I am relatively new to using Bayes rules for continuous variables and am having trouble setting up part of the formula and am looking for help. The example I am trying to work through is the following:
Suppose there are two numbers x and y that were chosen at random. We have a prior belief that x and y were chosen from a standard multivariate normal distribution. We are then given the following piece of information: x+y=1.0.
The question that I am interested in is how this should change our posterior belief about the values of x and y. From Bayes rule:
posterior distribution is proportional to P(x+y=1 | prior) * P(prior)
What I am stuck on is figuring out what to enter in the formula for P(x+y=1 | prior), would someone be willing to point me in the right direction?
My first thought was that it should be 0, because the volume of a line is zero but then the problem doesn't make much sense. But then I got to thinking that if I had a second such piece of information, say x+2y=5, then I could figure out x and y exactly and it wouldn't make sense for Bayes rule not to be able to make use of this kind of information. Thank you for your help.