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We will assume that the data were generated from a probability distribution
that is described by some parameters θ (not necessarily scalar). We treat θ
as a random variable. We will use the shorthand notation p(y |θ) to represent
the family of conditional density functions over y , parameterized by the ran
dom variable θ. We call this family p(y |θ) a likelihood function or likelihood
model for the data y , as it tells us how likely the data y are given the model
speciﬁed by any value of θ.
We specify a prior distribution over θ, denoted p(θ). This distribution rep
resents any knowledge we have about how the data are generated prior to
1 observing them. Our end goal is the conditional density function over θ, given the observed
data, which we denote as p(θ|y ). We call this the posterior distribution, and
it informs us which parameters are likely given the observed data.
We, the modeler, specify the likelihood function (as a function of y and θ)
and the prior (we completely specify this) using our know...
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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