

Journal of Machine Learning Research 5 (2004) 819-844 Submitted 1/04; Published 7/04

Probability Product Kernels Tony Jebara JEBARA@CS.COLUMBIA.EDURisi Kondor

RISI@CS.COLUMBIA.EDUAndrew Howard AHOWARD@CS.COLUMBIA.EDUComputer Science Department

Columbia UniversityNew York, NY 10027, USA

Editors: Kristin Bennett and Nicol`o Cesa-Bianchi

Abstract The advantages of discriminative learning algorithms and kernel machines are combined with gen-erative modeling using a novel kernel between distributions. In the probability product kernel, data

points in the input space are mapped to distributions over the sample space and a general innerproduct is then evaluated as the integral of the product of pairs of distributions. The kernel is straightforward to evaluate for all exponential family models such as multinomials and Gaussiansand yields interesting nonlinear kernels. Furthermore, the kernel is computable in closed form for latent distributions such as mixture models, hidden Markov models and linear dynamical sys-tems. For intractable models, such as switching linear dynamical systems, structured mean-field approximations can be brought to bear on the kernel evaluation. For general distributions, even ifan analytic expression for the kernel is not feasible, we show a straightforward sampling method to evaluate it. Thus, the kernel permits discriminative learning methods, including support vectormachines, to exploit the properties, metrics and invariances of the generative models we infer from each datum. Experiments are shown using multinomial models for text, hidden Markov models forbiological data sets and linear dynamical systems for time series data.

Keywords: Kernels, support vector machines, generative models, Hellinger divergence, Kullback-Leibler divergence, Bhattacharyya affinity, expected likelihood, exponential family, graphical models, latent models, hidden Markov models, dynamical systems, mean field

1. Introduction Recent developments in machine learning, including the emergence of support vector machines,have rekindled interest in kernel methods (Vapnik, 1998; Hastie et al., 2001). Typically, kernels

are designed and used by the practitioner to introduce useful nonlinearities, embed prior knowledgeand handle unusual input spaces in a learning algorithm (Sch "olkopf and Smola, 2001). In this article, we consider kernels between distributions. Typically, kernels compute a generalized innerproduct between two input objects x and x 0 which is equivalent to apply a mapping function \Phi  to each object and then computing a dot product between \Phi (x ) and \Phi (x 0) in a Hilbert space. Wewill instead consider the case where the mapping \Phi 

(x ) is a probability distribution p(x|x ), thusrestricting the Hilbert space since the space of distributions is trivially embedded in the Hilbert space

of functions. However, this restriction to positive normalized mappings is in fact not too limitingsince general \Phi 

(x ) mappings and the nonlinear decision boundaries they generate can often bemimicked by a probabilistic mapping. However, a probabilistic mapping facilitates and elucidates

cfl2004 Tony Jebara, Risi Kondor and Andrew Howard.

JEBARA, KONDOR AND HOWARD kernel design in general. It permits kernel design to leverage the large body of tools in statistical andgenerative modeling including Bayesian networks (Pearl, 1997). For instance, maximum likelihood or Bayesian estimates may be used to set certain aspects of the mapping to Hilbert space.

In this paper, we consider the mapping from datum x to a probability distribution p(x|x ) whichhas a straightforward inner product kernel k

(x , x 0) = R pr(x|x )pr(x|x 0)dx (for a scalar r which wetypically set to 1 or 1 /2). This kernel is merely an inner product between two distributions and, ifthese distributions are known directly, it corresponds to a linear classifier in the space of probability

distributions. If the distributions themselves are not available and only observed data is given, wepropose using either Bayesian or Frequentist estimators to map a single input datum (or multiple inputs) into probability distributions and subsequently compute the kernel. In other words, we mappoints as x ! p

(x|x ) and x 0 ! p(x|x 0). For the setting of r = 1/2 the kernel is none other than theclassical Bhattacharyya similarity measure (Kondor and Jebara, 2003), the affinity corresponding to

Hellinger divergence (Jebara and Kondor, 2003).1

One goal of this article is to explore a contact point between discriminative learning (supportvector machines and kernels) and generative learning (distributions and graphical models). Discriminative learning directly optimizes performance for a given classification or regression task.Meanwhile, generative learning provides a rich palette of tools for exploring models, accommodating unusual input spaces and handling priors. One approach to marrying the two is to estimateparameters in generative models with discriminative learning algorithms and optimize performance on a particular task. Examples of such approaches include conditional learning (Bengio and Fras-coni, 1996) or large margin generative modeling (Jaakkola et al., 1999). Another approach is to use kernels to integrate the generative models within a discriminative learning paradigm. The pro-posed probability product kernel falls into this latter category. Previous efforts to build kernels that accommodate probability distributions include the Fisher kernel (Jaakkola and Haussler, 1998),the heat kernel (Lafferty and Lebanon, 2002) and kernels arising from exponentiating KullbackLeibler divergences (Moreno et al., 2004). We discuss, compare and contrast these approaches tothe probability product kernel in Section 7. One compelling feature of the new kernel is that it is straightforward and efficient to compute over a wide range of distributions and generative mod-els while still producing interesting nonlinear behavior in, for example, support vector machine (SVM) classifiers. The generative models we consider include Gaussians, multinomials, the ex-ponential family, mixture models, hidden Markov models, a wide range of graphical models and (via sampling and mean-field methods) intractable graphical models. The flexibility in choosing adistribution from the wide range of generative models in the field permits the kernel to readily accommodate many interesting input spaces including sequences, counts, graphs, fields and so forth.It also inherits the properties, priors and invariances that may be straightforward to design within a generative modeling paradigm and propagates them into the kernel learning machine. This articlethus gives a generative modeling route to kernel design to complement the family of other kernel engineering methods including super-kernels (Ong et al., 2002), convolutional kernels (Haussler,1999; Collins and Duffy, 2002), alignment kernels (Watkins, 2000), rational kernels (Cortes et al., 2002) and string kernels (Leslie et al., 2002; Vishawanathan and Smola, 2002).

This paper is organized as follows. In Section 2, we introduce the probability product kerneland point out connections to other probabilistic kernels such as Fisher kernels and exponentiated

Kullback-Leibler divergences. Estimation methods for mapping points probabilistically to Hilbert

1. This has interesting connections to Kullback-Leibler divergence (Topsoe, 1999).

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PROBABILITY PRODUCT KERNELS space are outlined. In Section 3 we elaborate the kernel for the exponential family and show itsspecific form for classical distributions such as the Gaussian and multinomial. Interestingly, the kernel can be computed for mixture models and graphical models in general and this is elaboratedin Section 4. Intractable graphical models are then handled via structured mean-field approximations in Section 6. Alternatively, we describe sampling methods for approximately computing thekernel. Section 7 discusses the differences and similarities between our probability product kernels and previously presented probabilistic kernels. We then show experiments with text data setsand multinomial kernels, with biological sequence data sets using hidden Markov model kernels, and with continuous time series data sets using linear dynamical system kernels. The article thenconcludes with a brief discussion.

