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Unformatted text preview: ameter (β ).
In the UniformPareto 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 NormalNormal, 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.
 Spring '12
 CynthiaRudin

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