The Continuous Bayes' Rule

A radar reads a noisy speed; a market observes a noisy price signal; a detector receives a noisy transmitted bit. In every case, the quantity of interest is unobserved and continuous (or discrete), and only a noisy, related observation is available — exactly the inference structure Conditional Probability, Total Probability, and Bayes' Rule built for discrete events, now needed for continuous densities. This note inverts Joint Densities, Marginals, and Conditioning's "prior $\times$ likelihood $\to$ joint" construction to go the other way — from an observation back to a posterior belief about the unobserved quantity — covering both the fully-continuous case and the practically important mixed case of a discrete signal observed through continuous noise.

Learning objectives. By the end of this note you should be able to: