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The Value of Information for Populations in Varying Environments

Olivier Rivoire, Stanislas Leibler

arXiv:1010.5092v1q-bio.PEcond-mat.stat-mechcs.IT

TL;DR

The paper addresses how to define and value information in biological populations when fitness is difficult to formalize. It develops a mathematical model of growing populations with inherited and acquired information, then shows that individual-level stochasticity can exceed the mutual-information bound familiar from finance, motivating generalized uncertainty and information measures.

  • Problem

    Biology lacks a quantitative information measure grounded in reproductive fitness, especially when information is processed at both individual and population levels.

  • Method

    The paper analyzes a mathematical model of growing populations in varying environments with inherited information, environmental signals, individual stochasticity, and long-term growth as fitness.

  • Results

    Individual-level stochasticity can violate the mutual-information bound from the financial limit, requiring generalized measures of entropy and mutual information.

  • Takeaways & Limitations

    Information theory applied to biology must account for causality, functional value, and the distinct levels at which information is processed in populations.

  • Takeaways & Limitations

    Closed-form growth-rate expressions are generally unavailable when multiplication matrices are non-diagonal, except in special solvable cases.

Abstract

from arXiv · show

The notion of information pervades informal descriptions of biological systems, but formal treatments face the problem of defining a quantitative measure of information rooted in a concept of fitness, which is itself an elusive notion. Here, we present a model of population dynamics where this problem is amenable to a mathematical analysis. In the limit where any information about future environmental variations is common to the members of the population, our model is equivalent to known models of financial investment. In this case, the population can be interpreted as a portfolio of financial assets and previous analyses have shown that a key quantity of Shannon's communication theory, the mutual information, sets a fundamental limit on the value of information. We show that this bound can be violated when accounting for features that are irrelevant in finance but inherent to biological systems, such as the stochasticity present at the individual level. This leads us to generalize the measures of uncertainty and information usually encountered in information theory.

I. INTRODUCTION

The paper revisits how information can be quantified in biology by modeling growing populations, where fitness and distributed information processing matter. It also examines evolutionary strategies and when they correspond to Bayesian computation.

  • Motivation: Biological information requires a fitness-based measure because populations process information for reproductive value rather than merely encoding or transmitting signals.The population setting also enables bet-hedging through diversification.
  • Approach: The model connects biological regulation with engineering control and financial investment while retaining population structure and distributed decision-making.Unlike finance, biological information processing is distributed among potentially independent individuals.
  • Approach: Under assumptions of no memory, common environmental perception, and perfect adaptation, the value of information is expressed by mutual information.Relaxing these assumptions exposes limitations of the usual information measure.
  • Contributions: The paper also characterizes evolutionary stable strategies that optimize fitness and finds Bayesian computation under assumptions A2 and A3.When those assumptions fail, population-level features can make Bayesian computation irrelevant.

II. MODEL

The model represents growing populations whose types, reproduction, and environmental perceptions interact under varying environmental conditions. It separates common environmental signals from individual stochasticity and frames strategy choice and information value through long-term population growth.

  • Model ingredients: The model incorporates changing organisms, reproduction, environmental effects on organismal faculties, and environmental variation.These features are presented as commonly shared by living organisms.
  • Environment: Environmental states follow a stationary ergodic Markov chain, with i.i.d. environments as a special case.The chain has a unique stationary distribution independent of its initial state.
  • Individuals: An organism’s type determines its expected offspring number together with the current environmental state, including death when reproduction is zero.Types may represent phenotypes or genotypes.
  • Information processing: Two communication channels separate a population-wide environmental cue from independently perceived individual signals.The environmental channel produces x′_t, while the individual channel produces y_t and can model imperfect sensing or delays.
  • Population dynamics: Population dynamics are analyzed through conditional mean sizes governed by products of random matrices, whose asymptotic behavior controls population growth.The mean analysis can nevertheless overlook extinction events in discrete populations.
  • Financial correspondence: The financial interpretation removes individual stochasticity by making perceived signals common and treating currency units as strictly equivalent.Biological populations retain individual-level heterogeneity, producing qualitative differences from financial investment.
  • Questions and fitness: The model asks which transition strategy is most advantageous and what value is provided by environmental information.Answering these questions requires a non-arbitrary fitness measure emerging from long-term growth.
  • Assumptions: The simplifying assumptions restrict the model to open-loop control, common signals, and environments with a single perfectly adapted survivor type.These assumptions identify the conditions under which the financial analogy applies.

