arXiv subjects

Enes Makalic

Publications and source records attributed to Enes Makalic.

16 recordsLinked to original sources

arXiv2026

Entropic Strict Minimum Message Length and Its Connections to PAC-Bayes and NML

We introduce entropic strict minimum message length (SMML), a risk-sensitive generalization of strict minimum message length coding. The proposed criterion replaces expected two-part codelength under the prior predictive distribution with an exponential certainty equivalent, thereby defining a one-parameter family of coding rules that interpolates between Bayesian average-case coding and worst-case minimax coding. We show that ordinary SMML is recovered in the risk-neutral limit, while the extreme risk-sensitive limit yields a minimax codelength criterion. Applying the same entropic soft maximum to regret relative to the oracle maximum likelihood codelength recovers the normalized maximum likelihood (NML) minimax-regret principle. We further prove that entropic SMML admits a variational characterization as a Kullback--Leibler-regularized worst-case expected codelength, giving it a PAC--Bayes-type interpretation. We establish joint $n$--$τ$ asymptotics that identify how the risk parameter must scale with sample size in order to recover Bayesian average-case, intermediate robust, and worst-case minimax coding behavior. For regular exponential families, the fixed-codebook partition remains affine in sufficient-statistic space, while the codepoints satisfy a tilted moment-matching condition and admit an interpretation as tilted Bregman centroids. These results position entropic SMML as an information theoretic bridge between MML, PAC--Bayes, and MDL.

math.ST↗

arXiv2026

Asymptotic theory and first-order bias of the Wallace--Freeman estimator

The Wallace--Freeman estimator is a classical minimum message length estimator whose relationship with likelihood-based asymptotic theory has not been fully developed. We show that, in regular parametric models, the Wallace--Freeman criterion is equivalent, up to constants, to a penalised likelihood criterion with penalty weight $n^{-1}$. This representation places the estimator within the standard theory of penalised M-estimation and yields existence, consistency, an asymptotic linear expansion, and asymptotic normality under regularity conditions. We further derive the first-order difference between the Wallace--Freeman estimator and the maximum likelihood estimator, showing that it is an explicit $O(n^{-1})$ shift determined by the gradient of the Wallace--Freeman penalty. Combining this expansion with the Cox--Snell formula gives a first-order bias expansion for the Wallace--Freeman estimator. The result clarifies its relationship with maximum likelihood, Jeffreys-prior penalisation, and Firth-type bias reduction. We illustrate the theory for the Weibull model, where the penalty modifies the leading bias of the maximum likelihood estimator of the shape parameter.

math.ST↗

arXiv2026

Information Geometry and Asymptotic Theory for SMML Estimators

Strict minimum message length (SMML) is an information-theoretic coding principle that represents a continuous statistical model by a finite set of assertions and a partition of the sample space. We show that the SMML objective decomposes into assertion entropy and conditional cross-entropy, balancing the cost of identifying an assertion against the cost of encoding data under the assigned model. For any fixed partition, the optimal codepoint for each cell is the model distribution that minimises Kullback-Leibler divergence from the data distribution restricted to that cell. Using the local Fisher-Rao geometry of regular parametric models, we show that, under high-resolution regularity conditions, optimal SMML partitions are asymptotically the pullback, through the maximum likelihood estimator, of weighted Fisher-Rao Voronoi tessellations in parameter space, with assertion probabilities appearing as additive weights. For regular exponential families, SMML codepoints satisfy a moment-matching condition and admit an interpretation as KL/Bregman centroids, while exact SMML cells are pullbacks of convex polyhedra in sufficient-statistic space. Together, these results show that SMML induces a natural information-geometric quantisation linking entropy-based coding, KL projection, and divergence-based Voronoi geometry.

math.ST↗

arXiv2026

MML Probabilistic Principal Component Analysis

Principal component analysis (PCA) is perhaps the most widely used method for data dimensionality reduction. A key question in PCA is deciding how many factors to retain. This manuscript describes a new approach to automatically selecting the number of principal components based on the Bayesian minimum message length method of inductive inference. We derive a new estimate of the isotropic residual variance and demonstrate that it improves on the usual maximum likelihood approach. We also discuss extending this approach to finite mixture models of principal component analyzers.

stat.ME↗

arXiv2023

Minimum message length inference of the Weibull distribution with complete and censored data

