsets remain unevaluated.

major comments (2)
  1. [Abstract] Abstract: the statement that 'This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties' is unsupported; the manuscript contains no derivations, proofs, or closed-form verifications of the claimed equivalence between the penalized MIO objective and the MAP problem.
  2. [Abstract] Abstract: the central modeling choice (expert probabilities aggregated via Poisson binomial, win rates, or normalized ranks entering as a log-odds penalty) is presented without any analysis of bias, consistency, or conditions under which the prior improves rather than degrades subset recovery; this is load-bearing because the paper explicitly defers all empirical validation of solution quality.
minor comments (2)
  1. The reduction to best subsets when p_i = 1/2 is asserted but not written out explicitly as an equation; adding the simplified objective would improve clarity.
  2. Notation for the aggregated prior (e.g., how the three aggregation methods map to a single p_i vector) is described only at a high level; a short algorithmic box or explicit formula would help readers implement the penalty term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the two major comments point by point below, indicating the revisions we will undertake.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that 'This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties' is unsupported; the manuscript contains no derivations, proofs, or closed-form verifications of the claimed equivalence between the penalized MIO objective and the MAP problem.

    Authors: We agree that the submitted manuscript does not contain the promised derivations or closed-form verifications of the MAP equivalence. This was an omission in the current draft. In the revised version we will insert a dedicated section deriving the log-odds penalty from the expert-aggregated prior (via Poisson binomial marginals, win rates, and normalized ranks), proving the exact reduction to ordinary best-subsets when all p_i = 1/2, and characterizing the resulting MAP objective. The abstract will be rewritten to describe only the material that is actually present after revision. revision: yes

  2. Referee: [Abstract] Abstract: the central modeling choice (expert probabilities aggregated via Poisson binomial, win rates, or normalized ranks entering as a log-odds penalty) is presented without any analysis of bias, consistency, or conditions under which the prior improves rather than degrades subset recovery; this is load-bearing because the paper explicitly defers all empirical validation of solution quality.

    Authors: The current manuscript concentrates on the formulation and its MIO encoding. We concur that an examination of the statistical properties of the expert prior is necessary. The revision will add a theoretical section analyzing bias and consistency of the MAP estimator under the aggregated prior, together with sufficient conditions under which the expert term improves subset recovery relative to the uninformative case. While the original submission deferred full empirical results, we will also include a short set of synthetic experiments that illustrate the effect of the prior on recovery rates, thereby addressing the load-bearing concern raised. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents EBBS as a direct application of the standard MAP objective -log p(data|subset) - log p(subset|expert p_i) where p_i are external expert-derived inclusion probabilities aggregated via Poisson binomial, win rates, or ranks. The log-odds penalty is defined from these independent inputs and vanishes when all p_i = 1/2, recovering ordinary best subsets. No self-citations, fitted parameters, or self-definitional reductions appear; the analytic derivations referenced are of this standard equivalence and its MIO encoding, which remain self-contained against external expert data and the classical best-subsets problem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that expert probabilities can be treated as a valid Bayesian prior inside an MIO objective; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)
  • domain assumption Expert assessments can be aggregated into a single marginal probability per feature that functions as a legitimate prior for MAP estimation.
    Invoked when the paper states that probabilities enter the objective as a log-odds penalty.

pith-pipeline@v0.9.1-grok · 5753 in / 1265 out tokens · 20205 ms · 2026-06-30T07:35:16.396787+00:00 · methodology

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Reference graph

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