What are posterior probabilities?

Posterior probabilities are those probabilities that have been updated based on new evidence or sample information. In the context of Bayesian statistics, this concept is fundamental; it involves taking an initial belief or prior probability and revising it after observing new data.

When new information is acquired, the posterior probability represents a more informed understanding of certain parameters or events, reflecting both the prior probability and the likelihood of the observed data given that prior. This process allows for more accurate predictions or decisions as it incorporates both subjective beliefs (prior) and objective evidence (the sample information).

The other options refer to different concepts in probability. For instance, prior probabilities refer specifically to the initial beliefs before any evidence is considered, and constant probabilities suggest a static situation without updates based on new data. Probabilities that change after the initial assessment might imply some modifications or adjustments, but they do not specifically denote the process of updating based on new information, which is key in defining posterior probabilities. Thus, the emphasis on the updating process makes the correct choice essential in understanding how posterior probabilities function within statistical frameworks.