# How to properly put lower/upper limits on a Bayesian posterior

I have a situation where I compute a posterior probability using the vanilla Bayes' Theorem:

$$P(A|B) \propto P(B|A)P(A)$$

As a concrete example, $$P(B|A)$$ is a Normal distribution establishing the likelihood that a measured value plus random noise is consistent with a known expectation value (and variance), and $$P(A)$$ is a uniform prior that the measured value is a false positive. In this situation, large outliers will have a posterior probability of zero. Suppose I'd like to say "I can't really be so sure that the outlier is due to noise that I understand (the Normal distribution), so I want to hedge my bets and always leave the posterior with a tiny, minimum probability.

Naively, I can just clip/clamp the posterior to lie between $$[\epsilon,1-\epsilon]$$, but this won't pass the sniff test of people (like you) who clearly know much more than I do about probability, and will tell me "this is prior information so include it as a prior!" I just don't know enough about this field to see how to do it on the right-hand side of the equation. I'd love some guidance.

• You expression is wrong and should be $P(A\mid B) \propto P(B\mid A)P(A)$ or $P(B\mid A) \propto P(A\mid B)P(B)$ May 18, 2023 at 22:49
• So long as the likelihood of any actual observation is always positive, the posterior distribution will have the same support as the prior (though usually with an updated shape on that support). May 18, 2023 at 22:55
• If you don't want to that the outlier is due to noise from a normal distribution, you likely should not assume that the data is some measured value with normal noise. I don't really understand what you are trying to say here. May 19, 2023 at 19:00

The uniform distribution is often applied to situations where $$w \in [l,u]$$ represents a segment on the real line, with no preferred value for $$w$$ within this segment. However, on the one hand, this assumes that we are ignorant of any value of $$w$$, while, on the other hand, we seem to know precisely its boundary values. This is a contradiction in our state of knowledge. Therefore, one should be careful while using it.