# Robustness of Posterior distribution wrt likelihood function

Suppose we have $$X_1, \ldots, X_n \mid \theta \, \mathop{\sim}^{iid} \, L(\cdot \mid \theta), \quad \theta \sim \pi$$ By Bayes' theorem, the corresponding posterior distribution is $$\pi_n(\mathrm d \theta \mid X_{1:n}) \propto L_n(X_{1:n} \mid \theta) \pi(\mathrm d \theta)$$ where $$L_n(X_{1:n} \mid \theta) = \prod_{i=1}^n L(X_i \mid \theta)$$.

Suppose that instead of considering $$L(\cdot \mid \theta)$$ we perform inference, we consider an approximation $$\tilde L(\cdot \mid \theta)$$ such that $$d_{TV}\big(L(\cdot \mid \theta), \tilde L(\cdot \mid \theta)\big) \leq K(\theta)$$ where $$d_{TV}$$ is the Total Variation distance.

Let $$\tilde \pi_n(\mathrm d \theta) \propto \tilde L_n(X_{1:n} \mid \theta) \pi(\mathrm d \theta)$$ be the posterior distribution for this approximate model. I was wondering what is known on the relation between $$\pi_n$$ and $$\tilde \pi_n$$ and if there is some Lispchitz results (wrt to a suitable metric, possibly different from the TV one) to the likelihood function, or at least some bound on the distance between the two.

We can also assume that $$\sup_{\theta} K(\theta)$$ is bounded and, if it is more convenient, that $$L(\cdot \mid \theta)$$ is a distribution over the nonnegative integers.