In Bayesian statistics, usually we have some prior distribution $P(M)$, some observations $X$, and we compute a posterior $P(M|X)$. The observation allows us to gain information about the generating distribution, and learn about $M$.
My question is this: what if we gained less than the maximum amount of information about $M$ for some reason? This could happen if the update was somehow incomplete, like if the Bayesian update was implemented in some less-than-perfect system, or perhaps a system where there was an update cost. In that case, we would end up with a pseudo-posterior that was somewhere between $P(M)$ and $P(M|X)$. I have a few questions:
- Is this a thing that people have written about? Does it have a name?
- If so, is there a way to describe the degree of 'incompleteness' of this update?
- Is it equivalent to having a noisy observation of $X$? I suspect not.