Complexity and skeptical hypotheses

Suppose a strong epistemic preference for simpler theories of the world. One might then think that a simulation hypothesis is automatically more complex than the best physical story about our world, because in addition to all the complexity of our simulated cosmos, it includes the complexity of whatever physical cosmos houses the hardware running the simulation.

But this need not be the case. The best physical story about our world makes our world include vast amounts of information that would not need to be included in the simulation. To simulate the history of the human race, we at most need information on the particles wihtin a sphere of radius about a hundred thousand light-years, so basically just the Milky Way Galaxy, a very small fraction of the particles in the world. And even that is a vast overstatement. One can surely have a low simulation resolution for a lot of stuff, simulating things only on a macroscopic level, and only including particle-level information when the simulated humans peer into scientific instruments. So the information content of the simulation software could be much, much lower than the information content of the physical world that our best theories say we live in.

But what about the simulation hardware itself? Wouldn’t that need to live in a complex physical universe? Maybe, but that universe need not be as complex as our physical theories claim ours to be. It could be a universe that has a level of physical complexity optimized for running the computing hardware. The granularity of that universe could be much coarser than ours. For instance, instead of that universe being made of tiny subatomic particles like ours, requiring many (but fewer and fewer with progress in miniaturization) particles per logic gate, we could suppose a universe optimized for computing whose fundamental building blocks are logic gates, memory cells, etc.

I am dubious, thus, whether we can rule out simulation hypotheses by an epistemic preference for simpler theories. The same goes for Berkeleian skeptical hypotheses on which there is no physical world, but we are disembodied minds being presented with qualia.

And of course the “local five minute hypothesis”, on which the universe is five minutes old and has a radius of five light-minutes, posits a world with intuitively much less complexity than the world of our best theories, a world with vastly fewer particles.

But if we cannot avoid skeptical hypotheses on grounds of complexity, how can we avoid them?

My current view is that we simply have to suppose that our priors are normatively constrained by human nature (which on my Aristotelian view is a form, a real entity), and human nature requires us to have non-skeptical priors. This is a very anthropocentric account.

Species relativity of priors

1. It would be irrational for us to assign a very high prior probability to the thesis that spiky teal fruit is a healthy food.

2. If a species evolved to naturally assign a very high prior probability to the thesis that spiky teal fruit is a healthy food, it would not be irrational for them to do this.

3. So, what prior probabilities are rational is species relative.

More on credences of randomly chosen propositions

For a number of years I’ve been interested in what one might call “the credence of a random proposition”. Today, I saw that once precisely formulated, this is pretty easy to work out in a special case, and it has some interesting consequences.

The basic idea is this: Fix a particular rational agent and a subject matter the agent thinks about, and then ask what can be said about the credence of a uniformly randomly chosen proposition on that subject matter. The mean value of the credence will be, of course, 1/2, since for every proposition p, its negation is just as likely to be chosen.

It has turned out that on the simplifying assumption that all the situations (or worlds) talked about have equal priors, the distribution of the posterior credence among the randomly chosen propositions is binomial, and hence approximately normal. This was very easy to show once I saw how to formulated the question. But it still wasn’t very intuitive to me as to why the distribution of the credences is approximately normal.

Now, however, I see it. Let μ be any probability measure on a finite set Ω—say, the posterior credence function on the set of all situations. Let p be a uniformly chosen random proposition, where one identifies propositions with subsets of Ω. We want to know the distribution of μ(p).

Let the distinct members (“situations”) of Ω be ω₁, ..., ω_n. A proposition q can be identified with a sequence q₁, ..., q_n of zeroes and/or ones, where q_i is 1 if and only if ω_i ∈ q (“q is true in situation ω_i”). If p is a uniformly chosen random proposition, then p₁, ..., p_n will be independent identically distributed random variables with P(p_i = 0)=P(p_i = 1)=1/2, and p will be the set of the ω_i for which p_i is 1.

Then we have this nice formula:

1. μ(p)=μ(ω₁)p₁ + ... + μ(ω_n)p_n.

This formula shows that μ(p) is the sum of independent random variables, with the ith variable taking on the possible values 0 and μ(ω_i) with equal probability.

The special case in my first post today was one where the priors for all the ω_i are equal, and hence the non-zero posteriors are all equal. Thus, as long as there are lots of non-zero posteriors—i.e., as long as there is a lot we don’t know—the posterior credence is by (1) a rescaling of a sum of lots of independent identically distributed Bernoulli random variables. That is, of course, a binomial distribution and approximately a normal distribution.

But what if we drop the assumption that all the situations have equal priors? Let’s suppose, for simplicity, that our empirical data precisely rules out situations ω_m + 1, ..., ω_n (otherwise, renumber the situations). Let ν be the prior probabilities on Ω. Then μ is directly proportional to ν on {ω₁, ..., ω_m} and is zero outside of it, and:

2. μ(p)=c(ν(ω₁)p₁ + ... + ν(ω_m)p_m)

where c = 1/(ν(ω₁)+.... + ν(ω_m)). Thus, μ(p) is the sum of m independent but perhaps no longer identically distributed random variables. Nonetheless, the mean of μ(p) will still be 1/2 as is easy to verify. Moreover, if the ν(ω_i) do not differ too radically among each other (say, are the same order of magnitude), and m is large, we will still be close to a normal distribution by the Berry-Esseen inequality and its refinements.

In other words, as long as our priors are not too far from uniform, and there is a lot we don’t know (i.e., m is large), the distribution of credences among randomly chosen propositions is approximately normal. And to get estimates on the distribution of credences, we can make use of the vast mathematical literature on sums of independent random variables. This literature is available even without the "approximate uniformity" condition on the priors (which I haven't bothered to formulate precisely).

Priors, justification and rationalism

The rationalism of Leibniz and Spinoza worked like this: We figure out fundamental necessary metaphysical principles, and these principles determine everything else of necessity (with some qualifications on the Leibniz side as to the type of necessity).

But another rationalism is possible: We figure out fundamental necessary metaphysical principles, and these principles determine the basic probabilistic structure of reality. In Bayesian terms, the fundamental metaphysical principles yield the prior probabilities. A version of this was Descartes’ project in the Meditations.

And there is reason to engage in this probabilistic rationalist project. We cannot get out of the need to have something like prior probabilities. Moreover, priors need epistemic justification. Consider an empirical claim p that we assign a high enough credence for belief to, say 0.99, on the basis of total evidence e. Thus, P(p|e)=0.99. It follows by the axioms of probability that P(p ∨ ¬e)≥0.99. Hence we have a high enough prior credence for belief in p ∨ ¬e. Surely assigning a credence of 0.99 to something requires epistemic justification. Moreover, surely (though people who don’t like closure arguments may not like it) if we have posterior justification when we believe p, we have posterior justification when we believe the obviously entailed claim p ∨ ¬e. But this justification did not come from e. For P(p ∨ ¬e|e)=P(p|e)=0.99 and we have seen that P(p ∨ ¬e)≥0.99, so e is not evidence for p ∨ ¬e (in face, typically e will be evidence against this disjunction). Since e is our total evidence, the justification had to be there in the first place.

Thus we need epistemic justification for our priors. The priors encode genuine information about our world, information that we are justified in possessing. Where do we justifiedly get this information from? We don’t get it through logic, pace logical probability accounts. Metaphysics is one potential answer to this question and exploring this answer gives us good reason to engage in the probabilistic rationalist project. Another option is that the priors are a kind of innate knowledge built into our nature—my Aristotelian Bayesianism is a version of this.