statisticalrethinkin..

248 8. MARKOV CHAIN MONTE CARLO ESTIMATIONFIGURE 8.3. Samples from the posterior for 20 adult !Kung heights. evertical dashed line on the le shows the location of the slice through theposterior of σ shown on the right. ese slices are posterior densities conditionalon the value of the other parameter, µ in this case. Writing analyticalexpressions for these slices makes Gibbs sampling possible.the parameter values is that the proposal distribution in basic M-H knows nothing about thetarget distribution.To understand this, let’s focus on the size of the changes in parameter value that areallowed. In King Markov’s fable, he could only move to neighboring islands. But what if hecould move two or three islands on each side? Let’s call the distance of proposals from thecurrent parameter value the step size. If the step size is too large, then many proposalswill be rejected (the King won’t move), because they will jump too far outside the parametervalues with high posterior probability. If step size is instead too small, then most proposalswill be accepted, but it will take a very long time for the us to explore the tails of the posteriordistribution. If you set the step size just right, then you can tradeoff one of the concerns forthe other.But even if we optimize the step size, the algorithm will still be inefficient, because whatwe really want is to take big steps when we are far from the center of the posterior distributionand small steps when we are close to the center. In other words, we’d like the proposals to be afunction of how close we are to the target. Metropolis-Hastings, in its usual form, has no wayto do this. If we could make the proposal distribution a function of the current parametervalues, then we could improve the efficiency of the Markov chain.To work towards one solution to this problem, let’s consider the final samples from aMetropolis chain, one that has given us a good picture of the target posterior density. Supposefor example we are estimating the mean µ and standard deviation σ for 20 adult heightsfrom the !Kung data you met in Chapter 4. I display such a posterior in FIGURE 8.3. e lehandplot shows samples from the posterior for both parameters, or Pr(µ, σ|D). e verticaldashed line locates a particular slice through the posterior, at µ = 150. e righthand plotshows the profile of the posterior at this slice. What you are seeing is the posterior densityof σ, where µ = 150. Another way to say this is you are viewing Pr(σ|D, µ).