Conditional Probability

Conditional probabilities are supposed to earn their keep when the evidence or information that is given is more specific than what is captured by initial set of outcomes. This chapter explores various approaches to conditional probability, canvassing their associated mathematical and philosophical problems and numerous applications. It assess whether conditional probability can rightfully be regarded as the fundamental notion in probability theory after all. Conditional probability is near ubiquitous in both the methodology-in particular, the use of statistics and game theory-of the sciences and social sciences, and in their specific theories. Various central concepts in statistics are defined in terms of conditional probabilities: significance level, power, sufficient statistics, ancillarity, maximum likelihood estimation, and Fisher information. Game theorists use conditional probabilities for calculating the expected payoffs in correlated equilibrium; computing the Bayesian equilibrium in games of incomplete information; in certain Bayesian dynamic updating models of equilibrium selection in repeated games.