2. A Kernel Between Distributions Given a positive (semi-)definite kernel k : X *X 7! R on the input space X and examples x1, x2, . . . , xm 2X with corresponding labels y

1, y2, . . . , ym, kernel based learning algorithms return a hypothesis ofthe form h(x ) = a*m i=1 ai k(xi, x ) + b. The role of the kernel is to capture our prior assumptions ofsimilarity in X . When k(x , x 0) is large, we expect that the corresponding labels be similar or the

same.Classically, the inputs {x

i}mi=1 are often represented as vectors in Rn, and k is chosen to be oneof a small number of popular positive definite functions on Rn, such as the Gaussian RBF

k(x , x 0) = e-kx -x 0 k

2/(2s2). (1)

Once we have found an appropriate kernel, the problem becomes accessible to a whole array ofnon-parametric discriminative kernel based learning algorithms, such as support vector machines, Gaussian processes, etc. (Sch"olkopf and Smola, 2002).In contrast, generative models fit probability distributions p

(x ) to x1, x2, . . . , xm and base theirpredictions on the likelihood under different models. When faced with a discriminative task, approaching it from the generative angle is generally suboptimal. On the other hand, it is often easierto capture the structure of complex objects with generative models than directly with kernels. Specific examples include multinomial models for text documents, hidden Markov models for speech,Markov random fields for images and linear dynamical systems for motion. In the current paper we aim to combine the advantages of both worlds by

1. fitting separate probabilistic models p1(x), p2(x), . . . pm(x) to x1, x2, . . . , xm; 2. defining a novel kernel kprob(p, p0) between probability distributions on X ; 3. finally, defining the kernel between examples to equal kprob between the corresponding distri-butions:

k(x , x 0) = kprob(p, p0). We then plug this kernel into any established kernel based learning algorithm and proceed as usual.We define kprob in a rather general form and then investigate special cases.

Definition 1 Let p and p0 be probability distributions on a space X and r be a positive constant.Assume that pr

, p0r 2 L2(X ), i.e. that RX p(x)2rdx and RX p0(x)2rdx are well defined (not infinity).

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JEBARA, KONDOR AND HOWARD The probability product kernel between distributions p and p0 is defined

kprob(p, p0) = ZX p(x)r p0(x)r dx = \Omega pr, p0rffL2 . (2) It is well known that L2(X ) is a Hilbert space, hence for any set P of probability distributionsover X such that R

X p(x)2rdx is finite for any p2P , the kernel defined by (2) is positive definite. Theprobability product kernel is also a simple and intuitively compelling notion of similarity between

distributions.

2.1 Special Cases and Relationship to Statistical Affinities For r = 1/2

k(p, p0) = Z pp(x)pp0(x) dx,

which we shall call the Bhattacharyya kernel, because in the statistics literature it is known asBhattacharyya's affinity between distributions (Bhattacharyya, 1943), related to the better-known

Hellinger's distance

H(p, p0) = 12 Z ipp(x) - pp0(x)j2 dx

by H(p, p0) = p2 - 2k(p, p0) . Hellinger's distance can be seen as a symmetric approximation tothe Kullback-Leibler (KL) divergence, and in fact is a bound on KL, as shown in (Topsoe, 1999), where relationships between several common divergences are discussed. The Bhattacharyya kernelwas first introduced in (Kondor and Jebara, 2003) and has the important special property k

(x , x ) = 1.When r = 1, the kernel takes the form of the expectation of one distribution under the other:

k(x , x 0) = Z p(x) p0(x) dx = Ep[p0(x)] = Ep0 [p(x)]. (3) We call this the expected likelihood kernel.It is worth noting that when dealing with distributions over discrete spaces X

= {x1, x2, . . .},probability product kernels have a simple geometrical interpretation. In this case p can be represented as a vector p = (p1, p2, . . .) where pi = Pr (X = xi). The probability product kernel is thenthe dot product between ~p

= \Gamma pr1, pr2, . . .\Delta  and ~p0 = \Gamma p0r1, p0r2, . . .\Delta .In particular, in the case of the Bhattacharyya kernel, ~p and ~p0 are vectors on the unit sphere.

This type of similarity measure is not unknown in the literature. In text recognition, for exam-ple, the so-called "cosine-similarity" (i.e. the dot product measure) is well entrenched, and it has been observed that preprocessing word counts by taking square roots often improves performance(Goldzmidt and Sahami, 1998; Cutting et al., 1992). Lafferty and Lebanon also arrive at a similar kernel, but from a very different angle, looking at the diffusion kernel on the statistical manifold ofmultinomial distributions (Lafferty and Lebanon, 2002).

2.2 Frequentist and Bayesian Methods of Estimation Various statistical estimation methods can be used to fit the distributions p1, p2, . . . , pm to the exam-ples x

1, x2, . . . , xm. Perhaps it is most immediate to consider a parametric family {pq(x)}, take themaximum likelihood estimators ^q

i = arg maxq pq(xi), and set pi(x) = p^qi(x).

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PROBABILITY PRODUCT KERNELS

T (x) A(x) K (q) Gaussian ( q = u, s2 = 1 ) x - 12 xTx - D2 log(2p) 12 qTqGaussian ( q

= s2, u = 0 ) x - 12 log(2p) - 12 log qExponential ( q

= -1/b ) x 0 - log(-q)Gamma (q = a) log x - log x - x log \Gamma (q)Poisson ( q

= log p ) x - log(x!) exp qMultinomial ( q

i = log ai ) (x1, x2, . . .) log\Gamma \Gamma a*i xi\Delta !/\Gamma O~i xi!\Delta \Delta  1

Table 1: Some well-known exponential family distributions

The alternative, Bayesian, strategy is to postulate a prior on q, invoke Bayes' rule

p(q|x ) = p(x |q) p(q)R p(x |q) p(q) dq , and then use either the distribution p(x| ^qMAP) based on the maximum a posteriori estimator ^qMAP =arg max

q p(q|x ), or the true posterior

p(x|x ) = Z p(x|q) p(q|x ) dq. (4)

At this point the reader might be wondering to what extent it is justifiable to fit distributions tosingle data points. It is important to bear in mind that p

i(x) is but an intermediate step in forming thekernel, and is not necessarily meant to model anything in and of itself. The motivation is to exploit

the way that probability distributions capture similarity: assuming that our model is appropriate,if a distribution fit to one data point gives high likelihood to another data point, this indicates that the two data points are in some sense similar. This is particularly true of intricate graphical modelscommonly used to model structured data, such as times series, images, etc.. Such models can capture relatively complicated relationships in real world data. Probability product kernels allowkernel methods to harness the power of such models.

When x is just a point in Rn with no further structure, probability product kernels are less com-pelling. Nevertheless, it is worth noting that even the Gaussian RBF kernel (1) can be regarded as a probability product kernel (by setting setting r = 1 and choosing pi(x) to be a s/2 varianceGaussian fit to x

i by maximum likelihood). In particular situations when the point x does not yielda reliable estimate of p (typically because the class of generative models is too flexible relative to

the datum), we may consider regularizing the maximum likelihood estimate of each probability byfitting it to the neighbourhood of a point or by employing other more global estimators of the densities. Other related methods that use points to estimate local distributions including kernel densityestimation (KDE) where the global density is composed of local density models centered on each datum (Silverman, 1986). We next discuss a variety of distributions (in particular, the exponentialfamily) where maximum likelihood estimation is well behaved and the probability product kernel is straightforward to compute.

