Unformatted text preview: )mH +α−1 (1 − θt−1 )m−mH +β −1
∗ 30 which we can easily compute. This formula and a uniform random number
generator for the proposal distribution are all that is required to implement
the Metropolis-Hastings algorithm.
Consider the speciﬁc case of m = 25, mH = 6, and α = β = 5. The following
three ﬁgures show the time sequence of proposals θ∗ for chains with r = 0.01,
r = 0.1, and r = 1 respectively, with the colors indicating whether each
proposed θ∗ was accepted or not, and time along the x-axis. In this ﬁrst
ﬁgure we see that with r = 0.01, the step sizes are too small and after 2000
proposals we have not reached the stationary distribution. In the next ﬁgure, with r = 1 the steps are too large and we reject most of
the proposals. This leads to a small number of accepted draws after 2000
proposals. 31 The chain with r = 0.1 is the happy medium, where we rapidly reach the
stationary distribution, and accept most of the samples. To compare this to the analytic distribution that we obtain...
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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