Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.32.2: Exact sampling concepts 415Coupling from the pastHow can we use the coalescence property to find an exact sample from theequilibrium distribution of the chain? The state of the system at the momentwhen complete coalescence occurs is not a valid sample from the equilibriumdistribution; for example in figure 32.1b, final coalescence always occurs whenthe state is against one of the two walls, because trajectories merge only atthe walls. So sampling forward in time until coalescence occurs is not a validmethod.The second key idea of exact sampling is that we can obtain exact samplesby sampling from a time T 0 in the past, up to the present. If coalescencehas occurred, the present sample is an unbiased sample from the equilibriumdistribution; if not, we restart the simulation from a time T 0 further intothe past, reusing the same random numbers. The simulation is repeated at asequence of ever more distant times T 0 , with a doubling of T 0 from one run tothe next being a convenient choice. When coalescence occurs at a time before‘the present’, we can record x(0) as an exact sample from the equilibriumdistribution of the Markov chain.Figure 32.2 shows two exact samples produced in this way. In the leftmostpanel of figure 32.2a, we start twenty-one chains in all possible initial conditionsat T 0 = −50 and run them forward in time. Coalescence does not occur.We restart the simulation from all possible initial conditions at T 0 = −100,and reset the random number generator in such a way that the random numbersgenerated at each time t (in particular, from t = −50 to t = 0) will beidentical to what they were in the first run. Notice that the trajectories producedfrom t = −50 to t = 0 by these runs that started from T 0 = −100 areidentical to a subset of the trajectories in the first simulation with T 0 = −50.Coalescence still does not occur, so we double T 0 again to T 0 = −200. Thistime, all the trajectories coalesce and we obtain an exact sample, shown bythe arrow. If we pick an earlier time such as T 0 = −500, all the trajectoriesmust still end in the same point at t = 0, since every trajectory must passthrough some state at t = −200, and all those states lead to the same finalpoint. So if we ran the Markov chain for an infinite time in the past, from anyinitial condition, it would end in the same state. Figure 32.2b shows an exactsample produced in the same way with the Markov chains of figure 32.1b.This method, called coupling from the past, is important because it allowsus to obtain exact samples from the equilibrium distribution; but, as describedhere, it is of little practical use, since we are obliged to simulate chains startingin all initial states. In the examples shown, there are only twenty-one states,but in any realistic sampling problem there will be an utterly enormous numberof states – think of the 2 1000 states of a system of 1000 binary spins, forexample. The whole point of introducing Monte Carlo methods was to try toavoid having to visit all the states of such a system!MonotonicityHaving established that we can obtain valid samples by simulating forwardfrom times in the past, starting in all possible states at those times, the thirdtrick of Propp and Wilson, which makes the exact sampling method usefulin practice, is the idea that, for some Markov chains, it may be possible todetect coalescence of all trajectories without simulating all those trajectories.This property holds, for example, in the chain of figure 32.1b, which has theproperty that two trajectories never cross. So if we simply track the two trajectoriesstarting from the leftmost and rightmost states, we will know that