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Unformatted text preview: ce). If Θ
is a compact set, θ∗ is deﬁned as in (18), A is a neighborhood of θ∗ , and
p(θ∗ ∈ A) > 0, then p(θ ∈ A|y ) → 1 as m → ∞.
These theorems show that asymptotically, the choice of the prior does not
matter as long as it assigns non-zero probability to every θ ∈ Θ. They also
show that if the data were generated by a member of the family of likelihood
models, we will converge to the correct likelihood model. If not, then we will
converge to the model that is ‘closest’ in the KL sense.
We give these theorems along with a word of caution. These are asymptotic
results that tell us absolutely nothing about the sort of m we encounter in
practical applications. For small sample sizes, poor choices of the prior or
likelihood model can yield poor results and we must be cautious.
18 Coin Flip Example Part 6. Suppose that the coin ﬂip data truly came from
a biased coin with a 3/4 probability of Heads, but we restrict the likelihood
model to only include coins with probability in the interval [0, 1/2]:
p(y |θ) = θmH (1 − θ)m−mH , θ ∈ [0, 1/2]. This time we will use a uniform prior, p(θ) = 2, θ ∈ [0, 1/2]. The posterior
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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