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JEBARA, KONDOR AND HOWARD 3. Exponential Families A family of distributions parameterized by q 2 RD is said to form an exponential family if its mem-bers are of the form

pq(x) = exp(A(x) + qT T (x) - K (q)). Here A is called the measure, K is the the cumulant generating function and T are the sufficientstatistics.

Some of the most common statistical distributions, including the Gaussian, multinomial, Pois-son and Gamma distributions, are exponential families (Table 1). Note that to ensure that p

q(x) isnormalized, A, K and q must be related through the Laplace transform

K (q) = log Z exp \Gamma A(x) + qT T (x)\Delta  dx. The Bhattacharyya kernel (r=1/2) can be computed in closed form for any exponential family:

k(x , x 0) = k(p, p0) = Zx pq(x)1/2 pq0 (x)1/2 dx

= Zx exp iA(x) + \Gamma  12 q + 12 q0\Delta T T (x) - 12 K (q) - 12 K (q0)j dx = exp \Gamma K \Gamma  12 q + 12 q0\Delta  - 12 K (q) - 12 K (q0)\Delta  . For r6=1/2, k(x , x 0) can only be written in closed form when A and T obey some special properties.Specifically, if 2rA

(x) = A(hx) for some h and T is linear in x, then

log Z exp i2r A(x) + r \Gamma q + q0\Delta T T (x)j dx =

log ^ 1h Z expiA(hx) + rh \Gamma q + q0\Delta T T (hx)j d(hx) * = K \Gamma  rh \Gamma q + q0\Delta \Delta  - log h, hence k

(p, p0) = exp hK \Gamma  rh \Gamma q + q0\Delta \Delta  - log h - r2 K (q) - r2 K (q0)i .

In the following we look at some specific examples.

3.1 The Gaussian Distribution The D dimensional Gaussian distribution p(x) = N (u, \Sigma ) is of the form

p(x) = (2p)-D/2 | \Sigma  |-1/2 exp \Gamma - 12 (x-u)T \Sigma -1(x-u)\Delta  where \Sigma  is a positive definite matrix and | \Sigma  | denotes its determinant. For a pair of Gaussiansp

= N (u, \Sigma ) and p0 = N (u0, \Sigma 0), completing the square in the exponent gives the general probabilityproduct kernel

Kr(x , x 0) = Kr(p, p0) = ZRD p(x)r p0(x)rdx =

(2p)(1-2r)D/2 r-D/2 fifi\Sigma # fifi1/2fifi \Sigma  fifi-r/2fifi \Sigma 0 fifi-r/2 exp i- r2 iuT \Sigma -1u + u0T \Sigma 0-1u0 - u#T \Sigma #u#jj , (5)

824

PROBABILITY PRODUCT KERNELS where \Sigma # = \Gamma \Sigma -1+\Sigma 0-1\Delta -1 and u# = \Sigma -1u + \Sigma 0-1u0. When the covariance is isotropic and fixed,\Sigma 

= s2I, this simplifies to

Kr(p, p0) = (2r)-D/2 (2ps2)(1-2r)D/2 e-ku-u0 k

2/(4s2/r),

which, for r = 1 (the expected likelihood kernel) simply gives

k(p, p0) = 1(4ps2)D/2 e-ku0-uk

2/(4s2),

recovering, up to a constant factor, the Gaussian RBF kernel (1). 3.2 The Bernoulli Distribution The Bernoulli distribution p(x) = gx(1 - g)1-x with parameter g 2 (0, 1), and its D dimensionalvariant, sometimes referred to as Naive Bayes,

p(x) =

DO~

d=1 g

xdd (1 - gd)1-xd

with g 2 (0, 1)D, are used to model binary x 2{0, 1} or multidimensional binary x 2 {0, 1}D observa-tions. The probability product kernel

Kr(x , x 0) = Kr(p, p0) = a*

x2{0,1}D

DO~ d=1(gdg

0d)rxd ((1 - gd)(1 - g0d))r(1-xd)

factorizes as

Kr(p, p0) =

DO~

d=1 \Theta (gdg

0d)r + (1 - gd)r(1 - g0d)r\Lambda  .

3.3 The Multinomial Distribution The multinomial model

p(x) = s!x1! x2! . . . xD! ax11 ax22 . . . axDD

with parameter vector a = (a1, a2, . . . , aD) subject to a*Dd=1 ad = 1 is commonly used to modeldiscrete integer counts x

= (x1, x2, . . . , xD) with a*Di=1 xi = s fixed, such as the number of occurrencesof words in a document. The maximum likelihood estimate given observations x(1)

, x(2), . . . , x(n) is

^ad = a*

ni=1 x(i)d

a*ni=1 a*Dd=1 x(i)d

.

For fixed s, the Bhattacharyya kernel (r=1/2) can be computed explicitly using the multinomialtheorem:

k(p, p0) = a*

x=(x1,x2,...,xD)

a*i xi=s

s!x 1! x2! . . . xD!

DO~

d=1(ada

0d)xd/2 = " Da*

d=1(ada

0d)1/2#

s

(6)

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JEBARA, KONDOR AND HOWARD which is equivalent to the homogeneous polynomial kernel of order s between the vectors (pa1, pa2, . . . , paD)and

(pa01, pa02, . . . , pa0D). When s is not constant, we can sum over all its possible values

k(p, p0) =

e^a*

s=0 "

Da* d=1(ada

0d)1/2#

s

= 1 -

Da*

d=1(ada

0d)1/2!-

1

or weight each power differently, leading to a power series expansion. 3.4 The Gamma and Exponential Distributions The gamma distribution \Gamma (a, b) parameterized by a > 0 and b > 0 is of the form

p(x) = 1\Gamma (a)ba xa-1 e-x/b. The probability product kernel between p , \Gamma (a, b) and p0 , \Gamma (a0, b0) will then be

kr(p, p0) = \Gamma (a

#) b#a#h

\Gamma (a) ba \Gamma (a0) b0a0 ir where a# = r (a+a0-2) + 1 and 1/b# = r (1/b + 1/b0). When r = 1/2 this simplifies to a# = (a+a0) /2 and 1/b# = (1/b + 1/b0) /2.The exponential distribution p

(x) = 1b e-x/b can be regarded as a special case of the gammafamily with a = 1. In this case

kr(p, p0) = r (1/b + 1/b0)(bb0)r .

4. Latent and Graphical Models We next upgrade beyond the exponential family of distributions and derive the probability productkernel for more versatile generative distributions. This is done by considering latent variables and

structured graphical models. While introducing additional complexity in the generative model andhence providing a more elaborate probability product kernel, these models do have the caveat that estimators such as maximum likelihood are not as well-behaved as in the simple exponential familymodels and may involve expectation-maximization or other only locally optimal estimates. We first discuss the simplest latent variable models, mixture models, which correspond to a single unob-served discrete parent of the emission and then upgrade to more general graphical models such as hidden Markov models.Latent variable models identify a split between the variables in x such that some are observed and are denoted using xo (where the lower-case 'o' is short for 'observed') while others are hidden anddenoted using x

h (where the lower-case 'h' is short for 'hidden'). In the latent case, the probabilityproduct kernel is computed only by the inner product between two distributions p(x

o) and p0(xo)over the observed components, in other words k(p , p0) = R p(xo)p0(xo)dxo. The additional variablesx

h are incomplete observations that are not part of the original sample space (where the datum ordata points to kernelize exist) but an augmentation of it. The desired p(x

o) therefore, involvesmarginalizing away the hidden variables:

p(xo) = a*x

h p(xo

, xh) = a*x

h p(xh)p(xo|xh)

.

