CS B553: ALGORITHMS FOR OPTIMIZATION AND LEARNINGContinuous Probability Distributions and Bayesian Networks with Continuous Variables

AGENDA

Continuous probability distributions Common families:

The Gaussian distribution Linear Gaussian Bayesian networks

CONTINUOUS PROBABILITY DISTRIBUTIONS

Let X be a random variable in R, P(X) be a probability distribution over X P(x)0 for all x, “sums to 1”

Challenge: (most of the time) P(X=x) = 0 for any x

CDF AND PDF Probability density function (pdf) f(x)

Nonnegative, f(x) dx = 1 Cumulative distribution function (cdf) g(x)

g(x) = P(Xx)

g(-) = 0, g() = 1, g(x) = (-,x] f(y) dy, monotonic

f(x) = g’(x)

pdf f(x)

cdf g(x)1

Both cdfs and pdfs are complete representations of the probability

space over X, but usually pdfs are more intuitive to work with.

CAVEATS

pdfs may exceed 1 Deterministic values, or ones taking on a few

discrete values, can be represented in terms of the Dirac delta function a(x) pdf (an improper function) a(x) = 0 if x a a(x) = if x = a a(x) dx = 1

COMMON DISTRIBUTIONS

Uniform distribution U(a,b) p(x) = 1/(b-a) if x [a,b], 0 otherwise P(Xx) = 0 if x < a, (x-a)/(b-a) if x [a,b], 1

otherwise Gaussian (normal) distribution N(,)

= mean, = standard deviation

P(Xx) not closed form:(1+erf(x))/2 for N(0,1)

U(a1,b1)

U(a2,b2)

MULTIVARIATE CONTINUOUS DISTRIBUTIONS

Consider c.d.f. g(x,y) = P(Xx,Yy) g(-,y) = 0, g(x,-) = 0 g(,) = 1 g(x,) = P(Xx), g(,x) = P(Yy) g monotonic

Its joint density is given by the p.d.f. f(x,y) iff g(p,q) = (-,p] (-,q] f(x,y) dy dx

i.e. P(axXbx,ayYby) = [ax,bx] [ay,by] f(x,y) dy dx

MARGINALIZATION WORKS OVER PDFS Marginalizing f(x,y) over y:

If h(x) = (-,) f(x,y) dy, then h(x) is a p.d.f. for P(Xx) Proof:

P(Xa) = P(Xa,Y) = g(a,) = (-,a] (-,) f(x,y) dy dx h(a) = d/da P(Xa)

= d/da (-,a] (-,) f(x,y) dy dx (definition)= (-,) f(a,y) dy (fundamental theorem of

calculus)

So, the joint density contains all information needed to reconstruct the density of each individual variable

CONDITIONAL DENSITIES

We might want to represent the density P(Y|X=x)… but how? Naively, P(aYb|X=x) = P(aYb,X=x)/P(X=x), but

denominator is 0! Consider pdf p(x,y), consider taking limit

P(aYb|x+eXx+e) as e0

So p(x,y)/p(x) is the conditional density

TRANSFORMATIONS OF CONTINUOUS RANDOM VARIABLES

Suppose we want to compute the distribution of f(X), where X is a random variable distributed w.r.t. pX(x) Assume f is monotonic and invertible

Consider Y=f(X) a random variable P(Yy) = I[f(x)y]pX(x) dx

= = P(X f-1(y)) pY(y) = d/dy P(Yy)

= d/dy P(X f-1(y)) = p(f-1(y)) d/dy f-1(y) by chain rule

= pX(f-1(y))/f ’(f-1(y)) by inverse function derivative

NOTES:

In general, continuous multivariate distributions are hard to handle exactly

But, there are specific classes that lead to efficient exact inference techniques In particular, Gaussians

Other distributions usually require resorting to Monte Carlo approaches

MULTIVARIATE GAUSSIANS

Multivariate analog in N-D space Mean (vector) m, covariance (matrix) S

With a normalization factor

X ~ N(m,S)

INDEPENDENCE IN GAUSSIANS

If X ~ N(mX,SX) and Y ~ N(mY,SY) are independent, then

Moreover, if X~N( ,m S), then Sij=0 iff Xi and Xj are independent

LINEAR TRANSFORMATIONSLinear transformations of gaussians

If X~ N(m,S), y = A x + bThen Y ~ N(Am+b, ASAT) In fact,

Consequence: If X ~ N(mx,Sx), Y ~ N(my,Sy), Z=X+YThen Z ~ N(mx+my,Sx+Sy)

