Consider a misspecified model: $$P \sim \text{SegmentedUniform}(0,1) \\ Y \mid P \sim \text{Binomial}(N,P).$$ Where SegmentedUniform has uniform density on intervals $$(0.1, 0.2), (0.3, 0.4), (0.5, 0.6), (0.7, 0.8), (0.9,1.0)$$ and $0$ otherwise. Suppose I obtain posterior samples such that they concentrate around the segments $(0.3, 0.4), (0.5, 0.6)$.

Is it "valid" to derive a Bayes estimate $\hat p$ for $p$ that is not supported by the model? For example, $\hat p = 0.45$.

Here's my guess: A Bayes estimate by definition, is a minimizer of its expected posterior loss for some loss function. Although the expected loss is under the posterior, the minimizer itself is not constrained to the support, so $\hat p$ in this sense in the above example is valid. However, with this $\hat p$ being unsupported by the model, the resulting joint density would evaluate to $0$, and thus it is an "invalid" Bayes estimate.

As a follow-up to this, consider the model $$P \sim \text{Uniform}(0,1) \\ Y | P \sim \text{Binomial}(N,P)$$ whose posterior is inferred with a "misspecified sampler", such that it does not produce samples from the true posterior but instead from the SegmentedUniform model's posterior.

In this case, the Bayes estimate would be supported, but derived from an incorrect posterior. Is there any merit to an analysis performed as such? Is there any sense in studying this incorrectly-inferred posterior?

Consider a misspecified model: $$P \sim \text{SegmentedUniform}(0,1) \\ Y \mid P \sim \text{Binomial}(N,P).$$ Where SegmentedUniform has uniform density on intervals $$(0.1, 0.2), (0.3, 0.4), (0.5, 0.6), (0.7, 0.8), (0.9,1.0)$$ and $0$ otherwise. Suppose I obtain posterior samples such that they concentrate around the segments $(0.3, 0.4), (0.5, 0.6)$.

Is it "valid" to derive a Bayes estimate $\hat p$ for $p$ that is not supported by the model? For example, $\hat p = 0.45$.

Here's my guess: A Bayes estimate by definition, is a minimizer of its expected posterior loss for some loss function. Although the expected loss is under the posterior, the minimizer itself is not constrained to the support, so $\hat p$ in this sense in the above example is valid. However, with this $\hat p$ being unsupported by the model, the resulting joint density would evaluate to $0$, and thus it is an "invalid" Bayes estimate.

Edit: (see second comment)

In this case, the Bayes estimate would be supported, but derived from an incorrect posterior. Is there any merit to an analysis performed as such? Is there any sense in studying this incorrectly-inferred posterior?