Universal Darwinism As a Process of Bayesian Inference
This is exactly a form in which the free energy minimization principle may be cast (Moran et al., 2014): F(s,u)=DKL[q(ψ|μ)||p(ψ|m)]−Eq[logp(s|ψ,m)]
Where ψ are hidden states of the world or environment, s are their sensory consequences or samples (that can depend upon action), μ are internal states and m is the generative model. The distribution q is the current predictions of the states of the environment, the distribution p is the true states of the environment and the KL divergence is a measure of the distance between them. Crucially, free energy can also be expressed in terms of the surprise of sampled consequences: F(s,u)= DKL[q(ψ|μ)||p(ψ|s,m)]−logp(s|m)
This formulation of evolutionary change may appear quite different from that of Bayesian inference as it has a focus on model quality rather than fitness. However, a sustained decrease in free energy (or increase in log model evidence) is equivalent to a decrease in model entropy and therefore contravenes the spirit, if not the letter, of the second law. The letter of that law allows a decrease in entropy for dynamic systems only if an environmental swap is conducted where low entropy inputs are exchanged for high entropy outputs. In short, the second law forbids the existence or survival of low entropy dynamic systems lacking such an ability—an ability that mandates a model of the environment and Bayesian inference under that model. This provides a focus for the model's knowledge accumulation; it must entail knowledge of its environment as well as a strategy to perform the required entropy swaps within that environment. Thus, the drive to fitness, which is explicit in the Bayesian formulation, is also implicit in the free energy formulation.
As descriptions of evolutionary processes in terms of free energy minimization have great general applicability it may be useful to consider some specific examples. In biological evolution we can associate the model (m) with a genotype. This means the genotype corresponds to the sufficient statistics of the prior beliefs a phenotype is equipped with on entering the world. Keeping in mind that organisms may sense their environments through both chemical and neural means, we may associate sensory exchanges with the environment (s) with adaptive states. Finally, the sufficient statistics of the posterior (mμ) can be associated with a phenotype. In other words, the phenotype embodies probabilistic beliefs about states of its external milieu. This formulation tells us several fundamental things:
everything that can change will change to minimize free energy. Here, the only things that can change are the sufficient statistics; namely, the genotype and phenotype. This means there are two optimizations in play: adaptive changes in the phenotype over somatic time (i.e., changes in mμ) and adaptive changes in the genotype over evolutionary time (i.e., changes in m).
somatic changes will be subject to two forces: first, a maximization of accuracy that simply maximizes the probability of occupying adaptive states, and second, a minimization of complexity. This minimization corresponds to reducing the divergence between the beliefs about, or model of (hidden) environmental states (ψ) implicit in the phenotype and the prior beliefs implicit in the genotype. In other words, a good genotype will enable the minimization of free energy by equipping the phenotype with prior beliefs that are sufficient to maintain accuracy or a higher probability of adaptive states. Thus, the phenotype may be thought of as a type of experiment, which gathers evidence to test prior beliefs; i.e., gathers evidence for its own existence.
changes in the genotype correspond to Bayesian model selection (c.f., natural selection). This simply means selecting models or genotypes that have a low free energy or high Bayesian model evidence. Because the Bayesian model evidence is the probability of an adaptive state given a model or genotype (p(s|m)), natural selection's negative variational free energy becomes (free) fitness. At this level of free energy minimization, evolution is in the game of orchestrating multiple (phenotypic) experiments to optimize models of the (local) environment.
Another specific example of the general ability of the free energy minimization principle to describe evolutionary change is in neuroscience where it is fairly easy to demonstrate the centrality of this principle in explaining evolutionary, developmental and perceptual processes in a wide range of mental functions (Friston, 2010). The brain produces mental models which combine sensory information concerning the state of the environment, with possible actions with which the organism may intervene. The initiation of an action is a kind of experiment in the outside world testing the current beliefs about its hidden states. The overall drive of the free energy principle is to reduce the model complexity, while maximizing its accuracy in achieving the predicted outcome. Crucially, the ensuing self-organization can be seen at multiple levels of organization; from dendritic processes that form part of the single neuron—to entire brains. The principles are exactly the same, the only thing that changes is the way that the model is encoded (e.g., with intracellular concentrations of various substrates—or neuronal activity and connectivity in distributed brain systems). This sort of formulation has also been applied to self-organization and pattern formation when multiple systems jointly minimize their free energy (for example, in multi-agent games and morphogenesis at the cellular level).
Clearly, the application of variational (Bayesian) principles to ecological and cellular systems means we have to abandon the notion that only humans can make inferences. We will take up this theme below and see how freeing oneself from the tyranny of anthropomorphism leads us back to a universal Darwinism.