III. FITNESS AND OPTIMIZATION

The paper defines long-term growth as a fitness criterion for populations facing variable environments and examines optimal strategies under uncertainty. It also connects information value to entropy and mutual information under specific assumptions, then generalizes these quantities when those assumptions are relaxed.

  • Geometric-mean fitness: Maximizing geometric mean growth provides an asymptotically optimal strategy for almost every sequence of environmental outcomes.This strategy can outperform time-varying alternatives and is biologically interpreted as an evolutionary stable strategy.
  • Geometric-mean fitness: The long-term growth rate is an unambiguous fitness measure only in the infinite-horizon limit; finite horizons can favor different strategies.For a single time step, optimizing expected multiplication rate may be preferable.
  • Environmental processes: For stationary ergodic environments, geometric-mean growth remains the relevant long-term fitness criterion beyond independent environments.Self-averaging makes long environmental realizations share common statistical features.
  • Population dynamics: In the absence of inherited information, the growth analysis simplifies because the strategy depends only on current environmental information.This case reduces the population dynamics to a setting with direct recursion for population size.
  • Information measures: Under assumptions (A1)–(A3), uncertainty cost and acquired-information value correspond respectively to conditional entropy and mutual information.Relaxing (A1) or (A2) motivates generalized uncertainty and information measures independent of the multiplication matrix.

IV. KELLY’S HORSE RACES

The horse-race model applies Kelly-style geometric-growth optimization to uncertain environmental outcomes. It shows that proportional betting is optimal without side information, while acquired signals reduce uncertainty and provide value quantified by mutual information.

  • No acquired information: Proportional betting is the optimal strategy and does not depend on the numerical values of the returns.Its optimal growth rate separates the best growth achievable with perfect knowledge from the cost of environmental uncertainty.
  • No acquired information: The uncertainty term corresponds to Shannon entropy, which quantifies the cost of not knowing the realized sequence despite knowing outcome frequencies.Entropy also characterizes the growth rate of the number of typical environmental sequences.
  • No acquired information: Incorrectly estimating environmental frequencies incurs an additional cost measured by relative entropy, or Kullback–Leibler divergence.The divergence is nonnegative and vanishes only when the estimated and actual strategies coincide.
  • Acquired information: With acquired signals, the optimal strategy becomes conditional proportional betting based on the environmental distribution conditioned on the received signal.The strategy exactly amounts to a Bayesian computation.
  • Acquired information: Perfect side-information eliminates the entropic cost, leaving only the optimal growth rate with perfect information.This occurs when the signal equals the environmental state.
  • Acquired information: The gain in predictability from a signal is mutual information between the environmental state and the acquired information.The same quantity appears in channel capacity and rate-distortion theory.
  • Conclusion: For stationary ergodic environments, the decomposition extends by replacing the independent-environment distribution with the stationary distribution.The paper then examines how these relations change when assumptions about inheritance, information sharing, and multiplication rates are relaxed.

V. CAUSAL CONSTRAINTS AND INHERITED INFORMATION

Inherited information and causal access to environmental signals change the relevant uncertainty and information measures. The model identifies causally conditional entropy and directed information as the appropriate rates, with solvability limited outside special cases.