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PROBABILITY PRODUCT KERNELS For instance, we may consider a mixture of exponential families model (a direct generalization of theexponential family model) where x

h is a single discrete variable and where p(xo|xh) are exponentialfamily distributions themselves. The probability product kernel is then given as usual between two

such distributions p(xo) and p0(xo) which expand as follows:

k(p, p0) = a*x

o (p(x))

r \Gamma p0(x)\Delta r = a*

xo a*xh p(xh)p(xo|xh)!

r 0@a*

x0h p

0(x0h)p0(xo|x0h)1A

r

.

We next note a slight modification to the kernel which makes latent variable models tractable whenr 6

= 1. This alternative kernel is denoted ~k(p, p0) and also satisfies Mercer's condition using the sameline of reasoning as was employed for k

(p, p0) in the previous sections. Essentially, for ~k(p, p0) wewill assume that the power operation involving r is performed on each entry of the joint probability

distribution p(xo, xh) instead of on the marginalized p(xo) alone as follows:

~k(p, p0) = a*

xo a*xh (p(xh)p(xo|xh))

r a*

x0h \Gamma p

0(x0h)p0(xo|x0h)\Delta r .

While the original kernel k(p, p0) involved the mapping to Hilbert space given by x ! p(xo), theabove ~k

(p, p0) corresponds to mapping each datum to an augmented distribution over a more com-plex marginalized space where probability values are squashed (or raised) by a power of r. Clearly,

k(p, p0) and ~k(p, p0) are formally equivalent when r = 1. However, for other settings of r, we preferhandling latent variables with ~k

(p, p0) (and at times omit the , symbol when it is self-evident) sinceit readily accommodates efficient marginalization and propagation algorithms (such as the junction

tree algorithm (Jordan and Bishop, 2004)).One open issue with latent variables is that the ~k kernels that marginalize over them may produce different values if the underlying latent distributions have different joint distributions over hiddenand observed variables even though the marginal distribution over observed variables stays the same. For instance, we may have a two-component mixture model for p with both components having thesame identical emission distributions yet the kernel ~k

(p, p0) evaluates to a different value if wecollapse the identical components in p into one emission distribution. This is to be expected since

the different latent components imply a slightly different mapping to Hilbert space.We next describe the kernel for various graphical models which essentially impose a factorization on the joint probability distribution over the N variables xo and xh. This article focuseson directed graphs (although undirected graphs can also be kernelized) where this factorization implies that the joint distribution can be written as a product of conditionals over each node or vari-able x

i 2 {xo, xh}, i = 1 . . . N given its parent nodes or set of parent variables xpa(i) as p(xo, xh) =O~N i=1 p(xi|xpa(i)). We now discuss particular cases of graphical models including mixture models,hidden Markov models and linear dynamical systems.

4.1 Mixture Models In the simplest scenario, the latent graphical model could be a mixture model, such as a mixture ofGaussians or mixture of exponential family distributions. Here, the hidden variable x

h is a singlediscrete variable which is a parent of a single x o emission variable in the exponential family. Toderive the full kernel for the mixture model, we first assume it is straightforward to compute an

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(a) HMM for p. (b) HMM for p0. (c) Combined graphs.

Figure 1: Two hidden Markov models and the resulting graphical model as the kernel couples com-mon parents for each node creating undirected edges between them.

elementary kernel between any pair of entries (indexed via xh and x0h) from each mixture model asfollows:

~k(p(xo|xh), p0(xo|x0h)) = a*

xo \Gamma p(xo|xh)p

0(xo|x0h)\Delta r .

In the above, the conditionals could be, for instance, exponential family (emission) distributions.Then, we can easily compute the kernel for a mixture model of such distributions using these elementary kernel evaluations and a weighted summation of all possible pairings of components of themixtures:

~k(p, p0) = a*

xo a*xh (p(xh)p(xo|xh))

r a*

x0h \Gamma p

0(x0h)p0(xo|x0h)\Delta r

= a*

xh,x0h \Gamma p(xh)p

0(x0h)\Delta r ~k(p(xo|xh), p0(xo|x0h)).

Effectively, the mixture model kernel involves enumerating all settings of the hidden variables xhand x0

h. Taking the notation |.| to be the cardinality of a variable, we effectively need to performand sum a total of |x

h| * |x0h| elementary kernel computations ~kxh,x0h between the individual emissiondistributions.

4.2 Hidden Markov Models We next upgrade beyond mixtures to graphical models such as hidden Markov models (HMMs).Considering hidden Markov models as the generative model p allows us to build a kernel over sequences of variable lengths. However, HMMs and many popular Bayesian networks have a largenumber of (hidden and observed) variables and enumerating all possible hidden states for two distributions p and p0 quickly becomes inefficient since xh and x0h are multivariate and/or have largecardinalities. However, unlike the plain mixture modeling case, HMMs have an efficient graphical model structure implying a factorization of their distribution which we can leverage to compute ourkernel efficiently. A hidden Markov model over sequences of length T

+ 1 has observed variablesx

o = {x0, . . . , xT } and hidden states xh = {q0, . . . , qT }. The graphical model for an HMM in Fig-ure 1(a) reflects its Markov chain assumption and leads to the following probability density function

(note here we define q-1 = {} as a null variable for brevity or, equivalently, p(q0|q-1) = p(q0)):

p(xo) = a*x

h p(xo

, xh) = a*q

0

. . . a*q

T

TO~ t=0 p(xt |qt)p(qt |qt-1)

.

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 Figure 2: The undirected clique graph obtained from the kernel for efficiently summing over bothsets of hidden variables x

h and x0h.

To compute the kernel ~k(p, p0) in a brute force manner, we need to sum over all configurations ofq 0, . . . , qT and q00, . . . , q0T while computing for each each configuration its corresponding elementarykernel ~k(p(x

0, . . . , xT |q0, . . . , qT ), p0(x0, . . . , xT |q00, . . . , q0T )). Exploring each setting of the state spaceof the HMMs (shown in Figure 1(a) and (b)) is highly inefficient requiring |q

t|(T+1) * |q0t |(T+1)elementary kernel evaluations. However, we can take advantage of the factorization of the hidden

Markov model by using graphical modeling algorithms such as the junction tree algorithm (Jordanand Bishop, 2004) to compute the kernel efficiently. First, we use the factorization of the HMM in the kernel as follows:

~k(p, p0) = a*

x0,...,xT a*q0...qT

TO~ t=0 p(xt |qt )

r p(qt|qt-1)r a*

q00...q0T

TO~ t=0 p

0(xt|q0t )r p(q0t |q0t-1)r

= a*q

0...qT a*q00...q0T

TO~ t=0 p(qt|qt-1)

r p(q0t |q0t-1)r a*

xt p(xt|qt )

r p0(xt |q0t)r!

= a*q

0...qT a*q00...q0T

TO~ t=0 p(qt|qt-1)

r p(q0t |q0t-1)ry(qt, q0t )

= a*q

T a*q0T y(qT

, q0T )

TO~

t=1 a*qt-1 a*q0t-1 p(qt |qt-1)

r p(q0t|q0t-1)ry(qt-1, q0t-1)p(q0)r p0(q00)r.

