a Bayesian approach to statistical inference, one would assign a probability distribution to p regardless of the non-existence of any such "frequency" interpretation, and one would construe the probabilities as degrees of belief that p is in any interval to which a probability is assigned. In that model, the random variables X1, ..., Xn are not independent, but they are conditionally independent given the value of p. In particular, if a large number of the Xs are observed to be equal to 1, that would imply a high conditional probability, given that observation, that p is near 1, and thus a high conditional probability, given that observation, that the next X to be observed will be equal to 1. Rules of conditional independence A set of rules governing statements of conditional independence have been derived from the basic definition. The these rules were termed "Graphoid Axioms" by Pearl and Paz, because they hold in graphs, where is interpreted to mean: "All paths from X to A are intercepted by the set B". Symmetry Decomposition Proof (meaning of ) (ignore variable B by integrating it out) A similar proof shows the independence of X and B. Weak union Proof By assumption, . Due to the property of decomposition , . Combining the above two equalities gives , which establishes . The second condition can be proved similarly. Contraction Proof This property can be proved by noticing , each equality of which is asserted by and , respectively. Intersection For strictly positive probability distributions, the following also holds: Proof By assumption: Using this equality, together with the Law of total probability applied to : Since and , it follows that . Technical note: since these implications hold for any probability space, they will still hold if one considers a sub-universe by conditioning everything on another variable, say K. For example, would also mean that . See also Graphoid Conditional dependence de Finetti's theorem Conditional expectation References External links Independence (probability theory)