Some questions about the posterior distribution when the marginal distribution is zero

Question

Let $\{f(\cdot|\theta): \theta \in \Theta \}$ be a family of pdfs and let $\pi: \Theta \to \mathbb{R}$ be a prior. According to Bayes' theorem (as stated in, e.g., Casella and Berger), the posterior distribution $\pi(\cdot|x)$ is given by $$ \pi(\theta|x) = \frac{f(x|\theta) \pi(\theta)}{\int_{\Theta} f(x|\theta) \pi(\theta)\,d\theta}. $$

My questions:

1. How do we define the posterior distribution for values of $x$ such that $\int_{\Theta} f(x|\theta) \pi(\theta)\,d\theta = 0$ ? Or do we just leave it undefined?

2. If my reasoning is correct, $\int_{\Theta} f(x|\theta) \pi(\theta)\,d\theta > 0$ provided that the set $E_x := \{\theta \in \Theta: f(x|\theta)\, \pi(\theta) > 0 \}$ has positive measure. Is there anything else that we can say about the set of $x$ for which $\int_{\Theta} f(x|\theta) \pi(\theta)\,d\theta > 0$ ?

3. If one is working with a model such that $\int_{\Theta} f(x|\theta) \pi(\theta)\,d\theta = 0$ for certain values of $x$ in the sample space, does this indicate that there is a problem with the model (either in our choice of pdf $f(\cdot|\theta)$ or our choice of prior $\pi$) ?

Answer 1

If$$\int_{\Theta} f(x|\theta) \pi(\theta)\,\text d\theta = 0$$the Bayesian model is incompatible with the data $x$. For instance, if $$X\sim\mathcal U(0,\theta)\qquad\theta\sim\mathcal U(0,1)$$ and the realisation $x$ of $X$ is $x=2$, this realisation is incompatible with the model. The model must be modified.