We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, which are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models, and for the mean-field Ising model.
We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces. For periodic boundary conditions corresponding to isotropic compressive stress, we find that the probability distribution for single-contact forces decays faster than exponentially. This superexponential decay persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges in the force distribution, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.
Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph.
WC present two algorithms for uniformly sampling from the proper colorings of a graph using k colors. We use exact snmpling from the stationary distribution of a Markov chain with states that are the k-colorings of a graph with maximum degree A, As opposed to approximate sampling algorithms based on rapid mixing, these algorithms have terminntion criteria that allow them to stop on some inputs much more quickly than in the worst case running time bound. For the first algorithm we show that when k > A(A + 2), the algorithm has an upper limit on the expected running time that is polynomial, For the second algorithm we show that for b > rA, where r is an integer that satisfies rp > n, the running time is polynomial, Previously, Jet-rum showed that it was possible to approximately sample uniformly in polynomial time from the set of k-colorings when k 2 2A, but our algorithm is the first polynomial time e~uct sampling algorithm for this problem. Using approximate sampling, Jerrum also showed how to approximately count the number of b-colorings, We give a new procedure for approximntcly counting the number of k-colorings that improves the running time of the procedure of Jerrum by a factor of (m/n)2 when k 2 2A, where n is the number of nodes in the graph to be colored and m is the number of edges. In addition, WC present an improved analysis of the chain of Luby and Vigoda for exact sampling from the independent sets of a graph. Finally, we present the first polynomial time method for exactly sampling from the sink free orientations of a gmph. Bubley and Dyer showed how to approximately sample from this state space in O(m3 In(@)) time, our algorithm takes 0(m4) expected time.
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