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Algorithms to compute the probability that a Wishart-distributed matrix lies within an interval.

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Confidence Intervals for Wishart Random Matrices

This is a Python implementation of Algorithm 1 from (Chiani, 2017) to compute the probability that the eigenvalues of a standard Wishart-distributed random matrix lie within a given interval. This immediately also gives us the cumulative density functions for the distributions of the minimum and maximum eigenvalues. In addition, we provide optimization routines to evaluate the associated quantile functions. See the accompanying blog post.

Usage

Use Python 3. Ensure you have the dependencies listed in requirements.txt (this is just numpy, scipy, and matplotlib), then run the script using

python3 wishart_confidence_intervals.py

By default, the script replicates a few results from Chiani's papers and runs an example of confidence bounds for non-standard Wishart matrices. Feel free to play around with it.

Limitations

The Python implmentation uses standard floating point precision and so runs into numerical challenges when computing the results for matrices larger than, say, 10 by 10 or so.

References

  • Chiani, M. (2017). "On the Probability That All Eigenvalues of Gaussian, Wishart, and Double Wishart Random Matrices Lie Within an Interval," in IEEE Transactions on Information Theory, vol. 63, no. 7, pp. 4521-4531, July 2017, doi: 10.1109/TIT.2017.2694846, arxiv: 1502.04189.

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