probabilitiesremain

Probabilitiesremain is a coined term used to describe the property that certain probability assignments stay unchanged when specific information is updated or transformed. Although not a standard label in major textbooks, it serves to denote invariance phenomena in probabilistic reasoning and modeling.

Definition and interpretation: For a probability measure P over a set of events, probabilitiesremain for an event A with respect to a class S of conditioning events or information pieces means that P(A | G) = P(A) for every G in S where conditioning is defined (almost surely). In other words, A is invariant under the information in S. A common sufficient condition is independence: if A is independent of all conditioning events in S, then P(A | G) = P(A) for all G in S.

Characterizations and examples: A typical way probabilitiesremain arises is through independence. If A is independent of a random variable or of a sigma-algebra representing S, then conditioning on S does not change the probability of A. For example, in a fair coin toss, if A = "the first flip is heads" and E is a statement about the second flip, the events are independent, so P(A | E) = P(A).

Applications and relation to other concepts: The idea underlies discussions of invariance, sufficiency, and robustness in probability and statistics. It is related to, but distinct from, concepts such as independence, conditional probability, and Bayesian updating. In practice, probabilitiesremain highlights situations where updates do not affect certain probabilities, aiding theoretical analysis and robustness considerations.

Limitations: In real-world data, invariance conditions are uncommon and require specific model structures or independence assumptions. The term is mainly a descriptive tool rather than a universally adopted standard. See also: independence, conditional probability, Bayesian updating, invariance.