The Weibull distribution, with shape parameter $k>0$ and scale parameter $λ>0$, is one of the most popular parametric distributions in survival analysis with complete or censored data. Although inference of the parameters of the Weibull distribution is commonly done through maximum likelihood, it is well established that the maximum likelihood estimate of the shape parameter is inadequate due to the associated large bias when the sample size is small or the proportion of censored data is large. This manuscript demonstrates how the Bayesian information-theoretic minimum message length principle coupled with a suitable choice of weakly informative prior distributions, can be used to infer Weibull distribution parameters given complete data or data with type I censoring. Empirical experiments show that the proposed minimum message length estimate of the shape parameter is superior to the maximum likelihood estimate and appears superior to other recently proposed modified maximum likelihood estimates in terms of Kullback-Leibler risk. Lastly, we derive an extension of the proposed method to data with type II censoring.

stat.ME↗

arXiv2023

Maximum likelihood estimation of the Weibull distribution with reduced bias

In this short note, we derive a new bias adjusted maximum likelihood estimate for the shape parameter of the Weibull distribution with complete data and type I censored data. The proposed estimate of the shape parameter is significantly less biased and more efficient than the corresponding maximum likelihood estimate, while being simple to compute using existing maximum likelihood software procedures.

stat.ME↗

arXiv2022

Sparse Horseshoe Estimation via Expectation-Maximisation

The horseshoe prior is known to possess many desirable properties for Bayesian estimation of sparse parameter vectors, yet its density function lacks an analytic form. As such, it is challenging to find a closed-form solution for the posterior mode. Conventional horseshoe estimators use the posterior mean to estimate the parameters, but these estimates are not sparse. We propose a novel expectation-maximisation (EM) procedure for computing the MAP estimates of the parameters in the case of the standard linear model. A particular strength of our approach is that the M-step depends only on the form of the prior and it is independent of the form of the likelihood. We introduce several simple modifications of this EM procedure that allow for straightforward extension to generalised linear models. In experiments performed on simulated and real data, our approach performs comparable, or superior to, state-of-the-art sparse estimation methods in terms of statistical performance and computational cost.

stat.ML↗

arXiv2022

Introduction to minimum message length inference

The aim of this manuscript is to introduce the Bayesian minimum message length principle of inductive inference to a general statistical audience that may not be familiar with information theoretic statistics. We describe two key minimum message length inference approaches and demonstrate how the principle can be used to develop a new Bayesian alternative to the frequentist $t$-test as well as new approaches to hypothesis testing for the correlation coefficient. Lastly, we compare the minimum message length approach to the closely related minimum description length principle and discuss similarities and differences between both approaches to inference.

stat.ME↗

arXiv2020

Log-Scale Shrinkage Priors and Adaptive Bayesian Global-Local Shrinkage Estimation

Global-local shrinkage hierarchies are an important innovation in Bayesian estimation. We propose the use of log-scale distributions as a novel basis for generating familes of prior distributions for local shrinkage hyperparameters. By varying the scale parameter one may vary the degree to which the prior distribution promotes sparsity in the coefficient estimates. By examining the class of distributions over the logarithm of the local shrinkage parameter that have log-linear, or sub-log-linear tails, we show that many standard prior distributions for local shrinkage parameters can be unified in terms of the tail behaviour and concentration properties of their corresponding marginal distributions over the coefficients $β_j$. We derive upper bounds on the rate of concentration around $|β_j|=0$, and the tail decay as $|β_j| \to \infty$, achievable by this wide class of prior distributions. We then propose a new type of ultra-heavy tailed prior, called the log-$t$ prior with the property that, irrespective of the choice of associated scale parameter, the marginal distribution always diverges at $β_j = 0$, and always possesses super-Cauchy tails. We develop results demonstrating when prior distributions with (sub)-log-linear tails attain Kullback--Leibler super-efficiency and prove that the log-$t$ prior distribution is always super-efficient. We show that the log-$t$ prior is less sensitive to misspecification of the global shrinkage parameter than the horseshoe or lasso priors. By incorporating the scale parameter of the log-scale prior distributions into the Bayesian hierarchy we derive novel adaptive shrinkage procedures. Simulations show that the adaptive log-$t$ procedure appears to always perform well, irrespective of the level of sparsity or signal-to-noise ratio of the underlying model.

math.ST↗

arXiv2018

An efficient algorithm for sampling from $\sin^k(x)$ for generating random correlation matrices

In this note, we develop a novel algorithm for generating random numbers from a distribution with a probability density function proportional to $\sin^k(x)$, $x \in (0,π)$ and $k \geq 1$. Our algorithm is highly efficient and is based on rejection sampling where the envelope distribution is an appropriately chosen beta distribution. An example application illustrating how the new algorithm can be used to generate random correlation matrices is discussed.

