# What is the posterior distribution of a Bernoulli prior that gets updated with a continuous uniform signal?

I'm trying to figure out what the distribution of the posterior is after I update a Bernoulli prior with a continuous uniform signal, say:

P(D=G|u)=x where D{G,I} and u is uniformly distributed on some support

Thank you!

• Hi, is there any chance that you meant a uniform prior with Bernoulli data points? I struggle to see how one creates uniform data points with parameters that are Bernoulli distributed... – B.Liu Apr 12 at 16:10
• No, it's a theoretical signalling game, where the player has a Bernoulli prior such that P(D=G)=p and P(D=I)=1-p where D only takes values {G,I}. Then the player receives a signal u which is continuous and uniformly distributed with support [a,b] and updates their beliefs, and I need to figure out how that posterior is distributed.. hope this clarifies it – lost.econ.child Apr 12 at 16:25
• If Bernoulli distribution is a prior, then it's a parameter of the uniform distribution--how is that possible? Could you define what you mean by prior? Give us more details. – Tim Apr 12 at 16:41
• It looks to me like $P(D=G|u)$ is independent of $u$ if it just equals some $x$. In any case, you need to observe more than $u$, you need the triplet $(D,G,u)$, in order to update the probability that $D=G$. – jbowman Apr 12 at 17:04
• I equated it to x solely for notational ease, it's ultimately the distribution of that x that I'm trying to figure out. By prior I mean the initial beliefs of the player before the signal; for example, if the defendant's type is guilty or innocent, without observing anything at all, the probability that P(D=G)=1/2=P(D=I) since either state of the world is equally likely. However, after observing a signal u that's indicative of the state of the world but not fully revealing, we update the belief using Bayes Rule. What i'm interested in is the distribution of the posterior belief. thanks – lost.econ.child Apr 12 at 17:22