Universal Darwinism As a Process of Bayesian Inference
Many of the mathematical frameworks describing natural selection are equivalent to Bayes' Theorem, also known as Bayesian updating. By definition, a process of Bayesian Inference is one which involves a Bayesian update, so we may conclude that these frameworks describe natural selection as a process of Bayesian inference. Thus, natural selection serves as a counter example to a widely-held interpretation that restricts Bayesian Inference to human mental processes (including the endeavors of statisticians). As Bayesian inference can always be cast in terms of (variational) free energy minimization, natural selection can be viewed as comprising two components: a generative model of an “experiment” in the external world environment, and the results of that “experiment” or the “surprise” entailed by predicted and actual outcomes of the “experiment.” Minimization of free energy implies that the implicit measure of “surprise” experienced serves to update the generative model in a Bayesian manner. This description closely accords with the mechanisms of generalized Darwinian process proposed both by Dawkins, in terms of replicators and vehicles, and Campbell, in terms of inferential systems. Bayesian inference is an algorithm for the accumulation of evidence-based knowledge. This algorithm is now seen to operate over a wide range of evolutionary processes, including natural selection, the evolution of mental models and cultural evolutionary processes, notably including science itself. The variational principle of free energy minimization may thus serve as a unifying mathematical framework for universal Darwinism, the study of evolutionary processes operating throughout nature.
Although Darwin must be counted amongst history's greatest scientific geniuses, he had very little talent for mathematics. His theory of natural selection was presented in remarkable detail, with many compelling examples but without a formal or mathematical framework (Darwin, 1872). Darwin did not think in mathematical terms; he found mathematics repugnant and it comprised only a small part of his Cambridge education (Darwin, 1958).
Generally, mathematics is an aid to scientific theories because a theory whose basics are described through mathematical relationships can be expanded into a larger network of predictive implications and the entirety of the expanded theory subjected to the test of evidence. As a bonus, any interpretation of the theory must also conform to this larger network of implications to ensure some consistency.
Natural selection describes the change in frequency or probability of biological traits over succeeding generations. One might suppose that a mathematical description—complete with an insightful interpretation—would be straightforward, but even today this remains elusive. The current impasse involves conceptual difficulties arising from one of mathematics' bitterest interpretational controversies.
That controversy is between the Bayesian and Frequentist interpretations of probability theory. Frequentists assume probability or frequency to be a natural propensity of nature. For instance, the fact that each face of a dice will land with 1/6 probability is understood by frequentists to be a physical property of the dice. On the other hand, Bayesians understand that humans assign probabilities to hypotheses on the basis of the knowledge they have (and the hypotheses they can entertain); thus the probability of each side of a dice is 1/6 because the observer has no knowledge that would favor one face over the other; the only way that no face is favored is for each hypothesis to be assigned the same probability. Furthermore, the value 1/6 is conditioned upon the assumption that there are only six possible outcomes. This means that probabilities are an attribute of a hypothesis or model space—not of the world that is modeled.
The Bayesian framework is arguably more comprehensive and has been developed into the mathematics of Bayesian inference, at the heart of which is Bayes' theorem, which describes how probabilistic models gain knowledge and learn from evidence. In my opinion, the major drawback of the Bayesian approach is an anthropomorphic reliance on human agency, the assumption that inference is an algorithm performed only by humans that possess (probabilistic) beliefs. Despite this interpretational dispute there has been some progress in uniting Bayesian and frequentist mathematics (Bayarri and Berger, 2004).
Despite the lack of mathematics in Darwin's initial formulation it was not long before researchers began developing a mathematical framework describing natural selection. It is an historical curiosity that most of these frameworks involved Bayesian mathematics, yet no interpretations were offered, proposing natural selection as a process of Bayesian inference.
The first step in developing this mathematics was taken during Darwin's lifetime by his cousin, Francis Galton. Galton developed numerous probabilistic techniques for describing the variance in natural traits—as well as for natural selection in general. His conception of natural selection was intriguingly Bayesian; although he may never have heard of Bayes' theorem. Evidence of his Bayesian bent is provided by a visual aid that he built for a lecture on heredity and natural selection given to the Royal Society (Galton, 1877).
