truncnorm

TruncNorm

Arbitrary order moments for truncated multivariate normal distributions.

Introduction

Given

X ~ N(m, C), a <= X <= b

with mean vector m, covariance matrix C, lower limit vector a and upper limit vector b,

import truncnorm
truncnorm.moments(m, C, a, b, 4)

returns all the following moments of total order less or equal to 4 as a list:

[
  P(a<=X<=b),           (scalar)
  E[X_i],               (N vector)
  E[X_i*X_j],           (NxN matrix)
  E[X_i*X_j*X_k],       (NxNxN array)
  E[X_i*X_j*X_k*X_l],   (NxNxNxN array)
]

for all i, j, k and l. Note that the first element in the list is a bit of a special case. That's because E[1] is trivially 1 so giving the normalisation constant instead is much more useful.

TODO

• Double truncation
• Numerical stability could probably be increased by using logarithic scale in critical places of the algorithm
• Sampling (see Gessner et al below)
• Folded distribution
• Optimize recurrent integrals by using vector and index-mapping representation instead of arrays. Using arrays makes computations efficient and simple, but same elements are computed multiple times because of symmetry in the moments.

References

• "On Moments of Folded and Truncated Multivariate Normal Distributions" by Raymond Kan & Cesare Robotti, 2016

• "Integrals over Gaussians under Linear Domain Constraints" by Alexandra Gessner & Oindrila Kanjilal & Philipp Hennig, 2020