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Optimal prediction with resource constraints using the information bottleneck

Fig 8

The information bottleneck solution for a Wright Fisher process.

(a) The Wright-Fisher model of evolution can be visualized as a population of N parents giving rise to a population of N offspring. Genotypes of the offspring are selected as a function of the parents’ generation genotypes subject to mutation rates, μ, and selective pressures s. (b) Information bottleneck schematic with a discrete (rather than continuous) representation variable, . (c) Predictive information as a function of compression level. Predictive information increases with the cardinality, m, of the representation variable. The amount of predictive information is limited by log(m) (vertical dashed lines) for small m, and the mutual information between allele frequencies at time t + Δt and time t, I(X_t+Δt;X_t) (horizontal dashed line), for large m. Bifurcations occur in the amount of predictive information. For small , the encoding strategies for different m are degenerate and the degeneracy is lifted as ) increases, with large m schemes accessing higher ranges. Parameters: N = 100, Nμ = 0.2, Nμ = 0.2, Ns = 0.001, Δt = 1. (d-i) We explore information bottleneck solutions to Wright-Fisher dynamics under the condition that the cardinality of , m, is 2 and take β to be large enough that , β ≈ 4. Parameters: N = 100, Ns = 0.001, Δt = 1, and Nμ = 0.2, Nμ = 2, and Nμ = 40 (from left to right). (d-f) In blue, we plot the steady state distribution. In yellow and red, we show the inferred historical distribution of alleles based on the observed value of . Note that each distribution is corresponds to roughly non-overlapping portions of allele frequency space. (g-i) Predicted distribution of alleles based on the value of . We observe that as mutation rate increases, the timescale of relaxation to steady state decreases, so historical information is less useful and the predictions becomes more degenerate with the steady state distribution.

doi: https://doi.org/10.1371/journal.pcbi.1008743.g008