The free energy minimization principle may also be applied to processes of cultural evolution. A compelling example here is the evolution of scientific understanding itself. Science develops hypotheses or theoretical models of natural phenomena. These models are used to design experiments in the real world and the results of the experiment are used to update the probability of each hypothesis composing the model according to Bayes' theorem. In the process free energy is minimized through a balance which reduces the model's complexity (Occam's razor) while increasing the model's predictive accuracy and explanatory scope.
The evolutionary interaction between models and the systems they model, as described by the free energy minimization principle, may be applicable to additional natural phenomena beyond the examples above. Several attempts have been made to describe universal Darwinism in such terms. We have previously noted the wide range of scientific subject matter that has been identified within the literature as Darwinian processes—and have offered an interpretation in terms of inferential systems (Campbell, 2014); an interpretation closely related to that of the free energy minimization principle. Richard Dawkins offered a description of biological evolution in terms of replicators and vehicles (Dawkins, 1982), a description which Blackmore and Dennett have generalized to interpret universal Darwinism (Dennett, 1996; Blackmore, 1999). That description may also be understood as an interplay between internal models (replicators) and the experience of the “experiments” (vehicles) which they model in the external world.
The Price equation describing evolutionary change may be cast in a form which distinguishes between change due to selection and transmission. Changes due to selection tend to decrease model variation whereas changes due to transmission or copying of the model serve to increase variation. The transmission changes of biological models are often in the form of genetic mutations (Frank, 2011). From the perspective of universal Darwinism, we might expect a mechanism capable of increasing model variation within non-biological evolutionary processes that is analogous to biological mutation. As an example we might consider the process of evolutionary change in scientific models during transmission. These may appear less clear; there is less consensus on how new and sometimes improved scientific models are generated. It may seem this process has little in common with the somewhat random and undirected process of biological mutation.
The mental process by which researchers arrives at innovative models is largely hidden and might be considered closer to an art form than algorithmic but the development of inferential/Darwinian evolutionary computational processes have demonstrated a strong ability to discover innovative models in agreement with the evidence(Holland, 1975; Ibáñez et al., 2015). In some instances, these evolutionary approaches have inferred successful models for systems which have long eluded researchers (Lobo and Levin, 2015).
The reluctance of many researchers to endorse a Bayesian interpretation of evolutionary change may be somewhat puzzling. One reason for this is a peculiarity, and I would suggest a flaw, in the usual Bayesian interpretation of inference that renders it unfit as a description of generalized evolutionary change. The consensus Bayesian position is that probability theory only describes inferences made by humans. As Jaynes put it (Jaynes, 1988):
it is…the job of probability theory to describe human inferences at the level of epistemology.
Epistemology is the branch of philosophy that studies the nature and scope of knowledge. Since Plato the accepted definition of knowledge within epistemology has been “justified true beliefs” held by humans. In the Bayesian interpretation “justified” means justified by the evidence. “True belief” is the degree of belief in a given hypothesis which is justified by the evidence; it is the probability that the hypothesis is true within the terms of the model. Thus, knowledge is the probability, based on the evidence, that a given belief or model is true. I have proposed a technical definition of knowledge as 2−S where S is the entropy of the model (Campbell, 2014).
A perhaps interesting interpretation of this definition is that knowledge occurs within the confines of entropy or ignorance. For example, in a model composed of a family of 64 competing hypotheses, where no evidence is available to decide amongst them, we would assign a probability of 1/64 to each hypothesis. The model has an entropy of six bits and has knowledge of 2−6 = 1/64. Let's say some evidence becomes available and the model's entropy or ignorance is reduced to three bits. Then the knowledge of the updated model is 1/8, equivalent to the entropy of a model composed of only eight competing hypotheses that is maximally ignorant, which has no available evidence. The effect which evidence has on the model is to increase its knowledge by reducing the scope of its ignorance.
It is unfortunate that both Bayesian and Frequentist interpretations deny the existence of knowledge outside of the human realm because it forbids the application of Bayesian inference to phenomena other than models conceived by humans, it denies that knowledge may be accumulated in natural processes unconnected to human agency and it acts as a barrier in realizing our close relationship to the rest of nature. Thus, even though natural selection is clearly described in terms of the mathematics of Bayesian inference, neither Bayesians such as Jaynes nor frequentists such as Frank can acknowledge this fact due to another hard fact: natural selection is not dependent upon human agency. In both their views this may rule out a Bayesian interpretation.
