In computational statistics, the preconditioned Crank–Nicolson algorithm (pCN) is a Markov chain Monte Carlo (MCMC) method for obtaining random samples – sequences of random observations – from a target probability distribution for which direct sampling is difficult. The most significant feature of the pCN algorithm is its dimension robustness, which makes it well-suited for high-dimensional sampling problems. The pCN algorithm is well-defined, with non-degenerate acceptance probability, even for target distributions on infinite-dimensional Hilbert spaces. As a consequence, when pCN is implemented on a real-world computer in large but finite dimension N, i.e. on an N-dimensional subspace of the original Hilbert space, the convergence properties (such as ergodicity) of the algorithm are independent of N. This is in strong contrast to schemes such as Gaussian random walk Metropolis–Hastings and the Metropolis-adjusted Langevin algorithm, whose acceptance probability degenerates to zero as N tends to infinity. The algorithm as named was highlighted in 2013 by Cotter, Roberts, Stuart and White, and its ergodicity properties were proved a year later by Hairer, Stuart and Vollmer. In the specific context of sampling diffusion bridges, the method was introduced in 2008.
Description of the algorithm
Overview
The pCN algorithm generates a Markov chain ( X n ) n ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} on a Hilbert space H {\displaystyle {\mathcal {H}}} whose invariant measure is a probability measure μ {\displaystyle \mu } of the form
μ ( E ) = 1 Z ∫ E exp ( − Φ ( x ) ) μ 0 ( d x ) {\displaystyle \mu (E)={\frac {1}{Z}}\int _{E}\exp(-\Phi (x))\,\mu _{0}(\mathrm {d} x)}
for each measurable set E ⊆ H {\displaystyle E\subseteq {\mathcal {H}}} , with normalising constant Z {\displaystyle Z} given by
Z = ∫ H exp ( − Φ ( x ) ) μ 0 ( d x ) , {\displaystyle Z=\int _{\mathcal {H}}\exp(-\Phi (x))\,\mu _{0}(\mathrm {d} x),}
where μ 0 = N ( 0 , C 0 ) {\displaystyle \mu _{0}={\mathcal {N}}(0,C_{0})} is a Gaussian measure on H {\displaystyle {\mathcal {H}}} with covariance operator C 0 {\displaystyle C_{0}} and Φ : H → R {\displaystyle \Phi \colon {\mathcal {H}}\to \mathbb {R} } is some function. Thus, the pCN method applied to target probability measures that are re-weightings of a reference Gaussian measure. The Metropolis–Hastings algorithm is a general class of methods that try to produce such Markov chains ( X n ) n ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} , and do so by a two-step procedure of first proposing a new state X n + 1 ′ {\displaystyle X'_{n+1}} given the current state X n {\displaystyle X_{n}} and then accepting or rejecting this proposal, according to a particular acceptance probability, to define the next state X n + 1 {\displaystyle X_{n+1}} . The idea of the pCN algorithm is that a clever choice of (non-symmetric) proposal for a new state X n + 1 ′ {\displaystyle X'_{n+1}} given X n {\displaystyle X_{n}} might have an associated acceptance probability function with very desirable properties.
The pCN proposal The special form of this pCN proposal is to take
X n + 1 ′ = 1 − β 2 X n + β Ξ n + 1 , {\displaystyle X'_{n+1}={\sqrt {1-\beta ^{2}}}X_{n}+\beta \Xi _{n+1},}
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