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A statement I’ve seen without proof in many books and papers is that random probability measures obtained by normalizing a completely random measure, such as the Dirichlet process, are not dominated by a common $\sigma$-finite measure and therefore the posterior can’t be obtained using Bayes theorem. I can intuitively grasp why is that but don’t know how to prove it.

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elaborating on my comment, I think an example can be shown for basically every mixture model.

Suppose you have a Normalized Random Measure of the form $$ P(\cdot) = \sum_h w_h \delta_{\phi_h}(\cdot) $$ with the usual condition that the weights sum to $1$ and that $\phi_h$ are i.i.d from a non-atomic distribution (non atomicity is important)

Now suppose that you observe a single datum $X_1 = x^*$, living in a Polish space $(\mathbb X, \mathcal X)$. $$X_1 \mid P \sim P \quad P \sim \Pi$$ with $\Pi$ a measure on $\mathbb P_X$ the space of all probability measures on $(\mathbb X, \mathcal X)$.

Since the $\phi_h$s come from a diffuse measure, this entails that the set $A = \{\text{random measures that have an atom in $x^*$}\} \subset \mathbb P_X$ will have $0$ prior mass, i.e. $\Pi(A) = 0$.

On the other hand, we know that if $\Pi$ is for example a Dirichlet process, $\Pi(A \mid X_1 = x^*) = 1$.

So that the prior does not dominate the posterior, thus Bayes' theorem does not hold and the model cannot be dominated.

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