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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13 (Experiment) An experiment or trial is defined as the observation (realization) of an outcome based on the validity of the background information. 2 Worked example: The three-urn problem ✐ 25 An experiment need not be performed by humans. g. g. appropriately defined background information I). 14 (Random experiment) A random experiment is an experiment which can result in different outcomes under repetition, and for which the outcome or event is unknown in advance. Repeating a deterministic experiment, instead, is expected to yield identical outcomes.

For example, in the urn experiment with n draws with replacement, the cardinality of the sample space is given by |Gn | = 2n . 17 (Bernoulli trials) Repeated independent binary trials with constant probabilities are called Bernoulli binary trials or Bernoulli experiments. The above urn experiment can be considered as a series of Bernoulli trials, with 40 draws with the binary outcome of either red or green in each trial, where the probability for green is the same for all draws. 18 (Samples) n repeated independent trials (samples) can be bundled together into a sample of size n.

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.

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