Minzhao Liu

Notes for Handbook of MCMC

Notes on Handbook of MCMC

Authors(Editors): Steve Brooks, Andrew Gelman, Galin Jones, Xiao-li Meng

  1. Check Mixing trace plot and lag plot (acf)

  2. Monte Carlo standard error (MCSE) is $\hat{\sigma}/ \sqrt{n}$ , where

\[\sigma = \frac{1}{n} \sum\_i^n (X\_i - \mu)^2\]

  1. Multistart

  2. if (logr >= 0 || unif < exp(logr))

  3. random walk, y = x + e , where e is symmetric distributed.

  4. recommended optimal acceptance rate 0.234 for d > 1. (0.1 - 0.6) is fine. For d = 1, take 0.44.

  5. For mcmc R package, check mixing :

     plot(ts(out$batch))
  	acf(out$batch)

    lag 25 is sufficient to make the lag under 0.2.

     vignette("demo", package = "mcmc")

  6. scale is tuning the variance of proposal distribution \(N(0, \sigma^2)\).

  7. If $\sigma$ is too small, acceptance rate is high, not cover the whole posterior distribution . If $\sigma$ is too large, jumping around, rejecting too much. Not moving.

  8. For Metropolis-within-Gibbs, for every parameter, d = 1. The recommended optimal acceptance rate is fine when d >= 5.

  9. In Haario et al 2001, adaptive metropolis algorithm recommend the proposal distribution:

    \[N(Y\_{n-1}, 2.38^{2} / d \Sigma)\]

    where $\Sigma$ is the sample var-cov matrix.

  10. Tuning and adaptive metropolis within gibbs . In Robert and Rosenthal, 2005, monitor every 50 samples, make $\sigma \pm \delta(n)$ if acceptance rate $\neq 0.44$ , where $\delta(n) \to 0$ , one recommended is

    \[\delta(n) = min(0.01, 1/\sqrt{n})\]

  11. Inference:

    a. 3 and more parallel chains

    b. discard first half (conservative)

    c. mix after convergence

  12. thin

  13. when reporting $\theta_{\pi}$, MCSE should be included.

Notes from Steven Walker Lecture

f(y) = \int f(y x)f(x)dx
f(y x) = \int f(y u)f(u x)du

If f(y) = h(y)g(y), f(y, u) = I(u < h(y))g(y),

f(u y) = h(y)^-1 I(u< h(y))
f(y u) \propto g(y)I(u<h(y))

Instead of starting from beginning, M-H/gibbs make use of good samples, find new samples around previous one. Gibbs goes to stationary distribution by MC

\pi(mu_n+1, lambda_n+1) = \int \int pi(mu_n+1|lambda_n+1) pi(lambda_n+1|mu_n) pi( mu_n, lambda_n) dmu_n dlambda_n

q(y x) is the proposal density, if f(y x) is the transition density 
f(x)f(y|x) = f(y)f(x|y)

then

f(y) = \int f(y|x)f(x) dx

Methopolis Hasting

set the transition density :

f(y|x) = r(x) [a(y, x)q(y|x) /r(x)] + (1 - r(x)) I(y = x)

simplest choice of $\alpha$ is

a(y, x) = min{ 1, q(x|y)f(y)/(q(y|x)f(x))}

slice sampling

can be generalized to f(x)= g(x)h(x)

Published

28 October 2012

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