Convex Quantization Preserves Logconcavity

TL;DR

This paper proves that using convex quantization regions preserves the logconcavity of likelihood functions, which is crucial for statistical inference and optimization.

Contribution

It provides a general proof that quantized likelihoods remain logconcave under convex quantization regions, extending the understanding of likelihood models with quantization.

Findings

Logconcavity is preserved under convex quantization.

Quantized likelihoods maintain properties essential for inference.

The result applies broadly to models with convex quantization regions.

Abstract

A logconcave likelihood is as important to proper statistical inference as a convex cost function is important to variational optimization. Quantization is often disregarded when writing likelihood models, ignoring the limitations of the physical detectors used to collect the data. These two facts call for the question: would including quantization in likelihood models preclude logconcavity? are the true data likelihoods logconcave? We provide a general proof that the same simple assumption that leads to logconcave continuous-data likelihoods also leads to logconcave quantized-data likelihoods, provided that convex quantization regions are used.