In the above we see that we only need to compute the kernel for each setting of the hiddenvariable for a given time step on its own. Effectively, the kernel couples the two HMMs via their common children as in Figure 1(c). The kernel evaluations, once solved, form non-negative cliquepotential functions y

(qt , q0t) with a different setting for each value of qt and q0t. These couple thehidden variable of each HMM for each time step. The elementary kernel evaluations are easily

computed (particularly if the emission distributions p(xt|qt ) are in the exponential family) as:

~k(p(xt |qt), p0(xt |q0t)) = y(qt, q0t) = a*

xt p(xt |qt)

r p0(xt|q0t)r.

Only a total of (T + 1) * |qt| * |q0t| such elementary kernels need to be evaluated and form a lim-ited number of such clique potential functions. Similarly, each conditional distribution p

(qt |qt-1)rand p0 (q0t |q0t-1)r can also be viewed as a non-negative clique potential function, i.e. f(qt, qt-1) =

829

JEBARA, KONDOR AND HOWARD p(qt|qt-1)r and f(q0t , q0t-1) = p0(q0t|q0t-1)r. Computing the kernel then involves marginalizing overthe hidden variables x

h, which is done by running a forward-backward or junction-tree algorithmon the resulting undirected clique tree graph shown in Figure 2. Thus, to compute the kernel for

two sequences x and x 0 of lengths Tx + 1 and Tx 0 + 1, we train an HMM from each sequence usingmaximum likelihood and then compute the kernel using the learned HMM transition matrix and emission models for a user-specified sequence length T + 1 (where T can be chosen according tosome heuristic, for instance, the average of all T

x in the training data set). The following is pseudo-code illustrating how to compute the ~k(p , p0) kernel given two hidden Markov models (p, p0) anduser-specified parameters T and r.

\Phi (q0, q00) = p(q0)r p0(q00)r for t = 1 . . . T

\Phi (qt, q0t) = a*q

t-1 a*q0t-1 p(qt|qt-1)

r p0(q0t|q0t-1)ry(qt-1, q0t-1)\Phi (qt-1, q0t-1)

end~ k(p, p0) = a*q

T a*q0T \Phi (qT

, q0T )y(qT , q0T ).

Note that the hidden Markov models p and p0 need not have the same number of hidden states forthe kernel computation.

4.3 Bayesian Networks The above method can be applied to Bayesian networks in general where hidden and observed vari-ables factorize according to p

(x) = O~Ni=1 p(xi|xpa(i)). The general recipe mirrors the above deriva-tion for the hidden Markov model case. In fact, the two Bayesian networks need not have the same

structure as long as they share the same sample space over the emission variables xo. For instance,consider computing the kernel between the Bayesian network in Figure 3(a) and the hidden Markov model in Figure 1(b) over the same sample space. The graphs can be connected with their commonchildren as in Figure 3(b). The parents with common children are married and form cliques of hidden variables. We then only need to evaluate the elementary kernels over these cliques giving thefollowing non-negative potential functions for each observed node (for instance x

i 2 xo) under eachsetting of all its parents' nodes (from both p and p0):

y(xh,pa(i), x0h0,pa(i)) = ~k(p(xi|xh,pa(i)), p0(xi|xh0,pa(i)0 ))

= a*x

i p(xi|xh

,pa(i))r p0(xi|xh0,pa(i)0)r.

After marrying common parents, the resulting clique graph is then built and used with a junctiontree algorithm to compute the overall kernel from the elementary kernel evaluations. Although the resulting undirected clique graph may not always yield a dramatic improvement in efficiency aswas seen in the hidden Markov model case, we note that propagation algorithms (loopy or junction tree algorithms) are still likely to provide a more efficient evaluation of the kernel than the bruteforce enumeration of all latent configurations of both Bayesian networks in the kernel (as was described in the generic mixture model case). While the above computations emphasized discretelatent variables, the kernel is also computable over continuous latent configuration networks, where

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(a) Bayesian network for p. (b) Combined graphs.

Figure 3: The resulting graphical model from a Bayesian network and a hidden Markov modelas the kernel couples common parents for each node creating undirected edges between

them and a final clique graph for the junction tree algorithm.

xh contains scalars and vectors, as is the case in linear Gaussian models, which we develop in thenext subsections. 4.4 Linear Gaussian Models A linear Gaussian model on a directed acyclic graph G with variables x1, x2, . . . , xN associated to itsvertices is of the form

p (x1, x2, . . . , xN) = O~i N \Gamma  xi ; bi,0 +a* j2pa(i)bi, jx j, \Sigma i \Delta  where N (x; u, \Sigma ) is the multivariate Gaussian distribution, and pa(i) is the index set of parent ver-tices associated with the ith vertex. The unconditional joint distribution can be recovered from the conditional form and is itself a Gaussian distribution over the variables in the model p (x1, x2, . . . , xN).Hence, the probability product kernel between a pair of linear Gaussian models can be computed in closed form by using the unconditional joint distribution and (5).Latent variable models require an additional marginalization step. Variables are split into two sets: the observed variables xo and the hidden variables xh. After the joint unconditional distributionp

(xo, xh) has been computed, the hidden variables can be trivially integrated out:

k(p, p0) = Z `Z p(xo, xh) dxh'

r `Z p(x

o, x0h) dx0h'

r dx

o = Z p(xo)r p0(xo)r dxo,

so that the kernel is computed between Gaussians over only the observed variables. 4.5 Linear Dynamical Systems A commonly used linear Gaussian model with latent variables is the linear dynamical system (LDS)(Shumway and Stoffer, 1982), also known as the state space model or Kalman filter model. The

LDS is the continuous state analog of the hidden Markov model and shares the same conditionalindependence graph Figure 1(a). Thus, it is appropriate for modeling continuous time series data and continuous dynamical systems. The LDS's joint probability model is of the form

p(x0, . . . xT , s0, . . . sT ) = N (s0; u, \Sigma ) N (x0;Cs0, R)

TO~

t=1 N (st ; Ast-1

, Q) N (xt ;Cst , R),

831

JEBARA, KONDOR AND HOWARD where xt is the observed variable at time t, and st is the latent state space variable at time t, obeyingthe Markov property.

The probability product kernel between LDS models is

k(p, p0) = Z N (x; ux, \Sigma xx)r N (x; u0x, \Sigma 0xx)r dx, where ux and \Sigma xx are the unconditional mean and covariance, which can be computed from therecursions

ust = Aust-1 uxt = Cust \Sigma stst = A\Sigma st-1st-1A0 + Q \Sigma xtxt = C\Sigma ststC0 + R.

As with the HMM, we need to set the number of time steps before computing the kernel. Notethat the most algorithmically expensive calculation when computing the probability product kernel between Gaussians is taking the inverse of the covariance matrices. The dimensionality of thecovariance matrix will grow linearly with the number of time steps and the running time of the matrix inverse grows cubically with the dimensionality. However, the required inverses can beefficiently computed because \Sigma 

xx is block diagonal. Each extra time step added to the kernel willsimply add another block, and the inverse of a block diagonal matrix can be computed by inverting

the blocks individually. Therefore, the running time of inverting the block diagonal covariancematrix will only grow linearly with the number of time steps T and cubically with the (small) block size.

