Denition 4 the family of distributions f is an

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Unformatted text preview: ameter (β ). In the Uniform-Pareto example, the data come from a uniform distribution on [0, θ], but we don’t know θ. We choose a prior for θ that is a Pareto distribution with prior hyperparameters xs and k . Here, xs and k can be in­ terpreted as beginning the experiment with k observations, whose maximum value is xs (so we believe θ is at least xs ). In the posterior, we replace the maximum value xs with the new maximum value max{max yi , xs } including 12 −1 the observed data, and update the number of observations to k + m. For Normal-Normal, we see that the posterior mean is a weighted combination of the data and the prior, where the weights are the variances and correspond to our certainty in the prior vs. the data. The higher the variance in the data, the less certain we are in them and the longer it will take for the data to overwhelm the effect of the prior. (Might be helpful to think of µ0 as one prior example rather than a bunch of them.) 3.1 Exponential families and conjugate priors There is an important connection between exponential families (not to be confused with the exponential distribution) and conjugate priors. Definition 4. The family of distributions F is an e...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

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