I started looking at the Wiki entry for radical probabilism after I saw a paper from ArXiv this morning. The main idea is that it's an alternative to Bayes' rule for updating probabilities in light of some new information.
Bayes' rule says that, to form a new probability of some set $A$, given that you observed some set $B$, you can use $$ P_{\text{new}}(A) = P_{\text{old}}(A|B) = \frac{P_{\text{old}}(A \cap B)}{P_{\text{old}}(B)}. $$ When you draw the sigma field with a Venn diagram, you can see how you're eliminating all the area inside $B^c$ (the complement of $B$), and you re-normalize the measure.
Radical probabilism, or Jeffreys updating, or probability kinematics says you shouldn't eliminate everything inside $B^c$, but rather do $$ P_{\text{new}}(A) = P_{\text{old}}(A|B^c)P_{\text{new}}(B^c) + P_{\text{old}}(A|B)P_{\text{new}}(B). $$
Question:
How would/could this come into play for estimating models in the practice of statistics? I can find zero references right now (the word "kinematics" throws off my search engine a lot).
Would this do away with the need for a prior distribution? Or would this be applied to a distribution over the data space, for example a likelihood or a "posterior" predictive distribution?
The paper I was referring to (link here), seems to be using to be taking the second approach. In equation (3), they're applying it to all of the likelihoods (for each parameter in the parameter space) all at once.