MARGINALIZATION AND CONDITIONING If (X,Y) ~ N([mX mY],[SXX,SXY;SYX,SYY]), then:

Marginalization Summing out Y gives

X ~ N(mX , SXX)

Conditioning:On observing Y=y, we have

X ~ N(mX-SXYSYY-1(y-mY), SXX-SXYSYY

-1SYX)

LINEAR GAUSSIAN MODELS

A conditional linear Gaussian model has : P(Y|X=x) = N(0+Ax,S0) With parameters 0, A, and S0

LINEAR GAUSSIAN MODELS

A conditional linear Gaussian model has : P(Y|X=x) = N(0+Ax,S0) With parameters 0, A, and S0

If X ~ N(mX,SX), then joint distribution over is given by:

(Recall the linear transformation rule)

If X~ N(m,S) and y=Ax+b, thenY ~ N(Am+b, ASAT)

CLG BAYESIAN NETWORKS

If all variables in a Bayesian network have Gaussian or CLG CPTS, inference can be done efficiently!

X1

Y

X2

P(Y|x1,x2) = N(ax1+bx2,y)

P(X2) = N(2,2)P(X1) = N(1,1)

P(Z|x1,y) = N(c+dx1+ey,z)Z

CANONICAL REPRESENTATION All factors in a CLG Bayes net can be represented

as C(x;K,h,g) withC(x;K,h,g) = exp(-1/2 xT Kx + hTx + g)

Ex: if P(Y|x) = N(0+Ax,S0) thenP(y|x) = 1/Z exp(-1/2 (y-Ax-0)T S0

-1(y-Ax-0))=1/Z exp(-1/2 (y,x)T [I –A]T S0

-1 [I –A](y,x) + 0T

S0-1 [I –A](y,x) – ½ 0

TS0-10)

Is of form C((y,x);K,h,g) with K = [I –A]T S0

-1 [I –A]

h = [I –A]T S0

-1 0

g = log(1/Z) exp (–½ 0TS0

-10)

PRODUCT OPERATIONS

C(x;K1,h1,g1)C(x;K2,h2,g2) = C(x;K,h,g) with K=K1+K2

h=h1+h2

g = g1+g2

If the scopes of the two factors are not equivalent, just extend the K’s with 0 rows and columns, and h’s with 0 rows so that each row/column matches

SUM OPERATION C((x,y);K,h,g)dy = C(x;K’,h’,g’)with

K’=KXX-KXYKYY-1KYX

h’=hX-KXYKYY-1hY

g’ = g+1/2 (log|2pKYY-1|+hY

TKYY-1hY)

Using these two operations we can implement inference algorithms developed for discrete Bayes nets: Top-down inference, variable elimination (exact) Belief propagation (approximate)

MONTE CARLO WITH GAUSSIANS

Assume sample X ~ N(0,1) is given as a primitive RandN() To sample X ~ N(m,s2), simply + m s RandN()

How to generate a random multivariate Gaussian variable N( , )m S ?

MONTE CARLO WITH GAUSSIANS

Assume sample X ~ N(0,1) is given as a primitive RandN() To sample X ~ N(m,s2), simply set x + m s

RandN() How to generate a random multivariate

Gaussian variable N( , )m S ? Take Cholesky decomposition: S-1=LLT, L

invertible if S is positive definite Let y = LT(x-m) P(y) exp(-1/2 (y1

2 + … + yN2)) is isotropic, and

each yi is independent Sample each component of y at random Set x L-Ty+m

MONTE CARLO WITH LIKELIHOOD WEIGHTING

Monte Carlo with rejection has probability 0 of finding a continuous value given as evidence, so likelihood weighting must be used

X

Y=y

P(Y|x)=N(Ax+mY,SY)

P(X)=N(mX,SX)

Step 1: Sample x ~ N(mX,SX)Step 2: weight by P(y|x)

HYBRID NETWORKS

Hybrid networks combine both discrete and continuous variables

Exact inference techniques are hard to apply Result in Gaussian mixtures NP hard even in polytree networks

Monte Carlo techniques apply in straightforward way

Belief approximation can be applied (e.g., collapsing Gaussian mixtures to single Gaussians)

ISSUES

Non-gaussian distributions Nonlinear dependencies More in future lectures on particle filtering