34,503 research outputs found
On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences
Many scientific and engineering problems require to perform Bayesian
inferences for unknowns of infinite dimension. In such problems, many standard
Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh
refinement, which is referred to as being dimension dependent. To this end, a
family of dimensional independent MCMC algorithms, known as the preconditioned
Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional
parameters. In this work we develop an adaptive version of the pCN algorithm,
where the covariance operator of the proposal distribution is adjusted based on
sampling history to improve the simulation efficiency. We show that the
proposed algorithm satisfies an important ergodicity condition under some mild
assumptions. Finally we provide numerical examples to demonstrate the
performance of the proposed method
Representation Theorems for Quadratic -Consistent Nonlinear Expectations
In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a -expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic -consistent nonlinear
expectation with a certain domination property must be a quadratic
-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form
A hybrid adaptive MCMC algorithm in function spaces
The preconditioned Crank-Nicolson (pCN) method is a Markov Chain Monte Carlo
(MCMC) scheme, specifically designed to perform Bayesian inferences in function
spaces. Unlike many standard MCMC algorithms, the pCN method can preserve the
sampling efficiency under the mesh refinement, a property referred to as being
dimension independent. In this work we consider an adaptive strategy to further
improve the efficiency of pCN. In particular we develop a hybrid adaptive MCMC
method: the algorithm performs an adaptive Metropolis scheme in a chosen finite
dimensional subspace, and a standard pCN algorithm in the complement space of
the chosen subspace. We show that the proposed algorithm satisfies certain
important ergodicity conditions. Finally with numerical examples we demonstrate
that the proposed method has competitive performance with existing adaptive
algorithms.Comment: arXiv admin note: text overlap with arXiv:1511.0583
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