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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.372 29 — Monte Carlo Methods29.6 Terminology for Markov chain Monte Carlo methodsWe now spend a few moments sketching the theory on which the Metropolismethod and Gibbs sampling are based. We denote by p (t) (x) the probabilitydistribution of the state of a Markov chain simulator. (To visualize thisdistribution, imagine running an infinite collection of identical simulators inparallel.) Our aim is to find a Markov chain such that as t → ∞, p (t) (x) tendsto the desired distribution P (x).A Markov chain can be specified by an initial probability distributionp (0) (x) and a transition probability T (x ′ ; x).The probability distribution of the state at the (t+1)th iteration of theMarkov chain, p (t+1) (x), is given by∫p (t+1) (x ′ ) = d N x T (x ′ ; x)p (t) (x). (29.39)Example 29.6. An example of a Markov chain is given by the Metropolisdemonstration of section 29.4 (figure 29.12), for which the transition probabilityisT =¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¢1/ 2 1/ 2 · · · · · · · · · · · · · · · · · · ·1/ 2 · 1/ 2 · · · · · · · · · · · · · · · · · ·· 1/ 2 · 1/ 2 · · · · · · · · · · · · · · · · ·· · 1/ 2 · 1/ 2 · · · · · · · · · · · · · · · ·· · · 1/ 2 · 1/ 2 · · · · · · · · · · · · · · ·· · · · 1/ 2 · 1/ 2 · · · · · · · · · · · · · ·· · · · · 1/ 2 · 1/ 2 · · · · · · · · · · · · ·· · · · · · 1/ 2 · 1/ 2 · · · · · · · · · · · ·· · · · · · · 1/ 2 · 1/ 2 · · · · · · · · · · ·· · · · · · · · 1/ 2 · 1/ 2 · · · · · · · · · ·· · · · · · · · · 1/ 2 · 1/ 2 · · · · · · · · ·· · · · · · · · · · 1/ 2 · 1/ 2 · · · · · · · ·· · · · · · · · · · · 1/ 2 · 1/ 2 · · · · · · ·· · · · · · · · · · · · 1/ 2 · 1/ 2 · · · · · ·· · · · · · · · · · · · · 1/ 2 · 1/ 2 · · · · ·· · · · · · · · · · · · · · 1/ 2 · 1/ 2 · · · ·· · · · · · · · · · · · · · · 1/ 2 · 1/ 2 · · ·· · · · · · · · · · · · · · · ·£¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤1/ 2 · 1/ 2 · ·· · · · · · · · · · · · · · · · · 1/ 2 · 1/ 2 ·· · · · · · · · · · · · · · · · · · 1/ 2 · 1/ 2· · · · · · · · · · · · · · · · ·¥· · 1/ 2 1/ 2and the initial distribution wasp (0) (x) = [ · · · · · · · · · · 1 · · · · · · · · · · ] . (29.40)The probability distribution p (t) (x) of the state at the tth iteration is shownfor t = 0, 1, 2, 3, 5, 10, 100, 200, 400 in figure 29.14; an equivalent sequence ofdistributions is shown in figure 29.15 for the chain that begins in initial statex 0 = 17. Both chains converge to the target density, the uniform density, ast → ∞.p (0) (x)p (1) (x)p (2) (x)p (3) (x)p (10) (x)p (100) (x)p (200) (x)p (400) (x)0 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 20Figure 29.14. The probabilitydistribution of the state of theMarkov chain of example 29.6.Required propertiesWhen designing a Markov chain Monte Carlo method, we construct a chainwith the following properties:1. The desired distribution P (x) is an invariant distribution of the chain.A distribution π(x) is an invariant distribution of the transition probabilityT (x ′ ; x) if∫π(x ′ ) = d N x T (x ′ ; x)π(x). (29.41)An invariant distribution is an eigenvector of the transition probabilitymatrix that has eigenvalue 1.
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.29.6: Terminology for Markov chain Monte Carlo methods 3732. The chain must also be ergodic, that is,p (t) (x) → π(x) as t → ∞, for any p (0) (x). (29.42)A couple of reasons why a chain might not be ergodic are:(a) Its matrix might be reducible, which means that the state spacecontains two or more subsets of states that can never be reachedfrom each other. Such a chain has many invariant distributions;which one p (t) (x) would tend to as t → ∞ would depend on theinitial condition p (0) (x).The transition probability matrix of such a chain has more thanone eigenvalue equal to 1.(b) The chain might have a periodic set, which means that, for someinitial conditions, p (t) (x) doesn’t tend to an invariant distribution,but instead tends to a periodic limit-cycle.A simple Markov chain with this property is the random walk on theN-dimensional hypercube. The chain T takes the state from onecorner to a randomly chosen adjacent corner. The unique invariantdistribution of this chain is the uniform distribution over all 2 Nstates, but the chain is not ergodic; it is periodic with period two:if we divide the states into states with odd parity and states witheven parity, we notice that every odd state is surrounded by evenstates and vice versa. So if the initial condition at time t = 0 is astate with even parity, then at time t = 1 – and at all odd times– the state must have odd parity, and at all even times, the statewill be of even parity.The transition probability matrix of such a chain has more thanone eigenvalue with magnitude equal to 1. The random walk onthe hypercube, for example, has eigenvalues equal to +1 and −1.p (0) (x)p (1) (x)p (2) (x)p (3) (x)p (10) (x)p (100) (x)p (200) (x)p (400) (x)0 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 200 5 10 15 20Figure 29.15. The probabilitydistribution of the state of theMarkov chain for initial conditionx 0 = 17 (example 29.6 (p.372)).Methods of construction of Markov chainsIt is often convenient to construct T by mixing or concatenating simple basetransitions B all of which satisfy∫P (x ′ ) = d N x B(x ′ ; x)P (x), (29.43)for the desired density P (x), i.e., they all have the desired density as aninvariant distribution. These base transitions need not individually be ergodic.T is a mixture of several base transitions B b (x ′ , x) if we make the transitionby picking one of the base transitions at random, and allowing it to determinethe transition, i.e.,T (x ′ , x) = ∑ bp b B b (x ′ , x), (29.44)where {p b } is a probability distribution over the base transitions.T is a concatenation of two base transitions B 1 (x ′ , x) and B 2 (x ′ , x) if wefirst make a transition to an intermediate state x ′′ using B 1 , and then make atransition from state x ′′ to x ′ using B 2 .∫T (x ′ , x) = d N x ′′ B 2 (x ′ , x ′′ )B 1 (x ′′ , x). (29.45)
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