The distribution bias of direct sampling with continues variables

In direct sampling for the conditional simulation of random fields with continues distributions the probablity of finding exact matches of the local conditions in the training image is zero. We thus need to take samples with similar but different patterns. The maximum permissible difference is an algorithmic threshold parameter t controlling the speed, the reliability and the correctness of the simulation.

The contribution describes the effect that simulations sampled with this algorithm follow a conditional distribution, which is systematically biased with respect to the conditional distribution represented by the training images. This sampling bias is create by the fact that it is more likely to find conditioning events deviating from the observed conditioning shifted in the direction of the gradient of the marginal density of the conditioning events. These conditioning events however also typically have a conditional distribution shifted accordingly. The implicitly generated distribution of the conditional simulation is thus just not only (as it was always understood) a little smoothness with respect to the true distribution, but systematically biased. This sampling bias can be easily demostrated in simple Gaussian examples, where it introduces a regression to the mean type effect into a conditional simulation. The effect of the bias accumulates along the simulation path. The size of the sampling bias depends on the choosen tolerance, the choosen neighbourhood, the choosen iteration limit, and the local conditioning events.

The contribution will also discuss strategies to limit and control the effect of this sampling bias, by selecting appropriate algorithmic parameters, and by quantifying the DS sampling bias.