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Discussion led by Chunping Wang ECE, Duke University July 10, 2009

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Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data

Peter D. Hoffto appear in Journal of Computational and Graphical Statistics

Discussion led by Chunping Wang

ECE, Duke University

July 10, 2009

- Introduction and Motivations
- Sampling from the Vector Von Mises-Fisher (vMF) Distribution (existing method)
- Sampling from the Matrix Von Mises-Fisher (mMF) Distribution
- Sampling from the Bingham-Von Mises-Fisher (BMF) Distribution
- One Example
- Conclusions

1/21

Stiefel manifold: set of rank- orthonormal matrices, denoted

The matrix Bingham-von Mises-Fisher distribution

The matrix von Mises-Fisher distribution – linear term

The matrix Bingham distribution – quadratic term

2/21

Sampling orthonormal matrices from distributions is useful for many applications.

Examples:

- Factor analysis

observed matrix

latent

latent

Given uniform priors over Stiefel manifold,

3/21

- Principal components

observed matrix, with each row

with

Eigen-value decomposition

Likelihood

Posterior with respect to uniform prior

4/21

- Network data

, symmetric binary observed matrix, with the 0-1

indicator of a link between nodes i and j.

E: symmetric matrix of independent standard normal noise

Posterior with respect to uniform prior

5/21

the modal vector;

, concentration parameter

A distribution on the -sphere in

constant distribution for any given angle

defines the modal direction.

6/21

(1) A simple direction

( Proposal envelope )

(2) An arbitrary direction

For a fixed orthogonal matrix ,

7/21

Rejection sampling scheme 1: uniform envelope

rejection region

a bound

Sample

accept

when

Acceptance region

Extremely inefficient

8/21

Y

Y

Y

Rejection sampling scheme 2: based on sampling from vMF

Proposal samples are drawn from vMF density functions with parameter , constrained to be orthogonal to other columns of .

9/21

Y

Y

Y

Rejection sampling scheme 2: based on sampling from vMF

Proposal samples are drawn from vMF density functions with parameter , constrained to be orthogonal to other columns of .

9/21

Y

Y

Y

Rejection sampling scheme 2: based on sampling from vMF

Proposal samples are drawn from vMF density functions with parameter , constrained to be orthogonal to other columns of .

Rotate the modal direction

9/21

Y

Y

Y

Rejection sampling scheme 2: based on sampling from vMF

Rotate the sample to be orthogonal to the previous columns

9/21

Y

Y

Y

Rejection sampling scheme 2: based on sampling from vMF

Proposal distribution

9/21

Rejection sampling scheme 2: based on sampling from vMF

Sample scheme:

10/21

A Gibbs sampling scheme

Sample iteratively

- Note that . When .
- remedy: sampling two columns at a time
- Non-orthogonality among the columns of add to the autocorrelation in the Gibbs sampler.
- remedy: performing the Gibbs sampler on

11/21

The vector Bingham distribution

12/21

The vector Bingham distribution

12/21

The vector Bingham distribution

Better mixing

12/21

The vector Bingham distribution

From variable substitution, rejection sampling or grid sampling

12/21

The vector Bingham distribution

The density is symmetric about zero

12/21

The vector Bingham-von Mises-Fisher distribution

The density is not symmetric about zero any more, is no longer uniformly distributed on . The update of and should be done jointly. The modified step 2(b) and 2(c) are:

13/21

The matrix Bingham-von Mises-Fisher distribution

Rewrite

14/21

The matrix Bingham-von Mises-Fisher distribution

Sample two columns at a time

Parameterize 2-dimensional orthonormal matrices as

Uniform pairs on the circle

Uniform

15/21

The matrix Bingham-von Mises-Fisher distribution

16/21

17/21

, symmetric binary observed matrix, with the 0-1

indicator of a link between nodes i and j.

E: symmetric matrix of independent standard normal noise

Posterior with respect to uniform prior

BMF distribution with

18/21

Samples from two independent Markov chains with different starting values

19/21

20/21

- The sampling scheme of a family of exponential distributions over the Stiefel manifold was developed;
- This enables us to make Bayesian inference for those orthonormal matrices and incorporate prior information during the inference;
- The author mentioned several application and implemented the sampling scheme on a network data set.

21/21

- Andrew T. A. Wood. Simulation of the von Mises Fisher distribution. Comm. Statist. Simulation Comput., 23:157-164, 1994
- G. Ulrich. Computer generation of distributions on the m-sphere. Appl. Statist., 33, 158-163, 1984
- J. G. Saw. A family of distributions on the m-sphere and some hypothesis tests. Biometrika, 65, 69-74, 1978