I fitted a Bayesian logistic regression in WinBugs and it has an interaction term. Something like this: $$\mathrm{Prob}(y_{i}=1) = \mathrm{logit}^{-1} (a + b_{1}*x_{i} + b_{2}*w_{i} + b_{3}*x_{i}*w_{i})$$
where $x$ is a standardized continuous variable, and $w$ is a dummy variable. In reality the model is more complicated, but I wanna keep things simple.
It happens that the interaction term is "significant", but not the single predictors. For instance,
$\mathrm{mean}(b_{1}) = -.2$ and $95%$ quantile: $(-1.3$ and $.7)$
$\mathrm{mean}(b_{2}) = -.4$ and $95%$ quantile: $(-1.3$ and $.5)$
$\mathrm{mean}(b_{3}) = 1.4$ and $95%$ quantile: $(.4$ and $2.5)$
Do you guys have any advice on how to react to this finding? I thought that I could compute 95% credibility intervals for the whole effect of $x$ when $w=1$. This would be: 95% quantile for total effect of x, conditional on $w=1$: $(-1.3+.4$ and $.7+2.5) = (-.9 + 3.2)$
Is this correct? If not, what should I do? Any references on the subject?