Simulation of the matrix Bingham-von Mises- Fisher distribution, with applications to multivariate...

Simulation of the matrix Bingham-von Mises-

Fisher distribution, with applications to

multivariate and relational data

Discussion led by Chunping Wang

ECE, Duke University

July 10, 2009

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

Outline

• 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

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Introduction

The matrix Bingham distribution – quadratic term

The matrix von Mises-Fisher distribution – linear term

}{etr)|( XCCX TMFp

}{etr),|( AXBXBAX TBp

The matrix Bingham-von Mises-Fisher distribution

},{etr),,|( AXBXXCCBAX TTBMFp

0B0A ,

0C

Stiefel manifold: set of rank- orthonormal matrices, denotedRmR

X

2/21

MotivationsSampling orthonormal matrices from distributions is useful for many applications.

Examples:

• Factor analysis

),0(~ 2Niid

latent

latent

Given uniform priors over Stiefel manifold,

observed matrixpnR Y

UV

}{etr)|( XCCX Tp

3/21

Motivations

• Principal components

observed matrix, with each row

}2/)({etr),|( 1 TTp YUUΛYΛUY

),(~ Σ0y p

iid

i NTUUΛΣ Eigen-value decomposition

Likelihood

}2/)({etr),|( 1 UYYUΛΛYU TTp

Posterior with respect to uniform prior

pnR Y

with U

}{etr),|( AXBXBAX Tp

4/21

Motivations

• Network data, symmetric binary observed matrix, with the 0-1 mm:Y ijy

indicator of a link between nodes i and j.

U

}2/{etr),|( ZUUΛΛZU Tp


E: symmetric matrix of independent standard normal noise

}{etr),|( AXBXBAX Tp

5/21

Sampling from the vMF Distribution (wood, 1994)

},{exp),|( xξξx TMFp ,mSx

,mSξ the modal vector;

)cos()cos(|||||||| xξxξT

constant distribution for any given angle

, concentration parameter

A distribution on the -sphere in )1( m mR

ξx

defines the modal direction. ξ

6/21

http://upload.wikimedia.org/wikipedia/en/2/29/Von_mises_fisher.png

Sampling from the vMF Distribution (wood, 1994)

),(fromsampleabetoprovedis];1[ 2 ξvx MFww

).exp()1()(fromsampleaiswhen 2/)3(2 wwwfw m

( Proposal envelope ))1(2/)3(2 })1()1{()1()( mm wbbwwg

mTT x xξξ ],1,0,,0[ (1) A simple direction

,ddistributeuniformlyFor 1 mSv

For a fixed orthogonal matrix ,

P ).(~ PξPx MF

7/21

mm

(2) An arbitrary direction

Sampling from the mMF Distribution

Rejection sampling scheme 1: uniform envelope

XXCCX },{etr)|( TMFp

Mg

pMF )(

)|(

X

CX

X

)(XMg

)|( CXMFp

Acceptance region

rejection region

,)(

)|(

X

CX

Mg

pu MF accept

Sample )1,0(~),(~ Uug XX

when X

a bound

Extremely inefficient

u

0

1

8/21


Rejection sampling scheme 2: based on sampling from vMF

Y

Y

Y

9/21

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

][,rYY



Y

Y

Y

0)]1,,1([, rr YN

9/21


][,rYY



Y

Y

Y

Rotate the modal direction

9/21


][,rYY



Y

Y

Y Rotate the sample to be orthogonal to the previous columns

9/21


][,rYY




][,rYY

Y

Y

Y

}{etr)()|()|()(11

][,][,1

)]1,,1([,][, YHNHNYNYYY TR

rr

R

rr

Trr

TrMF

R

rrr Fppg

Proposal distribution

9/21



Sample scheme:

10/21


A Gibbs sampling scheme

Sample iteratively )|(~ ][,][,][, rrr p XXX

)1( RmSz• 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

}1,1{, zRm

C

TMF YVXUDY with),(~11/21

Sampling from the BMF Distribution

The vector Bingham distribution

symmetric, A

;2iy

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symmetric, A

;2iy

)exp(),|(1

2

m

iii yp ΛEy

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symmetric, A

;2iy Better mixing

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symmetric, A

;2iy

From variable substitution, rejection sampling or grid sampling

12/21



symmetric, A

;2iy

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:

is}1,1{

13/21

q, is

)exp(),,|(1

2

m

iii

T yp yddΛEy

),|( iip sq


The matrix Bingham-von Mises-Fisher distribution )( Rm

14/21

},{etr),,|( AXBXXCCBAX TTBMFp

},{ ]1[, XNzXRewrite

)exp()|( 1,1]1[,]1[, ANzNzNzCXz TTT bp



15/21

},{ )]2,1([, XNzXSample two columns at a time

Parameterize 2-dimensional orthonormal matrices as

]2[,)(Z

]1[,)(Z

1s 1s

]1[,)(Z

]2[,)(Z

Uniform pairs on the circle

Uniform )2,0(

)),((),( spsp Z



16/21

Example: Eigenmodel estimation for network data

17/21


indicator of a link between nodes i and j.

}2/{etr),|( ZUUΛΛZU Tp


, symmetric binary observed matrix, with the 0-1 mm:Y ijy

18/21

UE: symmetric matrix of independent standard normal noise

BMF distribution with 0CΛBZA ,,2/

270,3 mR

Samples from two independent Markov chains with different starting values


19/21


20/21

Conclusions

• 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

References

• 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

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