How to restrict positive posterior?

Question

Is there a way to restrict posterior, say, to only positive values or an interval of values? Let's say I want to estimate a linear model, y = a + Xb, using Bayesian techniques. I specify priors for a and b, a ~ N(0, 1), b ~ N(0,1). When estimating posteriors, I might get that b= -0.3 with some probability. What I want to do is to restrict somehow posterior values so that they can only be positive or within some interval.

Answer 1

You only put prior mass on the region that you want the posterior to have mass over. So in your situation, if you want the posterior of $b$ to only take on positive values, you'd specify a prior for $b$ that only takes positive values. You can accomplish this by assigning a truncated normal distribution as a prior instead of a normal distribution, or you can use a distribution already only defined on the positive reals, like a gamma.