I believe that the correct way out of this conundrum is to simply acknowledge that in many cases inference is performed by non-human agents as in the case of natural selection and that inference is an algorithm which we share with much of nature. The genome may for instance be understood as an example of a non-human conceived model involving families of competing hypotheses in the form of competing alleles within the population. Such models are capable of accumulating evidence-based knowledge in a Bayesian manner. The evidence involved is simply the proportion of traits in ancestral generations which make it into succeeding generations. In other words, we just need to broaden Jaynes' definition of probability to include non-human agency in order to view natural selection in terms of Bayesian inference.
In this view the accumulation of knowledge is a preoccupation we share with the rest of nature. It allows us to view nature as possessing some characteristics, such as surprise and expectations, previously thought by many as unique to humans or at least to animals. For instance, all organisms “expect” to find themselves in the type of environment for which they have been adapted and are “surprised” if they don't.
Bayesian probability, epistemology and science in general tend to draw a false distinction between the human and non-human realms of nature. In this view the human realm is replete with knowledge and thus, infused with meaning, purpose and goals, and Bayesian inference may be used to describe its knowledge-accumulating attributes. On the other hand, the non-human realm is viewed as devoid of these attributes and thus Bayesian inference is considered inapplicable.
However, if we recognize expanded instances, such as natural selection, in which nature accumulates knowledge then we may also recognize that Bayesian inference, as well as equivalent mathematical forms, provides a suitable mathematical description in both realms. Evolutionary processes, as described by the mathematics of Bayesian inference, are those which accumulate knowledge for a specific purpose, knowledge required for increased fitness, for increased chances of continued existence. Thus, the mathematics implies purpose, meaning and goals, and provides legitimacy for Daniel Dennett's interpretation of natural selection in those terms (Dennett, 1996). If we allow an expanded scope for Bayesian inference, we may view Dennett's poetic interpretation of Darwinian processes as having support from its most powerful mathematical formulations.
An important aspect of these mathematics is that they apply not only to natural selection but also to any generalized evolutionary processes where inherited traits change in frequencies between generations. As noted in a cosmological context by Gardner and Conlon (2013): Specifically, Price's equation of evolutionary genetics has generalized the concept of selection acting upon any substrate and, in principle, can be used to formalize the selection of universes as readily as the selection of biological organisms.
At the core of Bayesian inference, underlying both the Price equation and the principle of free energy minimization we find an extremely simple mathematical expression: Bayes' theorem: qi,=qi wiw
Simply put this equality says that the probabilities assigned to the hypotheses of a probabilistic model are updated by new data or experience according to a ratio, that of the probability of having the experience given that the specific hypothesis is correct to the average probability assigned by the model to having that experience. Those hypotheses supported by the data, those that assign greater than average probability to having the actual experience, will be updated to greater values and those hypotheses not supported by the data will be updated to lesser values. This simple equation describes the accumulation of evidence-based knowledge concerning fitness.
When Bayes' theorem is used to describe an evolutionary process the ratio involved is one of relative fitness, the ratio of the fitness of a specific form of a trait to the average fitness of all forms of that trait. It is thus extremely general in describing any entity able to increase its chances of survival or to increase its adaptiveness. When cast in terms of the principle of free energy minimization some further implications of this simple equation are revealed (see above).
In a biological evolutionary context, the Price equation is traditionally understood as the mathematics of evolutionary change. However, the Price equation may be derived from a form of Bayes' theorem (Gardner, 2008; Shalizi, 2009; Frank, 2012b) which means it describes a process of Bayesian inference, a very general form of Bayesian inference which according to Gardner (Gardner, 2008) applies to any group of entities that undergo transformations in terms of a change in probabilities between generations or iterations. Even with this great generality it provides a useful model as it partitions evolutionary change in terms of selection and transmission (Frank, 2012a).
There are numerous examples of these equivalent mathematical forms used in the literature to describe evolutionary change across a wide scope of scientific subject matter, specifically evolutionary change in biology (Gardner, 2008; Frank, 2012b), neuroscience (Friston, 2010; Fernando et al., 2012) and culture (Hull, 1988; Jaynes, 2003; Mesoudi et al., 2006; Gintis, 2007).
It is interesting to speculate on the similarity of these mathematical forms to those which may be used to describe quantum physics. Quantum physics is also based upon probabilistic models which are updated by information received through interactions with other entities in the world. Wojciech Zurek, the founder of the theory of quantum Darwinism (Zurek, 2009), notes that the update of quantum states may be understood in terms of ratios acting to update probabilistic models (Zurek, 2005).
Using this connection, we then infer probabilities of possible outcomes of measurements on S from the analogue of the Laplacian ‘ratio of favorable events to the total number of equiprobable events’, which we shall see in Section V is a good definition of quantum probabilities for events associated with effectively classical records kept in pointer states.