I have the following problem: I need to assess whether a given parameter $B$ is equal to 0. Let's consider the following model (my problem is more complicated but I think that this example is sufficient):
$$ Y_i \sim N(A \cdot X_i+B,\sigma^2) $$
the observations $Y_i$ being independent conditionally on $A$, $B$, and explanatory variables $(X_i)$. I used a non-informative prior for $A$, $\sigma$ and $B$. Then I estimate the 95% level hpd interval for $p(B|Y)$ and check whether $0$ belongs to this interval.
I have two questions regarding this strategy.
First, is it a correct way to answer the problem? If yes, are there some authoritative references? Perhaps I lack the keywords because I did not find anything. If not, what is a good alternative to my problem?
Second, in practice $B<0$ has no physical meaning (while analytically there is no problem to define the model for $B<0$). However, in practice if I restrict the domain of the prior of $B$ to non-negative values the described strategy becomes not feasible (I guess). Is it a real problem?