## Bayes’ Theorem and Falsifiability

03 Apr

The one lesson we should get from Bayes’ theorem is that it is the language of science. If you are thinking like a scientist, you are thinking like a Bayesian. That’s why I was pretty confident that I would end up with the numbers that I did even assuming a high initial probability of the Christian god’s existence for the last three posts on Bayes’ theorem and the existence of the Christian god.

If you notice throughout the last three posts, the conditional probabilities for the Christian god are almost always lower than the alternative. Why is that? Because the Christian god — an all-powerful god — is unfalsifiable. Sure, falsifiability is a good philosophical justification for doing science. But now we know why falsifiability is so important from a purely epistemic (i.e. probabilistic) point of view.

Generally, any hypothesis that is unfalsifiable will always have a lower conditional probability than a hypothesis that is falsifiable.

Falsifiable hypotheses tend to lean towards going all-in probability wise, and will tend to cluster its probability capital in a more bell-curve like way in a class of evidence. Unfalsifiable hypotheses spread their probability capital more evenly across all possible evidence of the same class.

Think of it this way. Say there are 10 possible doors to bet on for winning a prize, and you only have 100 dollars to bet. Someone who is trying not to be “proven wrong” in any sense will spread their money evenly across all 10 doors. Someone who wants a big payout will place the majority of their 100 dollars on very few doors. The unfalsifiable hypothesis is spreading its 100 dollars evenly across all of the doors, while the falsifiable hypothesis goes all in on one door or clusters around very few doors.

The total equation for determining conditional probabilities is P(E | H) + P(~E | H) = 1.00. An unfalsifiable hypothesis is attempting to equally explain everything. And as that equation shows, something that attempts to explain everything equally (i.e. E and everything that entails ~E), explains nothing. The more possible evidence (of the same class) that the unfalsifiable hypothesis attempts to explain equally, the closer the individual conditional probabilities move towards zero.

So, if someone says “You can’t prove that god doesn’t exist!” you know why they’ve already lost the debate: Just by the nature of being unfalsifiable, probability will favor the alternative, falsifiable, hypothesis more.

Besides the Christian god, what are some other unfalsifiable hypotheses? How about Solipsism, or philosophical zombies (P-Zombies). What observations could we see that would falsify solipsism or p-zombies? None. This means that every single possible observation is equal “confirmation” of p-zombies, even mutually exclusive observations. Which means that P(Evidence1 | P-Zombies) + P(Evidence2 | P-Zombies) + P(Evidence3 | P-Zombies) + P(Evidence4 | P-Zombies) + P(Evidencen | P-Zombies) = 1.00. All of the mutually exclusive P(Evidencen | P-Zombies) will be equal to each other; there’s no observation that is less likely than any other observation given p-zombies.

On the other hand, a falsifiable hypothesis would be something like “all swans are white” (from the wiki article). In Bayes’ theorem, the conditional probability is saying that P(E | H) = 1.00, or P(White Swan | All Swans Are White) = 1.00. This, in turn, means that P(Non White Swan | All Swans Are White) = 0 since P(E | H) + P(~E | H) = 1.00. Upon the event of seeing a non-white swan, this drops the prior probability of “all swans are white” to zero, yet seeing another white swan won’t change the prior probability much (depending on the conditionals of the alternative hypothesis).

In short, don’t posit an unfalsifiable hypothesis. If you do:

Posted by on April 3, 2012 in Bayes

### 2 responses to “Bayes’ Theorem and Falsifiability”

1. April 5, 2012 at 12:09 am

I really wish you would post some of this stuff in the LW discussion section to see what kind of response it gets. I'm still too much of a n00b at Bayescraft to analyze this, so I'm interested in what they have to say about it.

2. April 9, 2012 at 10:57 pm

I wish I had the time to follow a long thread on LW. I can only read bits and pieces at a time, and I lost the other discussion I posted.

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