  • Inherited information: Under assumptions (A2) and (A3), Bayesian inference using inherited state and current signal is optimal, with uncertainty cost independent of f.The strategy conditions on x_t−1 and y_t through P_Xt|Xt−1,Yt.
  • Inherited information: Inherited information reduces uncertainty cost from H(X_t) to H(X_t|X_t−1), with mutual information I(X_t;X_t−1) measuring its value.For Markov environments this conditional entropy is the environmental entropy rate; for i.i.d. environments it equals H(X_t).
  • Causal constraints: The value of acquired information is the directed-information rate I(Y→X), which is generally smaller than the mutual-information rate I(X;Y).The difference reflects information about current states contained in future signals, which is unavailable to strategies using only current signals and past memory.
  • Causal constraints: With acquired information, uncertainty cost is the rate of causally conditional entropy H(X∥Y), not the ordinary conditional-entropy rate H(X|Y).The causally conditional rate is greater than or equal to the ordinary conditional-entropy rate.
  • Scope and solvability: These causal quantities extend beyond Markov processes to ergodic processes when strategies allow arbitrarily long memory.Without diagonal f, combined inherited and acquired information is generally difficult to solve; closed-form growth-rate expressions are unavailable, except in special cases such as horse races.

VI. INDIVIDUAL STOCHASTICITY AND DISTRIBUTED INFORMATION

Individual-level stochasticity and distributed information produce generalized uncertainty and information measures without direct financial analogues. The model shows that population-level information can be more valuable than the information available to any individual and can exceed mutual information.

  • Distributed information: Allowing individuals to perceive different environmental signals yields a generalized entropy with no equivalent in financial-investment models.The analysis compares channels placed at the population level with channels placed at the individual level.
  • Distributed information: The same channel induces less uncertainty at the individual level than at the population level, as a consequence of Jensen’s inequality.The generalized entropy generically satisfies H(qenv,qin) < H(X_t|Y_t).
  • Value of information: Mutual information between environmental state and perceived signal is not an upper bound on acquired information’s value.The generalized value I(qenv,qin) can exceed the mutual information relevant to the source-signal pair.
  • Value of information: Collectively acquired information can exceed the value of information acquired by any individual member of the population.This contrasts with formulations in which I(X_t;Y_t) limits information value for control.
  • Optimal strategy: A critical noise level ε_c(p) may exist below which deterministic individual responses achieve optimal growth without requiring a stochastic response.At low error rates, deterministic responses to stochastic signals can still diversify the population optimally.

VII. GENERAL MULTIPLICATION RATES AND FUNCTIONAL INFORMATION

Relaxing the diagonal multiplication-rate assumption shows that uncertainty and information value depend on how environmental states map onto surviving types. The resulting measures refine entropy and mutual information, with individual-level information potentially exceeding the usual mutual-information bound.

  • Functional information: Non-diagonal multiplication rates make uncertainty measures depend on f(σ; x), because multiple types may survive in the same environment.Surviving non-optimal types can still contribute to population growth.
  • Functional information: The generalized uncertainty measure accounts for redundancy between environmental states and types encoded in f.It refines H(qenv,qin) beyond reductions caused only by unequal environmental probabilities.
  • Optimal strategies: For non-diagonal multiplication matrices, environmental states no longer have exclusive meanings, and optimal strategies may exclude some types.The paper gives analytically solvable two-type, two-state examples with homogeneous or switching strategies.
  • General conclusion: The paper generalizes entropy and mutual information to account for causality, processing level, and the meaning of information encoded in f.In the causality and functional-information cases, mutual information remains an upper limit that may be unattainable; at the individual level, the generalized bound can exceed it.
  • Value of information: At the population level, information value is bounded by mutual information, whereas individual-level information can yield a higher value.For the binary symmetric channel at p(1)=0.1, the optimal strategy changes from pure to mixed at εc(p)=0.1.

VIII. GENERALIZATIONS

The framework extends population control to broader information and control constraints, including feedback, sensing, switching, spatial heterogeneity, and general transition matrices. It also identifies implementation costs as a major unresolved limitation.

