arXiv subjects

Colin H. LaMont

Publications and source records attributed to Colin H. LaMont.

5 recordsLinked to original sources

arXiv2019

On the correspondence between thermodynamics and inference

We expand upon a natural analogy between Bayesian statistics and statistical physics in which sample size corresponds to inverse temperature. This analogy motivates the definition of two novel statistical quantities: a learning capacity and a Gibbs entropy. The analysis of the learning capacity, corresponding to the heat capacity in thermal physics, leads to new insight into the mechanism of learning and explains why some models have anomalously-high learning performance. We explore the properties of the learning capacity in a number of examples, including a sloppy model. Next, we propose that the Gibbs entropy provides a natural device for counting distinguishable distributions in the context of Bayesian inference. We use this device to define a generalized principle of indifference (GPI) in which every distinguishable model is assigned equal a priori probability. This principle results in a new solution to a long-standing problem in Bayesian inference: the definition of an objective or uninformative prior. A key characteristic of this new approach is that it can be applied to analyses where the model dimension is unknown and circumvents the automatic rejection of higher-dimensional models in Bayesian inference.

math.ST↗

arXiv2018

Information-based inference for singular models and finite sample sizes: A frequentist information criterion

In the information-based paradigm of inference, model selection is performed by selecting the candidate model with the best estimated predictive performance. The success of this approach depends on the accuracy of the estimate of the predictive complexity. In the large-sample-size limit of a regular model, the predictive performance is well estimated by the Akaike Information Criterion (AIC). However, this approximation can either significantly under or over-estimating the complexity in a wide range of important applications where models are either non-regular or finite-sample-size corrections are significant. We introduce an improved approximation for the complexity that is used to define a new information criterion: the Frequentist Information Criterion (QIC). QIC extends the applicability of information-based inference to the finite-sample-size regime of regular models and to singular models. We demonstrate the power and the comparative advantage of QIC in a number of example analyses.

stat.ML↗

arXiv2017

The Lindley paradox: The loss of resolution in Bayesian inference

There are three principle paradigms of statistics: Bayesian, frequentist and information-based inference. Although these paradigms are in agreement in some contexts, the Lindley paradox describes a class of problems, models of unknown dimension, where conflicting conclusions are generated by frequentist and Bayesian inference. This conflict can materially affect the scientific conclusions. Understanding the Lindley paradox---where it applies, why it occurs, and how it can be avoided---is therefore essential to the understanding of statistical analysis. In this paper, we revisit the Lindley paradox in the context of a simple biophysical application. We describe how predictive and postdictive measures of model performance provide a natural framework for understanding the Lindley paradox. We then identify methods which result in optimal experimental resolution for discovery.

math.ST↗

arXiv2015

An objective prior that unifies objective Bayes and information-based inference

There are three principle paradigms of statistical inference: (i) Bayesian, (ii) information-based and (iii) frequentist inference. We describe an objective prior (the weighting or $w$-prior) which unifies objective Bayes and information-based inference. The $w$-prior is chosen to make the marginal probability an unbiased estimator of the predictive performance of the model. This definition has several other natural interpretations. From the perspective of the information content of the prior, the $w$-prior is both uniformly and maximally uninformative. The $w$-prior can also be understood to result in a uniform density of distinguishable models in parameter space. Finally we demonstrate the the $w$-prior is equivalent to the Akaike Information Criterion (AIC) for regular models in the asymptotic limit. The $w$-prior appears to be generically applicable to statistical inference and is free of {\it ad hoc} regularization. The mechanism for suppressing complexity is analogous to AIC: model complexity reduces model predictivity. We expect this new objective-Bayes approach to inference to be widely-applicable to machine-learning problems including singular models.

stat.ML↗

arXiv2015

The development of an information criterion for Change-Point Analysis

Change-point analysis is a flexible and computationally tractable tool for the analysis of times series data from systems that transition between discrete states and whose observables are corrupted by noise. The change-point algorithm is used to identify the time indices (change points) at which the system transitions between these discrete states. We present a unified information-based approach to testing for the existence of change points. This new approach reconciles two previously disparate approaches to Change-Point Analysis (frequentist and information-based) for testing transitions between states. The resulting method is statistically principled, parameter and prior free and widely applicable to a wide range of change-point problems.

physics.data-an↗