We then use these quantities together with the data

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Unformatted text preview: ledge of the system at hand. We then use these quantities, together with the data, to compute the posterior. The likelihood, prior, and posterior are all related via Bayes’ rule: p(y |θ)p(θ) p(y |θ)p(θ) p(θ|y ) = = , (1) p(y ) p(y |θ' )p(θ' )dθ' where the second step uses the law of total probability. Unfortunately the integral in the denominator, called the partition function, is often intractable. This is what makes Bayesian analysis difficult, and the remainder of the notes will essentially be methods for avoiding that integral. Coin Flip Example Part 1. Suppose we have been given data from a se­ ries of m coin flips, and we are not sure if the coin is fair or not. We might assume that the data were generated by a sequence of independent draws from a Bernoulli distribution, parameterized by θ, which is the probability of flipping Heads. But what’s the value of θ? That is, which Bernoulli distribution generated these data? We could estimate θ as the proportion of the flips that are H...
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