In mathematics and computer science, the method of conditional probabilities[1][2] is a systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic algorithms that explicitly construct the desired object.[3]
Often, the probabilistic method is used to prove the existence of mathematical objects with some desired combinatorial properties. The proofs in that method work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probability. Consequently, they are nonconstructive — they don't explicitly describe an efficient method for computing the desired objects.
The method of conditional probabilities converts such a proof, in a "very precise sense", into an efficient deterministic algorithm, one that is guaranteed to compute an object with the desired properties. That is, the method derandomizes the proof. The basic idea is to replace each random choice in a random experiment by a deterministic choice, so as to keep the conditional probability of failure, given the choices so far, below 1.
The method is particularly relevant in the context of randomized rounding (which uses the probabilistic method to design approximation algorithms).
When applying the method of conditional probabilities, the technical term pessimistic estimator refers to a quantity used in place of the true conditional probability (or conditional expectation) underlying the proof.
^Spencer, Joel H. (1987), Ten lectures on the probabilistic method, SIAM, ISBN 978-0-89871-325-1
^
Raghavan, Prabhakar (1988), "Probabilistic construction of deterministic algorithms: approximating packing integer programs", Journal of Computer and System Sciences, 37 (2): 130–143, doi:10.1016/0022-0000(88)90003-7
^The probabilistic method — method of conditional probabilities, blog entry by Neal E. Young, accessed 19/04/2012 and 14/09/2023.
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