5. Sampling Approximation To capture the structure of real data, generative models often need to be quite intricate. Closedform solutions for the probability product kernel are then unlikely to exist, forcing us to rely on

approximation methods to compute k(p, p0).Provided that evaluating the likelihood p

(x) is computationally feasible, and that we can alsoefficiently sample from the model, in the r = 1 case (expected likelihood kernel) we may employthe Monte Carlo estimate

k(p, p0) ss bN

Na*

i=1 p

0(xi) + (1 - b)N0 Na*

i=1 p(x

0i),

where x1, . . . , xN and x01, . . . , x0N are i.i.d. samples from p and p0 respectively, and b 2 [0, 1] is aparameter of our choosing. By the law of large numbers, this approximation will converge to the true kernel value in the limit N ! e^ for any b. Other sampling techniques such as importance samplingand a variety of flavors of Markov chain Monte Carlo (MCMC), including Gibbs sampling, can be used to improve the rate of the sampling approximation's convergence to the true kernel.In some cases it may be possible to use the sampling approximation for an general r. This is possible if a normalizing factor Z can be computed to renormalize p(x)r into a valid distribution. Z iscomputable for most discrete distributions and some continuous distribution, such as the Gaussian.

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 Figure 4: Graph for the Switching Linear Dynamical System. The sampling approximation then becomes

k(p, p0) ss bN

Na*

i=1 Z p

0( ^xi)r + (1 - b)N0 Na*

i=1 Z

0 p( ^x0i)r,

where Z and Z0 are the normalizers of p and p0 after they are taken to the power of r. In thisformulation ^x1

, . . . , ^xN and ^x01, . . . , ^x0N are i.i.d. samples from ^p and ^p0 where ^p = p

r

Z and ^p0 = p

0r Z0 .In practice, for a finite number of samples, the approximate kernel may not be a valid Mercer

kernel. The non-symmetric aspect of the approximation can be alleviated by computing only thelower triangular entries in the kernel matrix and replicating them in the upper half. Problems associated with the approximation not being positive definite can be rectified by either using moresamples, or by using an implementation for the kernel based classifier algorithm that can converge to a local optimum such as sequential minimal optimization (Platt, 1999). 5.1 Switching Linear Dynamical Systems In some cases, part of the model can be evaluated by sampling, while the rest is computed in closedform. This is true for the switching linear dynamical system (SLDS) (Pavlovic et al., 2000), which

combines the discrete hidden states of the HMM and the continuous state space of the LDS:

p(x, s, q) = p(q0) p(s0|q0) p(x0|s0)

TO~

t=1 p(qt |qt-1) p(st |st-1

, qt ) p(xt|st ),

where qt is the discrete hidden state at time t, st is the continuous hidden state, and xt are observedemission variables (Figure 4). Sampling q1

. . . , qN according to p(q) and q01, . . . , q0N according top0 (q0), the remaining factors in the corresponding sampling approximation

1N Na*

i=1 Z Z p(s0|q

i0) p0(s00|q0i0) TO~

t=1 p(st|st-1

, qit ) p0(s0t|s0t-1, q0it )

TO~

t=0 p(xt |st) p

0(xt|s0t ) ds ds0!

r

dx

form a non-stationary linear dynamical system. Once the joint distribution is recovered, it can beevaluated in closed form by (5) as was the case with a simple LDS. For r 6

= 1 this is a slightlydifferent formulation than the sampling approximation in the previous section. This hybrid method

has the nice property that it is always a valid kernel for any number of samples since it simplybecomes a convex combination of kernels between LDSs. An SLDS model is useful for describing input sequences that have a combination of discrete and continuous dynamics, such as a time seriesof human motion containing continuous physical dynamics and kinematics variables as well as discrete action variables (Pavlovic et al., 2000).

833

JEBARA, KONDOR AND HOWARD 6. Mean Field Approximation Another option when the probability product kernel between graphical models cannot be computedin closed form is to use variational bounds, such as the mean field approximation (Jaakkola, 2000).

In this method, we introduce a variational distribution Q(x), and using Jensen's inequality, lowerbound the kernel:

k(p, p0) = Z p(x)r p0(x)r dx = Z \Psi (x)r dx = exp `log Z Q(x)Q(x) \Psi (x)r dx'

>= exp `Z Q(x) log \Psi (x)

r

Q(x) dx' = exp (r EQ [log \Psi (x)] + H(Q)) = B(Q),

where \Psi (x) = p(x) p0(x) is called the potential, EQ [ * ] denotes the expectation with respect to Q(x),and H

(Q) is the entropy. This transforms the problem from evaluating an intractable integral intothat of finding the sufficient statistics of Q

(x) and computing the subsequent tractable expectations.Note, using the mean field formulation gives a different (and arguably closer result) to the desired

intractable kernel than simply computing the probability product kernels directly between the ap-proximate simple distributions. It is interesting to note the following form of the bound when it is expressed in terms of the Kullback-Leibler divergence, D(Qkp) = R Q(x) log Q(x)p(x) dx, and an entropyterm. The bound on the probability product kernel is a function of the Kullback-Leibler divergence to p and p0:

k(p, p0) >= B(Q) = exp \Gamma -r D(Qkp) - r D(Qkp0) + (1 - 2r) H(Q)\Delta  . The goal is to define Q(x) in a way that makes computing the sufficient statistics easy. When thegraphical structure of Q

(x) has no edges, which corresponds to the case that the underlying variablesare independent, this is called the mean field method. When Q

(x) is made up of a number ofmore elaborate independent structures, it is called the structured mean field method. In either case,

the variational distribution factors in the form Q1(x1)Q2(x2) . . . QN(xN) with x1, x2, . . . , xN disjointsubsets of the original variable set x.

Once the structure of Q(x) has been set, its (locally) optimal parameterization is found by cy-cling through the distributions Q

1(x1), Q2(x2), . . . , QN(xN) and maximizing the lower bound B(Q)with respect to each one, holding the others fixed:

u"" log B(Q)

u"" Qn(xn) =

u"" u"" Qn(xn) ^ r Z Qn(xn) EQ [log \Psi (x)|xn] dxn + H(Qn) + constant * = 0.

This leads to the update equations

Qn(xn) = 1Zn exp (r EQ [log \Psi (x)|xn]) ,

where the conditional expectation EQ [ * | xn] is computed by taking the expectation over all variablesexcept x

n and Z

n = Z exp (r EQ [log \Psi (x)|xn]) dx

is the normalizing factor guaranteeing that Q(x) remains a valid distribution.

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 (a) FHMM P(x, q). (b) Potential \Psi (x, q, q0). (c) Variational Q(x, q, q0).

Figure 5: Graphs associated with the factorial hidden Markov model, its probability product kernelpotential, and its structured mean field distribution.

