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# 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`,
``` python
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
|
