When is the posterior distribution “continuous” in the prior?

If we consider Bayes rule in the continuous case, is there any conditions that guarantee a small change in the prior distribution will not change the posterior distribution too much?

If so, when can we characterize the convergence speed?

This is related to the field of sensitivity analysis. For example, suppose we had a normal likelihood:

$$X_i \overset{iid}{\sim}\textrm{N}(\mu,\sigma^2)$$

With $\sigma$ known, and with a normal prior on $\mu$:

$$\mu \sim \textrm{N}(\eta,\tau^2)$$

Then the posterior distribution is given by:

$$\mu|\{X_i\}_{i=1}^n \sim \textrm{N}(\frac{1}{(\frac{1}{\tau^2}+\frac{n}{\sigma^2})}(\frac{\eta}{\tau^2}+\frac{\sum_{i=1}^nX_i}{\sigma^2}),(\frac{1}{\tau^2}+\frac{n}{\sigma^2})^{-1})$$

Then, knowing that $\tau$ represents our prior uncertainty around $\mu$, we see that $(\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-1}$ represents our posterior certainty $\mu | \{X_i\}_{i=1}^n$.

Taking the derivative of $(\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-1}$ gives us:

$$\frac{\partial}{\partial\tau^2}(\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-1} = (\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-2}\frac{1}{\tau^4}$$

Which means that increasing the prior variance increases the posterior variance, but at a rate which is decreasing as $\tau \rightarrow \infty$. Additionally, because $\frac{1}{\tau^2}\rightarrow 0$ as $\tau \rightarrow \infty$, we have that as $\tau \rightarrow \infty$ the change in variance becomes more of a function of $n$ than of $\tau$.

One other point is that because:

$$\frac{\partial}{\partial\sigma^2}(\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-1} = (\frac{1}{\tau^2} + \frac{n}{\sigma^2})^{-2}\frac{n}{\sigma^4}$$

Our uncertainty around $\mu$ increases as our uncertainty around the original observations increases, but at a rate which is diminished with increasing sample sizes.

You could also do this with discrete observations, so long as the posterior and prior are continuous.

An obvious example would be a Binomial-Beta model:

We have $\{X_i\}\overset{iid}{\sim}\textrm{Binomial}(m,p)$ with $p\sim \textrm{Beta}(\omega,\kappa)$ where $\textrm{Beta}(\omega,\kappa)$ is the mode and concentration parameterization of the Beta distribution.

Then we get:

$$p|\{X_i\} \sim \textrm{Beta}(\omega(\kappa-2)+1+\sum_{i=1}^nX_i,(1-\omega)(\kappa-2)+1+b-\sum_{i=1}^nX_i)$$

Where the Beta in the expression is using the standard parameterization. Then the new concentration $\kappa_{p|\{X_i\}} = n +\kappa$, which means that:

$$\frac{\partial}{\partial \kappa}\kappa_{p|\{X_i\}} = n+1$$

Which means that increasing the prior concentration increases the posterior concentration, at a rate $n+1$.

The posterior mode is given by:

$$\omega_{p|\{X_i\}} = \frac{\omega(\kappa-2)+\sum_{i=1}^nX_i}{n+\kappa-1}$$

Then:

$$\frac{\partial}{\partial\omega}\omega_{p|\{X_i\}} = \frac{\kappa-2}{n+\kappa-1}$$

So we can see that for a highly concentrated prior $(\kappa\rightarrow \infty)$ we get $\frac{\partial}{\partial\omega} \approx 1$, giving us that a change in prior mode roughly corresponds to a change in the posterior mode at a 1-1 rate. Similarly, we see that as $n\rightarrow\infty$ the posterior mode derivative is $0$, giving us that for large samples the posterior and prior mode are roughly independent.