In practice this approximation may not give positive definite (PD) kernels. Much current re-search has been focused on kernel algorithms using non-positive definite kernels. Algorithms such as sequential minimal optimization (Platt, 1999) can converge to locally optimal solutions. Thelearning theory for non-PD kernels is currently of great interest. See Ong et al. (2004) for an interesting discussion and additional references. 6.1 Factorial Hidden Markov Models One model that traditionally uses a structured mean field approximation for inference is the factorialhidden Markov model (FHMM)(Ghahramani and Jordan, 1997). Such a model is appropriate for

modeling output sequences that are emitted by multiple independent hidden processes. The FHMMis given by

p(x0, x1, x2, . . . , xT ) =

CO~

c=1 " p(q

c0) TO~

t=1 p(q

ct |qct-1) # TO~

t=0 p(xt|q

1t , . . . , qCt ),

where qct is the factored discrete hidden state at time t for chain c and xt is the observed Gaussianemission variable at time t. The graphs of the joint FHMM distribution, the potential \Psi 

(x, q, q0),and the variational distribution Q (x, q, q0) are depicted in Figure 5. The structured mean field ap-proximation is parameterized as

Q( qc0 = i ) t, exp ir log fc(i) - r2 log | \Sigma i | - r2 EQ \Theta (x0-ui)T \Sigma -1i (x0-ui)\Lambda j Q( qct = i | qct-1 = j ) t, exp ir log \Phi c(i, j) - r2 log | \Sigma i | - r2 EQ \Theta (xt -ui)T \Sigma -1i (xt -ui)\Lambda j

Q(xt ) = N (xt ; ^ut, ^\Sigma t )

^\Sigma t = 1r ^ a*

i,c EQ [s

ct (i)] \Sigma -1i + a*

i,c EQ \Theta q

0ct (i)\Lambda  \Sigma 0-1i *-1

^ut = r ^\Sigma t^ a*i

,c EQ [q

ct (i)] \Sigma -1i ui + a*

i,c EQ \Theta q

0ct (i)\Lambda  \Sigma 0-1i u0i *.

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Figure 6: A statistical manifold with a geodesic path between two generative models q and q0 aswell as a local tangent space approximation at, for instance, the maximum likelihood

model q* for the aggregated data set.

It is necessary to compute the expected sufficient statistics to both update the mean field equa-tions and to evaluate the bound. The expectations E

Q [xt] and EQ \Theta xtxTt \Lambda  can easily be computedfrom Q(x t ), whereas Q( sct = i ) and Q( sct = i , sct-1 = j ) can be computed efficiently by means of ajunction tree algorithm on each independent chain in the approximating distribution.

7. Relationship to Other Probabilistic Kernels While the probability product kernel is not the only kernel to leverage generative modeling, it doeshave advantages in terms of computational feasibility as well as in nonlinear flexibility. For instance, heat kernels or diffusion kernels on statistical manifolds (Lafferty and Lebanon, 2002) area more elegant way to generate a kernel in a probabilistic setting, but computations involve finding geodesics on complicated manifolds and finding closed form expressions for the heat kernel, whichare only known for simple structures such as spheres and hyperbolic surfaces. Figure 6 depicts such a statistical manifold. The heat kernel can sometimes be approximated via the geodesic distancefrom p which is parameterized by q to p0 parameterized by q0. But, we rarely have compact closedform expressions for the heat kernel or for these geodesics for general distributions. Only Gaussianswith spherical covariance and multinomials are readily accommodated. Curiously, the probabilistic product kernel expression for both these two distributions seems to coincide closely with the moreelaborate heat kernel form. For instance, in the multinomial case, both involve an inner product between the square-rooted count frequencies. However, the probability product kernel is straight-forward to compute for a much wider range of distributions providing a practical alternative to heat kernels.Another probabilistic approach is the Fisher kernel (Jaakkola and Haussler, 1998) which approximates distances and affinities on the statistical manifold by using the local metric at the maximumlikelihood estimate q* of the whole data set as shown in Figure 6. One caveat, however, is that this approximation can produce a kernel that is quite different from the exact heat kernel above andmay not preserve the interesting nonlinearities implied by the generative model. For instance, in the exponential family case, all distributions generate Fisher kernels that are linear in the sufficientstatistics. Consider the Fisher kernel

k(x , x 0) = Ux I-1q* Ux 0

836

PROBABILITY PRODUCT KERNELS where I-1q* is the inverse Fisher information matrix at the maximum likelihood estimate and wherewe define

Ux = N~q log p(x |q)|q*. In exponential families, pq(x) = exp(A(x)+qT T (x)-K (q)) producing the following Ux = T (x )-G

(q*) where we define G (q*) = N~qK (q))|q*. The resulting Fisher kernel is then

k(x , x 0) = (T (x ) - G (q*))T I-1q* \Gamma T (x 0)T - G (q*)\Delta  , which is an only slightly modified form of the linear kernel in T (x ). Therefore, via the Fisher kernelmethod, a Gaussian distribution over means generates only linear kernels. A Gaussian distribution

over means and covariances generates quadratic kernels. Furthermore, multinomials generate log-counts unlike the square root of the counts in the probability product kernel and the heat kernel. In practical text classification applications, it is known that square root squashing functions on fre-quencies and count typically outperform logarithmic squashing functions (Goldzmidt and Sahami, 1998; Cutting et al., 1992). In (Tsuda et al., 2002; Kashima et al., 2003), another related probabilis-tic kernel is put forward, the so-called marginalized kernel. This kernel is again similar to the Fisher kernel as well as the probability product kernel. It involves marginalizing a kernel weighted by theposterior over hidden states given the input data for two different points, in other words the output kernel k(x, x0) = a*h a*h0 p(h|x)p(h0|x0)k((x, h), (x0, h0)). The probability product kernel is similar,however, it involves an inner product over both hidden variables and the input space x with a joint distribution.A more recently introduced kernel, the exponentiated symmetrized Kullback-Leibler (KL) divergence (Moreno et al., 2004) is also related to the probability product kernel. This kernel involvesexponentiating the negated symmetrized KL-divergence between two distributions

k(p, p0) = exp(-aD(pkp0) - aD(p0kp) + b) where D(pkp0) = Zx p(x) log p(x)p0(x) dx where a and b are user-defined scalar parameters. However, this kernel may not always satisfyMercer's condition and a formal proof is not available. Another interesting connection is that this form is very similar to our mean-field bound on the probability product kernel (7). However, unlikethe probability product kernel (and its bounds), computing the KL-divergence between complicated distributions (mixtures, hidden Markov models, Bayesian networks, linear dynamical systems andintractable graphical models) is intractable and must be approximated from the outset using numerical or sampling procedures which further decreases the possibility of having a valid Mercer kernel(Moreno et al., 2004).

8. Experiments In this section we discuss three different learning problems: text classification, biological sequenceclassification and time series classification. For each problem, we select a generative model that is

compatible with the domain which then leads to a specific form for the probability product kernel.For text, multinomial models are used, for sequences hidden Markov models are natural and for time series data we use linear dynamical systems. The subsequent kernel computations are fed to adiscriminative classifier (a support vector machine) for training and testing.

837

JEBARA, KONDOR AND HOWARD 8.1 Text Classification For text classification, we used the standard WebKB data set which consists of HTML documentsin multiple categories. Only the text component of each web page was preserved and HTML

markup information and hyper-links were stripped away. No further stemming or elaborate text pre-processing was performed on the text. Subsequently, a bag-of-words model was used where each document is only represented by the frequency of words that appear within it. This correspondsto a multinomial distribution over counts. The frequencies of words are the maximum likelihood estimate for the multinomial generative model to a particular document which produces the vectorof parameters ^a as in Section 3.

