Does Bayesian Epistemology Suffer Foundational Problems?


I recently had a discussion about whether Bayesian epistemology suffers from the problem of induction, and I think some interesting things came from it. If these words make you uncomfortable, think of epistemology as the study of how we form beliefs and gain knowledge. Bayesian epistemology means we model it probabilistically using Bayesian methods. This old post of mine talks a bit about it but is long and unnecessary to read to get the gist of this post.

I always think of the problem of induction in terms of the classic swan analogy. Someone wants to claim that all swans are white. They go out and see swan after swan after swan, each confirming the claim. Is there any point at which the person can legitimately say they know that all swans are white?

Classically, the answer is no. The problem of induction is crippling to classic epistemologies, because we can never be justified in believing any universal claim (at least using empirical methods). One of the great things about probabilistic epistemologies (not just Bayesian) is that it circumvents this problem.

Classical epistemologies require you to have 100% certainty to attain knowledge. Since you can’t ever be sure you’ve encountered every instance of a universal, you can’t be certain there is no instance that violates the universal. Hence the problem of induction is an actual problem. But note it is only a problem if your definition of knowledge requires you to have absolute certainty of truth.

Probabilistic epistemologies lower the threshold. They merely ask that you have 95% (or 98%, etc) confidence (or that your claim sits in some credible region, etc) for the justification. By definition, knowledge is always tentative and subject to change in these theories of knowledge.

This is one of the main reasons to use a probabilistic epistemology. It is the whole point. They were invented to solve this problem, so I definitely do not believe that Bayesian epistemology suffers from the problem of induction.

But it turns out I had misunderstood. The point the other person tried to make was much more subtle. It had to do with the other half of the problem of induction (which I always forget about, because I usually consider it an axiom when doing epistemology).

This other problem is referred to as the principle of the uniformity of nature. One must presuppose that the laws of nature are consistent across time and space. Recall that a Bayesian has prior beliefs and then upon encountering new data they update their beliefs factoring in both the prior and new data.

This criticism has to do with the application of Bayes’ theorem period. In order to consider the prior to be relevant to factor in at all, you must believe it is … well, relevant! You’ve implicitly assumed at that step the uniformity of nature. If you don’t believe nature is consistent across time, then you should not factor prior beliefs into the formation of knowledge.

Now a Bayesian will often try to use Bayesian methods to justify the uniformity of nature. We start with a uniform prior so that we haven’t assumed anything about the past or its relevance to the future. Then we merely note that billions of people across thousands of years have only ever observed a uniformity of nature, and hence it is credible to believe the axiom is true.

Even though my gut buys that argument, it is a bit intellectually dishonest. You can never, ever justify an axiom by using a method that relies on that axiom. That is the quintessential begging the question fallacy.

I think the uniformity of nature issue can be dismissed on different grounds. If you want to dismiss an epistemology based on the uniformity of nature issue, then you have to be willing to dismiss every epistemology that allows you to come to knowledge.

It doesn’t matter what the method is. If you somehow come to knowledge, then one second later all of nature could have changed and hence you no longer have that knowledge. Knowledge is impossible if you want to use that criticism. All this leave you with is radical skepticism, which of course leads to self-contradiction (if you know you can’t know anything, then you know something –><– ).

This is why I think of the uniformity of nature as a necessary axiom for epistemology. Without some form of it, epistemology is impossible. So at least in terms of the problem of induction, I do not see foundational problems for Bayesian epistemology.

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