# Bayesian Probability Theory: Applications in the Physical by von der Linden W., Dose V., von Toussaint U.

By von der Linden W., Dose V., von Toussaint U.

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What is the probability that the next draw will be black again? The required propositions are: • • • • • N: N balls have been drawn with replacement. n: n of the drawn balls are black. B The next draw will be a black ball. Eq : The intrinsic probability for a black ball in a single trial is q. e. all assumptions and all our prior knowledge. Invoking the marginalization rule, we can express the probability for B as 1 P (B|n, N, I) = dq P (B|Eq , n, N, I) P (Eq |n, N, I). 0 The first term is a sort of tautology, it is the probability that the next ball will be black, given the probability for a black ball in a single trial is q.

Consider an urn with black and white balls. Out of n previous draws with replacement, k balls are black. What is the probability that the next draw will be black again? The required propositions are: • • • • • N: N balls have been drawn with replacement. n: n of the drawn balls are black. B The next draw will be a black ball. Eq : The intrinsic probability for a black ball in a single trial is q. e. all assumptions and all our prior knowledge. Invoking the marginalization rule, we can express the probability for B as 1 P (B|n, N, I) = dq P (B|Eq , n, N, I) P (Eq |n, N, I).

20) with ng = 3 in the sequence c. We have used the sum rule to specify the probability for red in a single trial P (r|N, q, I) = 1 − q. 20) is valid for arbitrary colour sequences. As a matter of fact, we are not interested in the probability for a specific colour sequence, but rather in the probability that there are ng green balls. The two probabilities can be linked through the marginalization rule P (ng |N, q1 , I) = P (ng |c, N, q1 , I)P (c|N, q, I). 20) already. The first term is also very simple.