He used this device (see Figure 1 below) to explain natural selection in probabilistic terms. It contains three compartments: a top compartment representing the frequency of traits in the parent population, a middle one representing the application of “relative fitness” to the child generation and a third representing the normalization of the resulting distribution in the child generation. Beads are loaded in the top compartment to represent the distribution in the parent generation and then are allowed to fall into the second compartment. The trick is in the second compartment, which contains a vertical division, in the shape of the relative fitness distribution. Some of the beads fall behind this division and are “wasted”; they do not survive and are removed from sight. The remaining beads represent the distribution of the “survivors” in the child generation.
Galton's device has recently been rediscovered and employed by Stephan Stigler and others in the statistics community as a visual aid, not for natural selection, but for Bayes' theorem. The top compartment represents the prior distribution, the middle one represents the application of the likelihood to the prior, and the third represents the normalization of the resulting distribution. The change between the initial distribution and the final one is the Bayesian update.
Fisher further developed the mathematics describing natural selection during the 1920s and 1930s. He applied statistical methods to the analysis of natural selection via Mendelian genetics and arrived at the fundamental theorem of natural selection which states (Fisher, 1930): The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.
Although Fisher was a fierce critic of the Bayesian interpretation (which he considered subjective) he pioneered—and made many advances with—the frequentist interpretation.
The next major development in the mathematics of natural selection came in 1970 with the publication of the Price equation, which built on the fundamental theorem of natural selection (Harper, 2010; Frank, 2012a). Although the Price equation fully describes evolutionary change, its meaning has only recently begun to be unraveled, notably by Steven A. Frank in a series of papers spanning the last couple of decades. Frank's insights into the meaning of the Price equation culminated in a 2012 paper (Frank, 2012b) which derives a description of natural selection using the mathematics of information theory.
In my opinion, this paper represents a significant advance in the understanding of evolutionary change as it shifts the interpretation from the objective statistical description of frequentist probability to an interpretation in terms of Bayesian inference. Unfortunately, Frank does not share my appreciation of his accomplishment. While he understands that his mathematics are very close to those of Bayesian inference he does not endorse a Bayesian interpretation but prefers an interpretation in terms of information theory.
However, the mathematics of information theory and Bayesian probability are joined at the hip, as their basic definitions are in terms of one another. Information theory begins with a definition of information in terms of probability: I(hi|m)=−log(P(hi|m))
Here, we may view hi as the ith hypothesis or event in a mutually exclusive and exhaustive family of n competing hypotheses comprising a model m. I(hi|m) is the information gained, under the model, on learning that hypothesis hi is true. P(hi|m) is the probability that had previously been assigned by the model that the hypothesis hi is true. Thus, information is “surprise”; the less likely a model initially considers a hypothesis that turns out to be the case, the more surprise it experiences, and thus the more information it receives.
Information theory, starting with the very definition of information, is aligned with the Bayesian interpretation of probability; information is “surprise” or the gap between an existing state of knowledge and a new state of knowledge gained through receiving new information or evidence.
The model itself, composed of the distribution of the p(hi), may also be said to have an expectation. The information which the model “expects” is the weighted average of the information expected by the n individual p(hi), which is called the model's entropy.
Entropy is the amount of information that separates a model's current state of knowledge from certainty.
Bayes' theorem follows directly from the axioms of probability theory and may be understood as the implication that new evidence or information holds for the model described by the distribution of the p(hi). This theorem states that on the reception of new information (I) by the model (m) the probability of each component hypothesis (hi) making up the model updates according to: P(hi|I, m)=P(hi| m)P(I|him)P(I|m)
Bayesian inference is commonly understood as any process which employs Bayes' theorem to accumulate evidence based knowledge (Wikipedia1): the quantity P(I | m) is called (Bayesian) model evidence and corresponds to the probability of observing some new information, under a particular model, averaged over all hypotheses. This is a crucial quantity in practice and can be used to adjudicate between good and bad models in statistical analysis. It is also the quantity approximated by (variational) free energy—as we will see below. Effectively, this equation provides the formal basis for Bayesian belief updating: in which prior beliefs about the hypotheses P(hi | m) are transformed into posterior beliefsP(hi | I, m), which are informed by new information. This updating rests upon the likelihood model; namely the likelihood of observing new information given the i-th hypothesis P(I | him). This formalism highlights the information theoretic nature of Bayesian updating—and the key role of a (likelihood) model in accumulating evidence.
