I have a Multivariate Normal Distribution with the mean vector and the covariance matrix given as Now, I want to compute the probability that a realization of lies in a given polytopic set of the form where the matrix and the vector describes half-spaces and therefore a convex set. The probability which I want to compute is given as How can I compute/approximate this integration numerically in MATLAB for a given mean , a given covariance , and a given set . In my problem, the variable has around 10 to 100 dimensions. Edit: ->

Compute Probability of a Multivariate Normal Distribution over Polytope

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Michael Fink 2022-6-29

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评论： Torsten 2022-7-4

I have a Multivariate Normal Distribution with the mean vector

and the covariance matrix

given as

Now, I want to compute the probability that a realization of

lies in a given polytopic set of the form

where the matrix

and the vector

describes m half-spaces and therefore a convex set.

The probability which I want to compute is given as

How can I compute/approximate this integration numerically in MATLAB for a given mean μ, a given covariance Σ, and a given set 𝒫. In my problem, the variable x has around 10 to 100 dimensions.

Edit:

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Torsten 2022-6-29

编辑：Torsten 2022-7-1

It could be quite time-consuming, but maybe Monte-Carlo integration is an option:

https://en.wikipedia.org/wiki/Monte_Carlo_integration

Michael Fink 2022-6-30

Thank you @Torsten for the idea

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回答（1 个）

Paul 2022-6-30

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Hi Michael,

If A is nonsingular, perhaps a change of coordinates will work

% z = A*x
muz = A*mux;
Sigmaz = A*Sigmax*A.';
ProbAxLTb = mvncdf(-inf+b,b,muz,Sigmaz);

See the doc page for mvncdf for examples, info, options, etc.

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Michael Fink 2022-7-4

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Hello Paul,

yes, exactly, this is the probability. But in my case, the Polytope (i.e. matrix A and vector b) is given/is deterministic and not random.

For example an unitbox for n=2 (Vertecise: (1,1),(-1,1),(1,-1),(-1,1) ) would be given as

A = [eye(2);-eye(2)]
A = 4×2
     1     0
     0     1
    -1     0
     0    -1
b = ones(4,1)
b = 4×1
     1
     1
     1
     1

Torsten 2022-7-4

It was only an example.

Of course, you can directly use your values for sigma, mu, A and b in Paul's code instead of the randomly created.

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