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By Michael Halls Moore

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For every night that passes, the application of Bayesian inference will tend to correct our prior belief to a posterior belief that the Moon is less and less likely to collide with the Earth, since it remains in orbit. In order to demonstrate a concrete numerical example of Bayesian inference it is necessary to introduce some new notation. Firstly, we need to consider the concept of parameters and models. A parameter could be the weighting of an unfair coin, which we could label as θ. Thus θ = P (H) would describe the probability distribution of our beliefs that the coin will come up as heads when flipped.

Over the course of carrying out some coin flip experiments (repeated Bernoulli trials) we will generate some data, D, about heads or tails. ". A model helps us to ascertain the probability of seeing this data, D, given a value of the parameter θ. The probability of seeing data D under a particular value of θ is given by the following notation: P (D|θ). ". Thus we are interested in the probability distribution which reflects our belief about different possible values of θ, given that we have observed some data D.

A model helps us to ascertain the probability of seeing this data, D, given a value of the parameter θ. The probability of seeing data D under a particular value of θ is given by the following notation: P (D|θ). ". Thus we are interested in the probability distribution which reflects our belief about different possible values of θ, given that we have observed some data D. This is denoted by P (θ|D). Notice that this is the converse of P (D|θ). So how do we get between these two probabilities? It turns out that Bayes’ rule is the link that allows us to go between the two situations.

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