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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?

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    $\begingroup$ 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! $\endgroup$ – whuber Nov 9 '15 at 0:58
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Technically, by definition and Bayes theorem, the support of the posterior distribution must be included in the support of the prior distribution.

If your prior belief is that the true paramter value is most likely to be positive, but is possibly negative, you could (and should!) replace your prior appropriatlly. For example, asymmetric Laplace distribution.

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