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Non-Bayesian Post-Model-Selection Estimation as Estimation Under Model Misspecification

Nadav Harel, Tirza Routtenberg

TL;DR

This work addresses estimation after model selection when the true model is unknown and may be misspecified. It formalizes three interpretations of post-model-selection estimation as misspecification problems (naive, normalized, selective inference), derives corresponding misspecified ML estimators and pseudo-true parameter vectors, and introduces the PS-MCRB to bound estimation performance. The paper shows that the selective-inference interpretation yields coherent likelihoods and the PSML estimator, achieving the lowest MSE and providing the most informative bounds compared to the oracle CRB. Through a Gaussian linear-model example with channel-detection, the authors demonstrate that MSNL and PSML outperform the naive MSL approach, with PS-MCRBs offering tighter performance guarantees aligned with the selection procedure. Overall, the selective-inference framework provides a practical, theoretically grounded approach to post-model-selection estimation under misspecification with meaningful implications for signal processing and related domains.

Abstract

In many parameter estimation problems, the exact model is unknown and is assumed to belong to a set of candidate models. In such cases, a predetermined data-based selection rule selects a parametric model from a set of candidates before the parameter estimation. The existing framework for estimation under model misspecification does not account for the selection process that led to the misspecified model. Moreover, in post-model-selection estimation, there are multiple candidate models chosen based on the observations, making the interpretation of the assumed model in the misspecified setting non-trivial. In this work, we present three interpretations to address the problem of non-Bayesian post-model-selection estimation as an estimation under model misspecification problem: the naive interpretation, the normalized interpretation, and the selective inference interpretation, and discuss their properties. For each of these interpretations, we developed the corresponding misspecified maximum likelihood estimator and the misspecified Cram$\acute{\text{e}}$r-Rao-type lower bound. The relations between the estimators and the performance bounds, as well as their properties, are discussed. Finally, we demonstrate the performance of the proposed estimators and bounds via simulations of estimation after channel selection. We show that the proposed performance bounds are more informative than the oracle Cram$\acute{\text{e}}$r-Rao Bound (CRB), where the third interpretation (selective inference) results in the lowest mean-squared-error (MSE) among the estimators.

Non-Bayesian Post-Model-Selection Estimation as Estimation Under Model Misspecification

TL;DR

This work addresses estimation after model selection when the true model is unknown and may be misspecified. It formalizes three interpretations of post-model-selection estimation as misspecification problems (naive, normalized, selective inference), derives corresponding misspecified ML estimators and pseudo-true parameter vectors, and introduces the PS-MCRB to bound estimation performance. The paper shows that the selective-inference interpretation yields coherent likelihoods and the PSML estimator, achieving the lowest MSE and providing the most informative bounds compared to the oracle CRB. Through a Gaussian linear-model example with channel-detection, the authors demonstrate that MSNL and PSML outperform the naive MSL approach, with PS-MCRBs offering tighter performance guarantees aligned with the selection procedure. Overall, the selective-inference framework provides a practical, theoretically grounded approach to post-model-selection estimation under misspecification with meaningful implications for signal processing and related domains.

Abstract

In many parameter estimation problems, the exact model is unknown and is assumed to belong to a set of candidate models. In such cases, a predetermined data-based selection rule selects a parametric model from a set of candidates before the parameter estimation. The existing framework for estimation under model misspecification does not account for the selection process that led to the misspecified model. Moreover, in post-model-selection estimation, there are multiple candidate models chosen based on the observations, making the interpretation of the assumed model in the misspecified setting non-trivial. In this work, we present three interpretations to address the problem of non-Bayesian post-model-selection estimation as an estimation under model misspecification problem: the naive interpretation, the normalized interpretation, and the selective inference interpretation, and discuss their properties. For each of these interpretations, we developed the corresponding misspecified maximum likelihood estimator and the misspecified Cramr-Rao-type lower bound. The relations between the estimators and the performance bounds, as well as their properties, are discussed. Finally, we demonstrate the performance of the proposed estimators and bounds via simulations of estimation after channel selection. We show that the proposed performance bounds are more informative than the oracle Cramr-Rao Bound (CRB), where the third interpretation (selective inference) results in the lowest mean-squared-error (MSE) among the estimators.
Paper Structure (34 sections, 2 theorems, 102 equations, 3 figures)

This paper contains 34 sections, 2 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Post-Model-Selection Estimation
  4. Model: Estimation After Model Selection
  5. Post-Model-Selection Estimation as Estimation Under model misspecification
  6. Naive Interpretation
  7. Normalized Interpretation
  8. Selective Inference Interpretation
  9. Post-Model-Selection Estimators
  10. Maximum Selected Likelihood (MSL) Estimator
  11. Maximum Selected Normalized Likelihood (MSNL) Estimator
  12. Post-Selection ML (PSML) Estimator
  13. Post-Model-Selection Performance Analysis
  14. Post-model-selection MSE
  15. Post-Model-Selection Pseudo-True Parameter Vectors
  16. ...and 19 more sections

Key Result

Theorem 1

($k$th PS-MCRB) Let $f({\bf{x}};\boldsymbol \theta)$ be a general assumed pdf for a post-model-selection scheme with a selection rule $\Psi$ that satisfy cond1cond2cond3cond4. The $k$th-MSE of any finite variance, PSMS-unbiased estimator, $\hat{\boldsymbol \theta}^{(k)}$, satisfies where the $k$th-PS-MCRB is given by the pseudo-true parameter vector $\boldsymbol \vartheta^{(k)}$ and the post-mod

Figures (3)

  • Figure 1: Post-model-selection estimation scheme: first, a model is selected from a pool of candidate models based on a predetermined selection rule. Second, the unknown parameters of the selected model are estimated. Both stages are performed based on the same observation vector, ${\bf{x}}$.
  • Figure 2: The MSE of the MSL, MSNL, PSML, and the oracle MLs estimators, and PS-MCEBs and the oracle CRB versus the threshold, $\gamma$, where the true hypothesis is ${\cal H}_1$.
  • Figure 3: The bias (a) and MSE (b) of the MSL, MSNL, PSML, and the oracle MLs estimators, and PS-MCEBs and the oracle CRB (b) versus the threshold, $\gamma$, where the true hypothesis is ${\cal H}_2$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 3
  • Definition 4
  • Claim 1
  • Claim 2
  • Theorem 1
  • ...and 4 more