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Many approaches for conducting Bayesian inference on discretely observeddiffusions involve imputing diffusion bridges between observations. This can becomputationally challenging in settings in which the temporal horizon betweensubsequent observations is large, due to the poor scaling of algorithms forsimulating bridges as observation distance increases. It is common in practicalsettings to use a blocking scheme, in which the path is split into a(user-specified) number of overlapping segments and a Gibbs sampler is employedto update segments in turn. Substituting the independent simulation ofdiffusion bridges for one obtained using blocking introduces an inherenttrade-off: we are now imputing shorter bridges at the cost of introducing adependency between subsequent iterations of the bridge sampler. This is furthercomplicated by the fact that there are a number of possible ways to implementthe blocking scheme, each of which introduces a different dependency structurebetween iterations. Although blocking schemes have had considerable empiricalsuccess in practice, there has been no analysis of this trade-off nor guidanceto practitioners on the particular specifications that should be used to obtaina computationally efficient implementation. In this article we conduct thisanalysis and demonstrate that the expected computational cost of a blockedpath-space rejection sampler applied to Brownian bridges scales asymptoticallyat a cubic rate with respect to the observation distance and that this rate islinear in the case of the Ornstein-Uhlenbeck process. Numerical experimentssuggest applicability both of the results of our paper and of the guidance weprovide beyond the class of linear diffusions considered.

The computational cost of blocking for sampling discretely observed diffusions