We may conclude from this short overview of the relationship between information and Bayesian inference that information has little meaning outside a Bayesian context. Information depends upon a model that assigns probabilities to outcomes and which is updated on the reception of new information. In short, there is no information unless there is something that can be informed. This something is a model.
Thus, we see that, contrary to Frank's view, Bayesian inference and information theory have the same logical structure. However, it is instructive to follow Frank's development of the mathematics of evolutionary change in terms of information theory, while keeping in mind his denial of its relationship to Bayesian inference. Frank begins his unpacking of the Price equation by describing the “simple model” he will develop: A simple model starts with n different types of individuals. The frequency of each type is qi. Each type has wi offspring, where w expresses fitness. In the simplest case, each type is a clone producing wi copies of itself in each round of reproduction. The frequency of each type after selection is
Where w= ∑1nqiwi is the average fitness of the trait in the population. The summation is over all of the n different types indexed by the i subscripts.
Equation (1) is clearly an instance of Bayes' theorem, where the new evidence or information is given in terms of relative fitness and thus Frank's development of this simple model is in terms of Bayesian inference.
While Frank acknowledges an isomorphism between Bayes' theorem and his simple model, he does not find this useful and prefers to describe the relationship as an analogy. He makes the somewhat dismissive remark:
I am sure this Bayesian analogy has been noted many times. But it has never developed into a coherent framework that has contributed significantly to understanding selection.
On the contrary, I would suggest that Frank's paper itself develops a coherent framework for natural selection in terms of Bayesian inference. In particular, he highlights the formal relationships between the Price equation (or replicator equation) and Bayesian belief updating (e.g., Kalman Filtering). This is potentially interesting because many results in evolutionary theory can now be mapped to standard results in statistics, machine learning and control theory. Although we will not go into technical details, a nice example here is that Fisher's fundamental theorem corresponds to the increase in Kalman gain induced by random fluctuations (this variational principle is well-known in control theory and volatility theory in economics). Despite this, Frank dismisses Bayesian formulations because they do not appear to bring much to the table. This is understandable in the sense that the mathematics traditionally used to describe natural selection already has a Bayesian form and merely acknowledging this fact does not lead to a new formalism. However, this conclusion might change dramatically if biological evolution was itself a special case of a Universal Darwinism that was inherently Bayesian in its nature. In what follows, we pursue this line of argument by appealing to the variational principle of least free energy.
Baez and Pollard have recently demonstrated the similarities of a number of information-theoretic formulations, including the Bayesian replicator equation, evolutionary game theory, Markov processes and chemical reaction networks, that are applicable to biological systems as they approach equilibrium (Baez and Pollard, 2016). In general, any process of Bayesian inference may be cast in terms of (variational) free energy minimization (Roweis and Ghahramani, 1999; Friston, 2010) and—in this form—some important interpretative issues gain clarity. This approach has been used by Hinton, Friston, and others to describe the evolution of mental states as well as to describe pattern formation and general evolutionary processes. In its most general form, the free energy principle suggests that any weakly-mixing ergodic random dynamical system must be describable in terms of Bayesian inference. This means that the equivalence between classical formulations of evolution and Bayesian updating are both emergent properties of any random dynamical system that sustains measurable characteristics over time (i.e., is ergodic; Friston, 2013). This is quite important because it means that evolution is itself an emergent property of any such systems. Although conceptually intriguing, there may be other advantages to treating evolution in terms of minimizing variational free energy. In what follows, I will try to demonstrate this may be true.
In 1970 Ashby and Conant (Conant and Ashby, 1970) proved a theorem that any regulating mechanism for a complex system that is both successful and simple must be isomorphic with the system being regulated. In other words, it must contain a model of the system being regulated. As no model can be exactly isomorphic to its subject without being a clone and therefore exactly as complex as its subject, this theorem suggests a variational approach may be useful, one which optimizes the difference between the accuracy and the complexity of the model.