[Paper Review] Sex as Gibbs Sampling: a probability model of evolution

Chris Watkins, Yvonne Buttkewitz|arXiv (Cornell University)|Feb 12, 2014

Evolution and Genetic Dynamics28 references3 citations

TL;DR

This paper proposes a novel probability model of sexual evolution as Gibbs sampling, where populations evolve via reversible Markov chains satisfying detailed balance. The stationary distribution is a Gibbs distribution, enabling exact characterization of mutation-selection equilibrium as a Bayesian posterior, with genomes analogous to latent variables sampled from this posterior using standard MCMC techniques.

ABSTRACT

We show that evolutionary computation can be implemented as standard Markov-chain Monte-Carlo (MCMC) sampling. With some care, `genetic algorithms' can be constructed that are reversible Markov chains that satisfy detailed balance; it follows that the stationary distribution of populations is a Gibbs distribution in a simple factorised form. For some standard and popular nonparametric probability models, we exhibit Gibbs-sampling procedures that are plausible genetic algorithms. At mutation-selection equilibrium, a population of genomes is analogous to a sample from a Bayesian posterior, and the genomes are analogous to latent variables. We suggest this is a general, tractable, and insightful formulation of evolutionary computation in terms of standard machine learning concepts and techniques. In addition, we show that evolutionary processes in which selection acts by differences in fecundity are not reversible, and also that it is not possible to construct reversible evolutionary models in which each child is produced by only two parents.

Motivation & Objective

• To establish a formal connection between evolutionary processes and Bayesian inference using probability models.
• To develop reversible Markov chain models of evolution that satisfy detailed balance, enabling exact characterization of stationary distributions.
• To show that sexual evolution can be interpreted as Gibbs sampling over populations, with fitness as likelihood and breeding as prior.
• To demonstrate that such models are implementable for arbitrary fitness functions and yield closed-form stationary distributions.

Proposed method

• Model evolution as a Markov chain over populations of genomes under constant selection and mutation.
• Use exchangeable breeding distributions (EBMs) to define conditional probabilities for genome sampling given the rest of the population.
• Construct reversible Markov chains by ensuring detailed balance through symmetric transition mechanisms.
• Define the stationary distribution as proportional to the product of the breeding prior and population fitness, resembling a Bayesian posterior.
• Implement sampling via Gibbs sampling procedures that are equivalent to genetic algorithms with reversible transitions.
• Prove that mutation-selection equilibrium corresponds to a Gibbs distribution, with populations analogous to posterior samples.

Experimental results

Research questions

• RQ1 Can sexual evolution be modeled as a reversible Markov chain that satisfies detailed balance?
• RQ2 Is the stationary distribution of such evolutionary processes equivalent to a Gibbs distribution with a factorized form?
• RQ3 Can standard MCMC methods like Gibbs sampling be interpreted as evolutionary computation in a biologically plausible way?
• RQ4 Are there general, implementable sampling algorithms for arbitrary fitness functions that preserve detailed balance?
• RQ5 What is the relationship between evolutionary computation and Bayesian inference in this framework?

Key findings

• The stationary distribution of the proposed evolutionary model is a Gibbs distribution, factorizing as the product of a breeding prior and population fitness.
• Evolutionary processes satisfying detailed balance have a well-characterized equilibrium distribution, enabling exact analysis of mutation-selection balance.
• Genomes in the population are analogous to latent variables sampled from a Bayesian posterior, with fitness acting as likelihood.
• The model allows for fully implementable genetic algorithms that are reversible and satisfy detailed balance for arbitrary fitness functions.
• It is not possible to construct reversible evolutionary models where each child is produced by only two parents, nor are fecundity-based selection models reversible.
• The framework provides a tractable, insightful link between evolutionary computation and standard machine learning techniques like Gibbs sampling.