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In the multi-dimensional approach, the joint PDF is parametrized bin
by bin. If the correlation is small within each multi-dimensional
bin, the correlation between the input variables can be neglected and the product
of the marginal distributions is a good approximation of the joint PDF.
In the case of a vector of measurements
,
the marginal distributions depend on the bins
;
where
and
are the bin index associated with
and
, respectively.
Hence for any value of
the PDF is given by
the product of Monte Carlo 1-dimensional distribution functions
for the corresponding binned grid
:
 |
(3) |
In practice, one must rely on a finite number of bins so that
the PDFs are defined with enough Monte Carlo statistics. Binning
effects are an inherent limitation of the multi-D method.
Alain Bellerive
2006-05-19