# Increase in conditional probability for contradictory hypotheses in bayesian confirmation theory?

Although this question has a philosophical slant and my motivations for asking it are philosophical, I'm going to justify asking this in the mathematics stack exchange in two ways:

1) I've asked around the philosophy site and not really had much in the way of an answer. As in, no one answered the question. I think is is probably because:

2) This question can be better answered by people with better knowledge about probability theory and how Bayesian confirmation theory works mathematically.

Also a disclaimer: I'm a mathematics student but I'm fairly lay when it comes to more advance probability theory (I'm an A-level mathematics student from the UK, if that helps you formulate an answer).

As far as I can tell, Bayes' theorem allows you to calculate conditional probability of a hypothesis given a piece of evidence if you know a little bit about what that hypothesis predicts for that kind of evidence and your prior probabilities.

E.g. The classic balls out of the urn example where there are red balls and blue balls placed in an urn (maybe 100, maybe 1000, it doesn't really matter) and the red balls have some distribution between 0 and 1 and you have to hypothesise about what this distribution is. Say, for example, you have taken out 10 red balls at this point and 0 blue balls. We have two hypotheses:

1: "All the balls in the urn are red". 2: "All the balls found in the urn until the 20th ball will be red, then all of them are blue".

Both of these hypotheses seem to predict the evidence (that all balls so far observed have been red) equally. Does our confidence in both of them really increase equally? Or is there some nuance that I'm not getting here?

I'm asking this because it seems to have some serious implications for confirmation in science. E.g. we have observed many instances of the universe obeying some specific law of physics. Then we have two hypotheses:

1: "The universe will always obey this law" and 2: "The universe will obey this law until a future time t where it will then obey something else".

It seems intuitively that our confidence the 2nd hypothesis in both case shouldn't increase like the 1st does. But does it? Am I not understand something? Thanks for any answers in advance.

P.S. Of course someone might be tempted to simply lower prior probabilities for the 2nd type of hypothesis but is there really any mathematical way to justify being more confident in 1 than 2 that doesn't involve arbitrary alteration of priors?

• "Of course someone might be tempted to simply lower prior probabilities for the 2nd type of hypothesis but is there really any mathematical way to justify being more confident in 1 than 2 that doesn't involve arbitrary alteration of priors?" I would go with this. After all, if there are infinitely many hypotheses and their probability sums to $1$, then their probability has to diminish as they become more complex. – Oscar Cunningham May 20 '18 at 21:38
• Thanks for a suggestion. Although I kind of agree with an intuition about more complex hypotheses deserving lower prior probabilities, why would that kind of hypothesis deserve a lower one? Usually if I believe a hypothesis deserves some lower prior, it's because extra assumptions are made or auxiliary hypotheses are added in order to have it entail the evidence. How can I say that the second kind of hypothesis is more complex in this way? – Joe Lee-Doktor May 21 '18 at 6:17
• That's how I interpret the principle of parsimony generally, by the way. It seems like a mathematically 'proper' definition of 'simple' since an assumption is by definition something that might not be true and as assumptions stack upon one another, basic probability theory says that the complete truth of a hypothesis that includes all of those assumptions is lower than one that includes fewer if we believe the truth of those assumptions are independent of one another. – Joe Lee-Doktor May 21 '18 at 6:19