  • General framework: Population growth rate provides an unambiguous long-term fitness criterion for evaluating regulation under uncertainty.The model treats acquired information as feedforward and inherited information as feedback information.
  • General constraints: General constraints are represented by restricting τ(σt|σt−1, xt) to a subset C of conceivable transition matrices.This formulation can enforce different information patterns specifying who knows what and when.
  • Extensions: The framework can incorporate phenotype-dependent sensing, transmission errors, continuous states, continuous time, and communication or sexual reproduction.These extensions are expressed through additional constraints or transition matrices.
  • General constraints: Constraint costs and the value of relaxing constraints are generalized by H(C) and I(C;C′), respectively.The constrained optimal growth rate is compared with the unconstrained optimum.
  • Limitation: The major unresolved issue is specifying constraints and characterizing the growth-rate costs of implementing strategies.Potential costs include sensing, switching types, estimation, decision, and actuation, creating trade-offs between accuracy and growth.
  • Extensions: Spatial heterogeneity can be reduced to an effective multiplication rate when local sensing and reproduction are conditionally independent.The effective rate may be non-integer and non-diagonal even when the original multiplication rate is diagonal.

IX. CONCLUSION

The analysis integrates directionality and value into a biological information model, and shows that population-level information processing requires generalized entropy and mutual information.

  • The model addresses critiques that biological information measures overlook causality and the value of information.
  • Accounting for different information-processing levels in populations leads to generalized measures of entropy and mutual information.

Appendix A: Definition and properties of the model

The appendix defines a multi-type branching-process model in random environments and characterizes population growth and extinction through reproduction matrices and their dominant growth rates.

  • Model definition: The model is a multi-type branching process in a stationary, ergodic random environment with finite environmental and individual-state spaces.
  • Model definition: Individuals reproduce independently according to a distribution conditioned on the current environmental state, while population counts track each type over time.
  • Constant environment: In a constant environment, the population either becomes extinct or grows without bound, and the matrix A determines extinction and growth through its largest eigenvalue.
  • Constant environment: Λ > 0 is equivalent to a positive probability of avoiding extinction for every type, defining the supercritical regime assumed by the model.
  • Constant environment: Under the X log X condition and non-extinction, asymptotic population composition is described by the dominant eigenvector of A.
  • Varying environments: In varying environments, the Lyapunov exponent of the random-matrix product gives the typical growth rate, although population composition need not converge.

2. Two-state diagonal model with a binary erasure channel

For the two-state diagonal model with a binary erasure channel, optimization of growth rate reduces to a one-variable problem whose solution changes with channel noise.

  • Model setup: The binary erasure channel introduces an erasure probability ε while preserving the diagonal reproduction structure.
  • Optimization: The optimized strategy sets the correctly informed allocations to one, leaving the allocation after erasure as the single independent variable.
  • Optimization: The resulting optimization is formally equivalent to the no-information problem after reparameterizing noise by γ = ε/(1 − ε).
  • Results: For p(1) ≤ p(2), the critical noise level is εc(p) = (1 − 2p(1))/(1 − p(1)); below it, the optimum has π̂ = 0 and H(δ,qe)_p = −p(1) ln(1 − ε).

3. Two-state model diagonal with a binary symmetric channel

For the two-state diagonal model with a binary symmetric channel, the optimal strategy is generally boundary-valued, except in the fully noisy blind case where proportional betting is optimal.

  • Model setup: The binary symmetric channel flips the perceived state with probability ε, with correct transmission probability 1 − ε and 0 ≤ ε ≤ 1/2.
  • Optimization: The optimization depends on two independent correct-state allocations, π1 = π(1|1) and π2 = π(2|2).
  • Results: Only the blind case ε = 1/2 permits both optimized allocations to lie strictly between zero and one, yielding proportional betting π̂1 = p(1) and π̂2 = p(2).
  • Results: In every other case, at least one optimized allocation equals one, reducing the problem to a single-variable optimization.
  • Results: For p(1) ≤ p(2), the critical noise level is εc(p) = min(p(1), p(2)); the solution changes according to whether εc(p) is below or above p(1).

