# Truncated prior leads to non-intuitive posterior

I am setting up a linear regression model for continuous data that is Normally distributed. For this model, I want to assume that my $\beta$ predictor is truncated to be positive, that is $$\beta \sim N(0, \tau^{-1})\textbf{1}(\beta>0)$$ $$\tau^{-1} \sim Ga(a,b)$$

I perform a Gibbs sampling to find the posterior distribution of $\beta$. The standard approach for finding the full conditionals of $\beta$ and $\tau$ lead to closed, conjugate distributions. My problem, however, is that the posterior for $\beta$ is always positive, regardless of the data.

I understand the bayesian logic - if my prior belief is that $\beta$ cannot take negative values, then I assign 0 probability to that ever happening. So it can't. However, intuitively, shouldn't my data pull my $\beta$ values in one direction or another? Eventually, if my data suggests that my prior was incorrect, shouldn't it overwhelm my prior belief and somehow get reflected in my posterior? Am I missing something?

• Your assumption that $\beta$ has no chance of being negative implies there will be no negative data. If there are negative data, your model is obviously incorrect! – whuber Nov 9 '15 at 0:58