Solve a little Bayesian updating problem

Suppose that our prior information is that

[23, 33],

[112, 150], and

[2000, 3200].

Using Bayes’ rule, what is the posterior if we want to update this with the additional knowledge that W

H = A? In this case, any triples (W, H, A) such that W H A will get excluded with a likelihood of zero, and the probability mass gets concentrated onto manifold of feasible combinations. In other words, given

Likelihood: L(A – W

H | W, H, A) = (A – W H)

Prior:

Posterior: f(W, H, A | A – W

H) A – W H) W, H, A)

where

() is the Dirac delta function and I( ) is the indicator function that’s unity when its argument is true and zero otherwise, solve for f, and plot the marginal cumulative distributions for the three variables.

How does the formulation change if our prior knowledge were that W

H = A and the three interval ranges were the new information?