Not strictly relevant. If your hypothesis is extreme, a single observation can make its posterior probability small. For instance, suppose that Alice gives you a coin and says "it lands on heads 99% of the time" (hypothesis A) but Bob calls BS and says "it looks like a regular quarter, so it should land on heads 50% of the time" (hypothesis B). You assign prior probabilities of 50% to A,B because on one hand you can't imagine how anyone could engineer a coin to land on heads 99% of the time, but on the other hand Alice is a great engineer, so you don't know what to believe. If you flip the coin a single time and get tails, your experiment strongly supports hypothesis B over A. OTOH, if it landed on heads, A would be weakly supported over B. We started with an even prior, so the posterior probability of A,B are proportional to their likelihoods: P(A|t)≈2% and P(B|t)≈98%, while P(A|h)≈66% and P(B|h)≈33%.

Let's say that a typical planet remains habitable for 8b years and it takes 4b years from the emergence of life to the emergence of intelligent life, so according to the anthropic principle life had a 4b year window in which to emerge. Under the model where P(emergence)/time is tiny, the probability of life emerging in the first billion years is ~.25. Under the model where P(emergence)/time≈1, the probability of life emerging in the first billion years is 1. With even priors, the posterior of P(emergence)/time≈1 is 80% and the posterior of P(emergence)/time≈0 is 20%.

It's not scientific proof of anything, but it's enough to make me consider alternative explanations for why life is rare.