Recombination operators and selection strategies for evolutionary Markov Chain Monte Carlo algorithms

Markov Chain Monte Carlo (MCMC) methods are often used to sample from intractable target distributions. Some MCMC variants aim to improve the performance by running a population of MCMC chains. In this paper, we investigate the use of techniques from Evolutionary Computation (EC) to design population-based MCMC algorithms that exchange useful information between the individual chains. We investigate how one can ensure that the resulting class of algorithms, called Evolutionary MCMC (EMCMC), samples from the target distribution as expected from any MCMC algorithm. We analytically and experimentally show—using examples from discrete search spaces—that the proposed EMCMCs can outperform standard MCMCs by exploiting common partial structures between the more likely individual states. The MCMC chains in the population interact through recombination and selection. We analyze the required properties of recombination operators and acceptance (or selection) rules in EMCMCs. An important issue is how to preserve the detailed balance property which is a sufficient condition for an irreducible and aperiodic EMCMC to converge to a given target distribution. Transferring EC techniques to population-based MCMCs should be done with care. For instance, we prove that EMCMC algorithms with an elitist acceptance rule do not sample the target distribution correctly

Contributions on evolutionary computation for statistical inference

Evolutionary Computation (EC) techniques have been introduced in the 1960s for dealing with complex situations. One possible example is an optimization problems not having an analytical solution or being computationally intractable; in many cases such methods, named Evolutionary Algorithms (EAs), have been successfully implemented. In statistics there are many situations where complex problems arise, in particular concerning optimization. A general example is when the statistician needs to select, inside a prohibitively large discrete set, just one element, which could be a model, a partition, an experiment, or such: this would be the case of model selection, cluster analysis or design of experiment. In other situations there could be an intractable function of data, such as a likelihood, which needs to be maximized, as it happens in model parameter estimation. These kind of problems are naturally well suited for EAs, and in the last 20 years a large number of papers has been concerned with applications of EAs in tackling statistical issues. The present dissertation is set in this part of literature, as it reports several implementations of EAs in statistics, although being mainly focused on statistical inference problems. Original results are proposed, as well as overviews and surveys on several topics. EAs are employed and analyzed considering various statistical points of view, showing and confirming their efficiency and flexibility. The first proposal is devoted to parametric estimation problems. When EAs are employed in such analysis a novel form of variability related to their stochastic elements is introduced. We shall analyze both variability due to sampling, associated with selected estimator, and variability due to the EA. This analysis is set in a framework of statistical and computational tradeoff question, crucial in nowadays problems, by introducing cost functions related to both data acquisition and EA iterations. The proposed method will be illustrated by means of model building problem examples. Subsequent chapter is concerned with EAs employed in Markov Chain Monte Carlo (MCMC) sampling. When sampling from multimodal or highly correlated distribution is concerned, in fact, a possible strategy suggests to run several chains in parallel, in order to improve their mixing. If these chains are allowed to interact with each other then many analogies with EC techniques can be observed, and this has led to research in many fields. The chapter aims at reviewing various methods found in literature which conjugates EC techniques and MCMC sampling, in order to identify specific and common procedures, and unifying them in a framework of EC. In the last proposal we present a complex time series model and an identification procedure based on Genetic Algorithms (GAs). The model is capable of dealing with seasonality, by Periodic AutoRegressive (PAR) modelling, and structural changes in time, leading to a nonstationary structure. As far as a very large number of parameters and possibilites of change points are concerned, GAs are appropriate for identifying such model. Effectiveness of procedure is shown on both simulated data and real examples, these latter referred to river flow data in hydrology. The thesis concludes with some final remarks, concerning also future work

Contributions on evolutionary computation for statistical inference

Evolutionary Computation (EC) techniques have been introduced in the 1960s for dealing with complex situations. One possible example is an optimization problems not having an analytical solution or being computationally intractable; in many cases such methods, named Evolutionary Algorithms (EAs), have been successfully implemented. In statistics there are many situations where complex problems arise, in particular concerning optimization. A general example is when the statistician needs to select, inside a prohibitively large discrete set, just one element, which could be a model, a partition, an experiment, or such: this would be the case of model selection, cluster analysis or design of experiment. In other situations there could be an intractable function of data, such as a likelihood, which needs to be maximized, as it happens in model parameter estimation. These kind of problems are naturally well suited for EAs, and in the last 20 years a large number of papers has been concerned with applications of EAs in tackling statistical issues. The present dissertation is set in this part of literature, as it reports several implementations of EAs in statistics, although being mainly focused on statistical inference problems. Original results are proposed, as well as overviews and surveys on several topics. EAs are employed and analyzed considering various statistical points of view, showing and confirming their efficiency and flexibility. The first proposal is devoted to parametric estimation problems. When EAs are employed in such analysis a novel form of variability related to their stochastic elements is introduced. We shall analyze both variability due to sampling, associated with selected estimator, and variability due to the EA. This analysis is set in a framework of statistical and computational tradeoff question, crucial in nowadays problems, by introducing cost functions related to both data acquisition and EA iterations. The proposed method will be illustrated by means of model building problem examples. Subsequent chapter is concerned with EAs employed in Markov Chain Monte Carlo (MCMC) sampling. When sampling from multimodal or highly correlated distribution is concerned, in fact, a possible strategy suggests to run several chains in parallel, in order to improve their mixing. If these chains are allowed to interact with each other then many analogies with EC techniques can be observed, and this has led to research in many fields. The chapter aims at reviewing various methods found in literature which conjugates EC techniques and MCMC sampling, in order to identify specific and common procedures, and unifying them in a framework of EC. In the last proposal we present a complex time series model and an identification procedure based on Genetic Algorithms (GAs). The model is capable of dealing with seasonality, by Periodic AutoRegressive (PAR) modelling, and structural changes in time, leading to a nonstationary structure. As far as a very large number of parameters and possibilites of change points are concerned, GAs are appropriate for identifying such model. Effectiveness of procedure is shown on both simulated data and real examples, these latter referred to river flow data in hydrology. The thesis concludes with some final remarks, concerning also future work

Studying the dark universe with galaxies

This work presents two different but connected projects that study the dark components of the universe. We show the first full shape analysis of the eBOSS Luminous Red Galaxy (LRG) sample, which has an effective redshift of $z_{eff}\sim0.72$ and used the data from the 14th data release of SDSS (DR14). Amongst other parameters, we constrain the growth rate of the universe to have the value $f(z_{eff})\sigma_8(z_{eff}) = 0.454 \pm 0.134$. Our results are in full agreement with the current $\Lambda$-Cold Dark Matter cosmological model under the Planck cosmology. This study was followed up with a later data release that has found comparable results (DR16 Gil-Marín et al., 2020). The second project uses sparse regression methods (SRM) to model the stellar masses of galaxies inside the EAGLE hydrodynamical simulation as a function of the properties of their host dark matter halos, without using prior knowledge of the underlying physics. Our model is designed to be an accurate and simple equation of the host halo properties, which makes it modifiable if one is interested in fitting to a set of statistics. An advantage of SRMs is that they are designed to remove unnecessary terms, our method discarded all parameters related to the angular momentum of the host halo, suggesting that they are not required to explain the stellar mass halo mass relation to the accuracy considered. Using an appropriate formulation of input parameters, our methodology can model satellite and central galaxies at the same time using a simpler model than when they are treated separately. Our models accurately reproduce the stellar mass function and the correlation function of EAGLE galaxies, which makes them an encouraging approach for the construction of realistic mock galaxy catalogs to interpret results from galaxy surveys