stat.CO↗

arXiv2018

A Minimum Message Length Criterion for Robust Linear Regression

This paper applies the minimum message length principle to inference of linear regression models with Student-t errors. A new criterion for variable selection and parameter estimation in Student-t regression is proposed. By exploiting properties of the regression model, we derive a suitable non-informative proper uniform prior distribution for the regression coefficients that leads to a simple and easy-to-apply criterion. Our proposed criterion does not require specification of hyperparameters and is invariant under both full rank transformations of the design matrix and linear transformations of the outcomes. We compare the proposed criterion with several standard model selection criteria, such as the Akaike information criterion and the Bayesian information criterion, on simulations and real data with promising results.

stat.ME↗

arXiv2018

Minimum message length inference of the Poisson and geometric models using heavy-tailed prior distributions

Minimum message length is a general Bayesian principle for model selection and parameter estimation that is based on information theory. This paper applies the minimum message length principle to a small-sample model selection problem involving Poisson and geometric data models. Since MML is a Bayesian principle, it requires prior distributions for all model parameters. We introduce three candidate prior distributions for the unknown model parameters with both light- and heavy-tails. The performance of the MML methods is compared with objective Bayesian inference and minimum description length techniques based on the normalized maximum likelihood code. Simulations show that our MML approach with a heavy-tail prior distribution provides an excellent performance in all tests.

stat.ME↗

arXiv2017

Bayesian Sparse Global-Local Shrinkage Regression for Selection of Grouped Variables

Most estimates for penalised linear regression can be viewed as posterior modes for an appropriate choice of prior distribution. Bayesian shrinkage methods, particularly the horseshoe estimator, have recently attracted a great deal of attention in the problem of estimating sparse, high-dimensional linear models. This paper extends these ideas, and presents a Bayesian grouped model with continuous global-local shrinkage priors to handle complex group hierarchies that include overlapping and multilevel group structures. As the posterior mean is never a sparse estimate of the linear model coefficients, we extend the recently proposed decoupled shrinkage and selection (DSS) technique to the problem of selecting groups of variables from posterior samples. To choose a final, sparse model, we also adapt generalised information criteria approaches to the DSS framework. To ensure that sparse groups, in which only a few predictors are active, can be effectively identified, we provide an alternative degrees of freedom estimator for sparse Bayesian linear models that takes into account the effects of shrinkage on the model coefficients. Simulations and real data analysis using our proposed method show promising performance in terms of correct identification of active and inactive groups, and prediction, in comparison with a Bayesian grouped slab-and-spike approach.

stat.ME↗

arXiv2016

High-Dimensional Bayesian Regularised Regression with the BayesReg Package

Bayesian penalized regression techniques, such as the Bayesian lasso and the Bayesian horseshoe estimator, have recently received a significant amount of attention in the statistics literature. However, software implementing state-of-the-art Bayesian penalized regression, outside of general purpose Markov chain Monte Carlo platforms such as STAN, is relatively rare. This paper introduces bayesreg, a new toolbox for fitting Bayesian penalized regression models with continuous shrinkage prior densities. The toolbox features Bayesian linear regression with Gaussian or heavy-tailed error models and Bayesian logistic regression with ridge, lasso, horseshoe and horseshoe$+$ estimators. The toolbox is free, open-source and available for use with the MATLAB and R numerical platforms.

stat.CO↗

arXiv2015

A simple sampler for the horseshoe estimator

In this note we derive a simple Bayesian sampler for linear regression with the horseshoe hierarchy. A new interpretation of the horseshoe model is presented, and extensions to logistic regression and alternative hierarchies, such as horseshoe$+$, are discussed. Due to the conjugacy of the proposed hierarchy, Chib's algorithm may be used to easily compute the marginal likelihood of the model.

stat.CO↗

arXiv2014

Genome disorder and breast cancer susceptibility

Many common diseases have a complex genetic basis in which large numbers of genetic variations combine with environmental and lifestyle factors to determine risk. However, quantifying such polygenic effects and their relationship to disease risk has been challenging. In order to address these difficulties we developed a global measure of the information content of an individual's genome relative to a reference population, which may be used to assess differences in global genome structure between cases and appropriate controls. Informally this measure, which we call relative genome information (RGI), quantifies the relative "disorder" of an individual's genome. In order to test its ability to predict disease risk we used RGI to compare single nucleotide polymorphism genotypes from two independent samples of women with early-onset breast cancer with three independent sets of controls. We found that RGI was significantly elevated in both sets of breast cancer cases in comparison with all three sets of controls, with disease risk rising sharply with RGI (odds ratio greater than 12 for the highest percentile RGI). Furthermore, we found that these differences are not due to associations with common variants at a small number of disease-associated loci, but rather are due to the combined associations of thousands of markers distributed throughout the genome. Our results indicate that the information content of an individual's genome may be used to measure the risk of a complex disease, and suggest that early-onset breast cancer has a strongly polygenic basis.

q-bio.GN↗