Loss functions in restricted parameter spaces and their Bayesian applications

Abstract

Squared error loss remains the most commonly used loss function for constructing a Bayes estimator of the parameter of interest. However, it can lead to sub-optimal solutions when a parameter is defined on a restricted space. It can also be an inappropriate choice in the context when an extreme overestimation and/or underestimation results in severe consequences and a more conservative estimator is preferred. We advocate a class of loss functions for parameters defined on restricted spaces which infinitely penalize boundary decisions like the squared error loss does on the real line. We also recall several properties of loss functions such as symmetry, convexity and invariance. We propose generalizations of the squared error loss function for parameters defined on the positive real line and on an interval. We provide explicit solutions for corresponding Bayes estimators and discuss multivariate extensions. Four well-known Bayesian estimation problems are used to demonstrate inferential benefits the novel Bayes estimators can provide in the context of restricted estimation.

Improving safety of the continual reassessment method via a modified allocation rule

“…Indeed, the squared distance between such linearized objects was proposed as a distance between p and γ and is known as the Aitchison distance. Although the Aitchison distance was proven to be a useful tool in the compositional data analysis, it was recently noted that the Aitchison distance lacks some important properties such as convexity and a closed form solution for the corresponding minimizer . Moreover, one can argued that clinicians might encounter particular difficulties with interpreting the distance in an actual trial.…”

“…Moreover, one can argued that clinicians might encounter particular difficulties with interpreting the distance in an actual trial. Instead, the convex unit‐interval‐symmetric divergence was proposed . The symmetry of the divergence is defined in terms of the squared distance after the logit transformation.…”

“…Although the Aitchison distance was proven to be a useful tool in the compositional data analysis, 12 it was recently noted that the Aitchison distance lacks some important properties such as convexity and a closed form solution for the corresponding minimizer. 9 Moreover, one can argued that clinicians might encounter particular difficulties with interpreting the distance (4) in an actual trial. Instead, the convex unit-interval-symmetric divergence…”

An Information Theoretic Phase I–II Design for Molecularly Targeted Agents That Does Not Require an Assumption of Monotonicity

“…Clearly, δ (·) ⩾ 0, δ (·)=0 if and only if θ = γ and boundary values θ i =0, i =1,2, or θ 1 + θ 2 =1 correspond to infinite values of δ (·). The former property is argued to be fundamental for restricted parameter spaces (Aitchison, ; Mateu i Figueras et al ., 2013; Mozgunov et al ., ) and enables us to avoid parametric assumptions. Terms in result (6) have the following interpretations.…”

“…We derive a novel trade‐off criterion that governs the regimen selection during the study. The form of the trade‐off function proposed is motivated by recent developments in the theory of weighted information measures (Mozgunov and Jaki, ) and in the estimation on restricted parameter spaces (Mozgunov et al ., ). Although model‐based estimates can be used together with the proposed trade‐off function, we show that highly accurate optimal and correct regimens recommendations can be achieved without parametric assumptions because of the trade‐off function's special properties.…”