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Quantitative Finance · Glossaire

Qu'est-ce que « Bayesian update » ?

Definition 30.3 Markets II: Rates, FX and Credit · Chapitre 30 — Market-Making Games

A Bayesian update revises a probability distribution over an unknown value after an observation, in proportion to how likely the observation would be under each value: posterior ∝\propto prior ×\times likelihood.

Distribution of the sum of five cards dealt from a 52-card deck, aces counting 1 and kings 13: mean 35 (dashed), standard deviation 8.03, from 6 to 64. Exact, by counting. Data: the chapter’s tutorial.
Figure 30.1. Distribution of the sum of five cards dealt from a 52-card deck, aces counting 1 and kings 13: mean 35 (dashed), standard deviation 8.03, from 6 to 64. Exact, by counting. Data: the chapter’s tutorial.
One game at a width of 11.5: the maker’s expected value of the sum before each round, updated by Bayes’ rule after every trade and pass, against the true sum, 31. Early buys push the estimate up; the sales that follow bring it down towards the truth. The maker ends the game with a profit of 18.6. Data: the chapter’s tutorial, seeded.
Figure 30.2. One game at a width of 11.5: the maker’s expected value of the sum before each round, updated by Bayes’ rule after every trade and pass, against the true sum, 31. Early buys push the estimate up; the sales that follow bring it down towards the truth. The maker ends the game with a profit of 18.6. Data: the chapter’s tutorial, seeded.

Exemples

Example 30.4 (The first trade)

The maker opens 30 at 40, a width of 10. An uninformed trader trades with probability e−10/8=0.29e^{-10/8} = 0.29. If someone buys at 40, it was an informed trader with probability 0.31; the maker’s expected value of the sum rises from 35 to 38.13, and its next market is centred there. After a sale at 30 it falls to 31.87; after a pass it stays at 35, since a pass is equally likely from a high or a low sum.

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