*Bootstrapping Objective Probabilities for Evolutionary Biology*

### Marshall Abrams (University of Alabama at Birmingham)

While there are difficult philosophical problems concerning quantum mechanics that are relevant to our understanding of the nature of quantum mechanical probabilities, it’s clear that (a) the probabilities seem to be fundamental to quantum mechanics, and (b) the numerical values of probabilities are given by the mathematical theory of quantum mechanics (perhaps via some extension such as GRW). There are also philosophical problems concerning the nature of probabilities that arise in statistical mechanics, but the mathematical theories of statistical mechanics that are relevant to these problems usually constrain the numerical values of these probabilities fairly narrowly.

The situation is different in evolutionary biology, where widespread uses of models and frequentist statistical inference in empirical research seem to depend on the fact that evolving populations realize (something like) objective probabilities. (These might be objective imprecise probabilities.) But we have had little guidance about how to think about the fundamental nature of these “evolutionary” probabilities.

- Some authors have suggested that evolutionary probabilities might derive from quantum mechanical probabilities, but it’s completely unclear how quantum mechanical probabilities would connect to the probabilities assumed in empirical research.
- Some authors have suggested that evolutionary probabilities might be analogous to probabilities in statistical mechanics, but this faces similar problems.
- Strevens has suggested that evolutionary probabilities arise as microconstant probabilities, where individual organisms play a role analogous to wheels of fortune. In a forthcoming book, I suggest something similar under the name of “measure map complex causal structure probability” (MM-CCS), but treat an entire population in its environment as analogous to a wheel of fortune. However, even if we put aside well-known problems about measures of initial conditions that challenge views like Strevens’ and mine, it’s unclear why it’s reasonable to think that the causal structure of an organism in an environment (Strevens) or a population in an environment (me) should give probabilities the numerical values that seem to be required by empirical research in evolutionary biology.

There is a common problem with all of these proposals which is due to the fact that there is an enormous amount of heterogeneity among the interacting components in an evolving population in an environment, with many interactions between different levels. In my book, I call this property of evolutionary processes “lumpy complexity”. This kind of complex causal structure of an evolving populations makes them extremely difficult to understand well, and difficult to model except very approximately. This makes it difficult to spell out details of the above proposals in a way that allows us to understand what gives rise to the numerical values of probabilities in evolutionary biology.

I argue that at least in many cases, there is a different sort of explanation of the probabilistic character of biological populations in environments. This strategy can explain some important numerical values of relevant probabilities, and sidesteps the sorts of problems highlighted above. It is a strategy that is not available for explaining objective probability in physical sciences. I outline the strategy below.

Empirical researchers have argued that many animals engage in a kind of random walk known as a Lévy walk or a Lévy flight: at each time step, the direction of motion is randomly chosen, and the length *d* of travel in that direction has probability proportional to *d*^{μ}, where μ is near a particular value (usually 2).

Modeling arguments and a small amount of empirical research have been used to argue for the “Lévy Flight Foraging hypothesis” (LFF), which says that when food (or other resources) are sparsely distributed, it’s adaptive for organisms to search randomly for the food by following a Lévy walk with parameter μ near 2. That is, the LFF is the claim that Lévy flight foraging, or more precisely foraging using Lévy walks, is the result of natural selection on internal mechanisms because that pattern of foraging is adaptive.

My argument can be outlined as follows.

- When members of species S1 engage in Lévy walk foraging, this has a probabilistic effect on members of a species S2 (e.g. prey of S1) that applies an equal probability of (e.g.) survival to members of S2. That is, random foraging by members of S1 is a source of probabilities of survival and reproduction for members of S2. For example, in the case of animals of species S1 searching for plants of species S2, members of S2 that are equally perceptible by members of S1 will have equal probability of being eaten by members of S1.
- Where the LFF is correct, the random foraging behavior by members of S1 is the result of natural selection. This claim is independent of details about the internal mechanisms that give rise to Lévy walk foraging, or details about the fundamental character of the probabilities determined by the mechanism. For example, if an animal’s Lévy walk foraging is guided by neural circuits, the probabilistic character of the animal’s foraging choices could make use of quantum mechanical or statistical mechanical effects within individual neurons, or could depend on neural circuits that function in ways that are broadly analogous to deterministic pseudorandom number generators. While it’s worthwhile to investigate further questions about the nature of the mechanisms that generate Lévy walk forging behavior, natural selection explains why such mechanisms are present, and will often give rise to them—one way or another. So in a sense it is unimportant how these mechanisms work; they will exist, and will explain Lévy walk foraging regardless.
- These facts can explain, at least in part, the (approximately) equal probabilities of survival that are assumed as a null hypothesis by many empirical studies in evolutionary biology. They do so despite the lumpy complexity of evolving populations, and despite the fact that we might not know the fundamental character (e.g. quantum mechanical or pseudorandom) of the relevant mechanisms that have been selected for.
- Although I argue research in evolutionary biology depends on the existence of (something like) objective probability, the bare fact of natural selection does not depend on probability. This means that natural selection itself can give rise to at least some of the objective probabilities that evolutionary biology uses to make inferences about natural selection.

Where such an explanation of evolutionary probabilities is applicable (and I argue that it is more widely applicable than one might think), we have an explanation of objective probability (or objective imprecise probability, or behavior that seems to involve probability) in evolving populations. This explanation avoids the problems of proposals that evolutionary probabilities should be understood as directly resulting from quantum mechanical, statistical mechanical, or microconstant/MM-CCS probabilities.

There are some reasons to think that human behavior is sometimes randomized by internal mechanisms that result either from natural selection or learning. If this is correct, people who interact with those whose behavior is randomized by such a mechanism would in turn experience randomized consequences of that behavior. Thus the explanatory strategy that I describe for evolutionary biology may also be relevant in social sciences.