Computational Bayesian Methods Applied to Complex Problems in Bio and Astro Statistics

In this dissertation we apply computational Bayesian methods to three distinct problems. In the first chapter, we address the issue of unrealistic covariance matrices used to estimate collision probabilities. We model covariance matrices with a Bayesian Normal-Inverse-Wishart model, which we fit with Gibbs sampling. In the second chapter, we are interested in determining the sample sizes necessary to achieve a particular interval width and establish non-inferiority in the analysis of prevalences using two fallible tests. To this end, we use a third order asymptotic approximation. In the third chapter, we wish to synthesize evidence across multiple domains in measurements taken longitudinally across time, featuring a substantial amount of structurally missing data, and fit the model with Hamiltonian Monte Carlo in a simulation to analyze how estimates of a parameter of interest change across sample sizes

Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice

We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigate the critical region around the phase transition, and the behavior of two-time correlation functions. We finally underline the close relationship between this approach and the dynamical replica theory under the assumption of replica symmetry.Comment: 21 pages, 5 figure

Bayesian statistics and modelling

Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data. The background knowledge is expressed as a prior distribution and combined with observational data in the form of a likelihood function to determine the posterior distribution. The posterior can also be used for making predictions about future events. This Primer describes the stages involved in Bayesian analysis, from specifying the prior and data models to deriving inference, model checking and refinement. We discuss the importance of prior and posterior predictive checking, selecting a proper technique for sampling from a posterior distribution, variational inference and variable selection. Examples of successful applications of Bayesian analysis across various research fields are provided, including in social sciences, ecology, genetics, medicine and more. We propose strategies for reproducibility and reporting standards, outlining an updated WAMBS (when to Worry and how to Avoid the Misuse of Bayesian Statistics) checklist. Finally, we outline the impact of Bayesian analysis on artificial intelligence, a major goal in the next decade

Explosive Percolation: Unusual Transitions of a Simple Model

In this paper we review the recent advances on explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (Science, 2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in "Achlioptas processes". Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdos networks), but with very unusual finite size scaling. We present a review of the field, indicate open "problems" and propose directions for future research.Comment: 27 pages, 4 figures, Review pape

Simulations in statistical physics and biology: some applications

One of the most active areas of physics in the last decades has been that of critical phenomena, and Monte Carlo simulations have played an important role as a guide for the validation and prediction of system properties close to the critical points. The kind of phase transitions occurring for the Betts lattice (lattice constructed removing 1/7 of the sites from the triangular lattice) have been studied before with the Potts model for the values q=3, ferromagnetic and antiferromagnetic regime. Here, we add up to this research line the ferromagnetic case for q=4 and 5. In the first case, the critical exponents are estimated for the second order transition, whereas for the latter case the histogram method is applied for the occurring first order transition. Additionally, Domany's Monte Carlo based clustering technique mainly used to group genes similar in their expression levels is reviewed. Finally, a control theory tool --an adaptive observer-- is applied to estimate the exponent parameter involved in the well-known Gompertz curve. By treating all these subjects our aim is to stress the importance of cooperation between distinct disciplines in addressing the complex problems arising in biology. Contents: Chapter 1 - Monte Carlo simulations in stat. physics; Chapter 2: MC simulations in biology; Chapter 3: Gompertz equationComment: 82 pages, 33 figures, 4 tables, somewhat reduced version of the M.Sc. thesis defended in Jan. 2006 at IPICyT, San Luis Potosi, Mx. (Supervisers: Drs. R. Lopez-Sandoval and H.C. Rosu). Last sections 3.3 and 3.4 can be found at http://lanl.arxiv.org/abs/physics/041108