MIT15_097S12_lec15

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Unformatted text preview: n practice. Unfortunately, there is no way to be entirely certain that we are truly drawing from the posterior and we must be very cautious. Coin Flip Example Part 8. Let us return to our coin ﬂip example. We will draw from the posterior using the Metropolis-Hastings algorithm. Our model has a scalar parameter θ ∈ [0, 1] which is the probability of heads. The proposal distribution J (θt−1 , θ∗ ) will be uniform on an interval of size r around θt−1 : 1 r r if θ∗ ∈ [θt−1 8 2 , θt−1 ⊕ 2 ] Jt (θt−1 , θ∗ ; r) = r 0 otherwise where ⊕ and 8 represent modular addition and subtraction on [0, 1], e.g., 0.7 ⊕ 0.5 = 0.2. Notice that this proposal distribution is symmetric: J (θt−1 , θ∗ ; r) = J (θ∗ , θt−1 ; r). We accept the proposed θ∗ with probability α(θ t−1 p(y |θ∗ )p(θ∗ )J (θ∗ , θt−1 ; r) , θ ) = min ,1 p(y |θt−1 )p(θt−1 )J (θt−1 , θ∗ ; r) p(y |θ∗ )p(θ∗ ) = min ,1 p(y |θt−1 )p(θt−1 ) (θ∗ )mH +α−1 (1 − θ∗ )m−mH +β −1 = min ,1 , (θt−1...
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