Smarter sampling methods
DIVIDING THE SAMPLING POINTS INTO SETS
Using well-adapted phase space mappings may be not enough in order to
achieve quick convergence of the Monte-Carlo integration. For this purpose,
improved sampling methods may be paramount. There are two pretty general
ways to improve the sampling, which are quite orthogonal to each other.
Both, however, base on the same observation: When a sampling procedure
is divided such that different independent samplings contribute to the overall
result, the overall variance is minimised when it is "spread" equally over all
individual contributions. To rephrase this: The overall variance and hence the
error is minimised when the variances of the individual contributions is equal.
The reason for this is simple, it is due to the fact that the variance is
constructed from quadratic contributions.
The two ways to use this fact can be scetched as follows:
Division of phase space
Divide the phase space into disjoint regions (bins). Attach a weight,
i.e. probability, to each region to select a phase space point in it.
Distribute the regions and weights such that the variances in each region
become identical. There are different realisations of this idea, one of
the most popular ones being VEGAS.
Layers of phase space: Multi-channeling
Define a set of different mappings (channels) covering the whole phase space,
according to a p.d.f.. Give an a-priori weight to each channel, steering
the probability to generate a sampling point with this channel. Distribute the
weights accordingly, such that the overall variance is minimised. In this second
approach usually more or less specific "template" channels are employed,
this, of course, implies that there is some knowledge about how the
integrand could behave. More details on this method can be found
here, the discussion is based on
this article.