Appendix C: A solvable model in non i.i.d. environments

The model quantifies how environmental timing and population adjustment interact in non-i.i.d. environments. Its uncertainty cost separates environmental unpredictability from delays in reallocating population types.

  • Model and time scales: The model makes explicit the time scales governing short-term adjustment to current conditions and longer-term anticipation of environmental changes.The framework connects an earlier model to the paper’s treatment of uncertainty in timing.
  • Model and time scales: Adjustment depends on the duration τ(ϵ) of an environmental state and the time α(ϵ; ϵ′) required for its best-adapted type to dominate.The adjustment time depends on the population composition when the environment changes.
  • Adiabatic regime: In the adiabatic regime, environmental periods exceed adjustment times, allowing the growth rate to be expressed using the population’s near-equilibrium response to each state.The regime requires α(ϵ; ϵ′) ≪ τ(ϵ) for all distinct environmental states.
  • Uncertainty cost: The uncertainty cost contains an optimal Lyapunov-exponent contribution and a second term interpreted as an uncertainty cost depending on environmental transitions and multiplication rates.The two contributions can be separated in cases with no sensor or a reliable sensor.
  • Uncertainty cost: A transition-dependent term is analogous to the horse-race cost, while Γ(ϵ; ϵ′) represents delay in transferring most of the population between types.This delay term is absent from horse-race models because capital transfers occur instantaneously.
  • Uncertainty decomposition: Environmental uncertainty has separate components for the next state’s identity and the timing of environmental changes.The timing component is represented through the entropy of a Bernoulli distribution with parameter 1/τ(ϵ).

Appendix D: Proof of the entropic bound

The appendix proves an entropic upper bound on information-related uncertainty costs. The proof handles both injective and non-injective mappings from environmental states to optimal population types through direct construction and coarse-graining.

  • Bound setup: The appendix proves a bound on the uncertainty cost for the model under i.i.d. environments and arbitrary multiplication rates.The proof establishes the entropic bound in the general case.
  • Injective case: When the mapping from environmental states to optimal types is injective, an optimal transition matrix can be constructed so the bound’s right-hand side equals the generalized entropy H(q).The construction assigns each mapped environmental state its minimizing transition probability.
  • Non-injective case: For a non-injective mapping, equivalent environmental states are grouped into quotient classes defined by sharing the same optimal type.The resulting process is coarse-grained over those equivalence classes.
  • Non-injective case: Coarse-graining always reduces the generalized entropy, which yields an even tighter upper bound in the non-injective case.The inequality follows from the concavity of the logarithm.

Appendix E: Proof of the maladjustment bound

The appendix bounds the cost of using a strategy optimized for one environmental distribution when another distribution is correct. The argument combines optimality conditions with logarithmic concavity and the model’s population dynamics.

  • Bound statement: The appendix proves a bound on the cost of following a non-optimal strategy in an i.i.d. environment with arbitrary multiplication rates.It generalizes the corresponding bound for financial investment models with q_in = δ.
  • Bound statement: The mismatch between environmental distributions p and p′ is quantified by their relative entropy, ∑_x p(x) ln[p(x)/p′(x)].The proof embeds the two distributions into environmental processes with shared transition structure.
  • Proof strategy: The proof characterizes the strategy optimized for p′ using Kuhn–Tucker conditions and nonnegative multipliers λ_y.These conditions account for inequality constraints on strategy components.
  • Proof strategy: The argument extends an earlier proof by combining a union of environmental state spaces with a sequence of inequalities.The first inequality uses Jensen’s inequality, and the second uses the Kuhn–Tucker conditions.
  • Population dynamics: The population dynamics are represented by a population vector whose components count mean individuals of each type, with changes driven by multiplication rates and type switching.The perturbative treatment assumes composition changes are primarily caused by multiplication-rate differences rather than switching.
  • Population dynamics: The adiabatic approximation requires environmental periods to exceed adjustment times, so the population becomes approximately aligned with the dominant type for each environment.The relevant delay is determined by the preceding state associated with the second-largest eigenvalue.
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