A priori probability

An a priori probability is a probability that is derived purely by deductive reasoning. One way of deriving a priori probabilities is the principle of indifference, which has the character of saying that, if there are N mutually exclusive and exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/N. Similarly the probability of one of a given collection of K events is K/N.

One disadvantage of defining probabilities in the above way is that it applies only to finite collections of events.

In Bayesian inference, the terms "uninformative priors" or "objective priors" refer to particular choices of a priori probabilities. Note that "prior probability" is a broader concept.

Similar to the distinction in philosophy between a priori and a posteriori, in Bayesian inference a priori denotes general knowledge about the data distribution before making an inference, while a posteriori denotes knowledge that incorporates the results of making an inference.

The a priori probability has an important application in statistical mechanics. The classical version is defined as the ratio of the number of elementary events (e.g. the number of times a die is thrown) to the total number of events. In the case of the die each elementary event has the same probability -- thus the probability of each outcome of throwing a (perfect) die is 1/6. Each face of the die appears with equal probability -- probability being a measure defined for each event.

In statistical mechanics, e.g. that of a gas contained in a finite volume, both the spatial coordinates ${\displaystyle q_{i}}$ and the momentum coordinates ${\displaystyle p_{i}}$ of the individual gas elements (atoms or molecules) are finite in the phase space spanned by these coordinates. In analogy to the case of the die, the a priori probability is here (in the case of a continuum) proportional to the phase space volume element ${\displaystyle \Delta q\Delta p}$, and is the number of standing waves (``states´´) therein, where ${\displaystyle \Delta q}$ is the range of the variable ${\displaystyle q}$ and ${\displaystyle \Delta p}$ is the range of the variable ${\displaystyle p}$ (here for simplicity considered in one dimension). An important consequence is a result known as Liouville's theorem (frequently also under different names), i.e. the time independence of this phase space volume element and thus of the a priori probability. A time dependence of this quantity would imply known information about the dynamics of the system, and hence would not be an a priori probability. Thus the region

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