Genome-Wide Association Studies: Information Theoretic Limits of Reliable Learning

In the problems of Genome-Wide Association Study (GWAS), the objective is to associate subsequences of individual's genomes to the observable characteristics called phenotypes. The genome containing the biological information of an individual can be represented by a sequence of length $G$. Many observable characteristics of the individuals can be related to a subsequence of a given length $L$, called causal subsequence. The environmental affects make the relation between the causal subsequence and the observable characteristics a stochastic function. Our objective in this paper is to detect the causal subsequence of a specific phenotype using a dataset of $N$ individuals and their observed characteristics. We introduce an abstract formulation of GWAS which allows us to investigate the problem from an information theoretic perspective. In particular, as the parameters $N, G$, and $L$ grow, we observe a threshold effect at $\frac{Gh{(L/G)}}{N}$, where $h(.)$ is the binary entropy function. This effect allows us to define the capacity of recovering the causal subsequence by denoting the rate of the GWAS problem as $\frac{Gh(L/G)}{N}$. We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the $\epsilon$ -error-rate.