0 1 2 3 4 5 6 7 80 0.05

0.1 0.15

0.2 0.25

Regularization log(C) Error Rate

RBF s=1/4RBF s=1 RBF s=4Linear Bhattacharyya X=1

0 1 2 3 4 5 6 7 80 0.05

0.1 0.15

0.2 0.25

Regularization log(C) Error Rate

RBF s=1/4RBF s=1 RBF s=4Linear Bhattacharyya X=1

(a) Training Set Size = 77 (b) Training Set Size = 622 Figure 7: SVM error rates (and standard deviation) for the probability product kernel (set at r = 1/2as a Bhattacharyya kernel) for multinomial models as well as error rates for traditional

kernels on the WebKB data set. Performance for multiple settings of the regularizationparameter C in the support vector machine are shown. Two different sizes of training sets are shown (77 and 622). All results are shown for 20-fold cross-validation.

A support vector machine (which is known to have strong empirical performance on such textclassification data) was then used to build a classifier. The SVM was fed our kernel's evaluations via a gram matrix and was trained to discriminate between faculty and student web pages in theWebKB data set (other categories were omitted). The probability product kernel was computed for multinomials under the parameter settings r = 1/2 (Bhattacharyya kernel) and s = 1 as in Section 3.Comparisons were made with the (linear) dot product kernel as well as Gaussian RBF kernels. The data set contains a total of 1641 student web pages and 1124 faculty web pages. The data foreach class is further split into 4 universities and 1 miscellaneous category and we performed the usual training and testing split as described by (Lafferty and Lebanon, 2002; Joachims et al., 2001)where testing is performed on a held out university. The average error was computed from 20-fold cross-validation for the different kernels as a function of the support vector machine regularizationparameter C in Figure 7. The figure shows error rates for different sizes of the training set (77 and 622 training points). In addition, we show the standard deviation of the error rate for the Bhat-tacharyya kernel. Even though we only explored a single setting of s

= 1, r = 1/2 for the probability

838

PROBABILITY PRODUCT KERNELS product kernel, it outperforms the linear kernel as well as the RBF kernel at multiple settings of theRBF s parameter (we attempted s

= {1/4, 1, 4}). This result confirms previous intuitions in thetext classification community which suggest using squashing functions on word frequencies such

as the logarithm or the square root (Goldzmidt and Sahami, 1998; Cutting et al., 1992). This alsorelated to the power-transformation used in statistics known as the Box-Cox transformation which may help make data seem more Gaussian (Davidson and MacKinnon, 1993). 8.2 Biological Sequence Classification For discrete sequences, natural generative models to consider are Markov models and hidden Markovmodels. We obtained labeled gene sequences from the HS3D data set2 The sequences are of variable

lengths and have discrete symbol entries from the 4-letter alphabet (G,A,T,C). We built classifiers todistinguish between gene sequences of two different types: introns and exons using raw sequences of varying lengths (from dozens of characters to tens of thousands of characters). From the orig-inal (unwindowed) 4450 introns and 3752 exons extracted directly from GenBank, we selected a random subset of 500 introns and 500 exons. These 1000 sequences were then randomly splitinto a 50% training and a 50% testing subset. In the first experiment we used the training data to learn stationary hidden Markov models whose number of states M was related to the length Tn of agiven sequence n as follows: M

= floor( 12 pO2 + 4(T z + O + 1) - 12 O) + 1. Here, O is the numberof possible emission symbols (for gene sequences, O

= 4) and z is the ratio of parameters to thenumber of symbols in the sequence T (we used z = 1/10 although higher values may help avoidover-parameterizing the models). Each hidden Markov model was built from each sequence using

the standard Expectation-Maximization algorithm (which is iterated for 400 steps to ensure conver-gence, this is particularly crucial for longer sequences). Gram matrices of size 1000 * 1000 were then formed by computing the probability product kernel between all the hidden Markov models.It was noted that all Gram matrices were positive definite by a singular value decomposition. We also used the following standard normalization of the kernel (a typical pre-processing step in theliterature):

~k(p, p0)  ~k(p, p0)q~

k(p, p)q~k(p0, p0)

.

Subsequently, we trained a support vector machine classifier on 500 example sequences andtested on the remaining 500. Figure 8(a) depicts the resulting error rate as we vary C, the regularization parameter on the SVM as well as T the number of time steps used by the kernel. Throughout,these experiments, we only used the setting r

= 1. Note that these models were 1st order hiddenMarkov models where each state only depends on the previous state. Varying T , the length of the

hidden Markov models used in the kernel computation has a slight effect on performance and itseems T

= 9 at an appropriate C value performed best with an error rate as low as 10-11%.For comparison, we also computed more standard string kernels such as the k-mer counts or

spectrum kernels on the same training and testing gene sequences as shown in Figure 8(b). Thesebasically correspond to a fully observed (non-hidden) stationary Markov model as in Figure 9. The zeroth order (K = 1) Markov model does poorly with its lowest error rate around 34%. The first

2. This is a collection of unprocessed intron and exon class gene sequences referred to as the homo sapiens splice sitesdata set and was downloaded from www.sci.unisannio.it/docenti/rampone.

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JEBARA, KONDOR AND HOWARD 1 2 3 4 5 60.05 0.1 0.15

0.2 0.25

0.3 0.35

0.4

Regularization log(C) Error Rate

T=7T=8 T=9T=10 T=11

1 2 3 4 5 60.05 0.1 0.15

0.2 0.25

0.3 0.35

0.4

Regularization log(C) Error Rate

K=1K=2 K=3K=4 K=5

(a) Hidden Markov Models (1st Order) (b) Markov Models (0th to 4th Order) Figure 8: SVM error rates for the probability product kernel at r = 1 for hidden Markov modelsand Markov models (equivalent to string or spectrum kernels). In (a) test error rate under

various levels of regularization is shown for 5 different settings of T in the probabilityproduct kernel. In (b) test error rate under various levels of regularization is shown for 5 different settings of the order K of the Markov model.%%[ ProductName: ESP Ghostscript ]%%


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 Figure 9: A higher order (2nd order or K = 3) Markov model.

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PROBABILITY PRODUCT KERNELS order Markov model (K = 2) performs better with its lowest error rate at 18% yet is slightly worsethan first order hidden Markov models. This helps motivate the possibility that dependence on the past in first order Markov models could be better modeled by a latent state as opposed to just theprevious emission. This is despite the maximum likelihood estimation issues and local minima that we must contend with when training hidden Markov models using expectation-maximization. Thesecond order Markov models with K

= 3 fare best with an error rate of about 7-8%. However,higher orders (K = 4 and K = 5) seem to reduce performance. Therefore, a natural next experimentwould be to consider higher order hidden Markov models, most notably second order ones where

the emission distributions are conditioned on the past two states as opposed to the past two emittedsymbols in the sequence.

8.3 Time Series Experiments In this experiment we compare the performance of the probability product kernel between LDSs andthe sampling approximation of the SLDS kernel with more traditional kernels and maximum likelihood techniques on synthetic data. We sampled from 20 exemplar distributions to generate synthetictime series data. The sampled distributions were 5 state, 2 dimensional SLDS models generated at random. Each model was sampled 5 times for 25 time steps with 10 models being assigned to eachclass. This gave 50 time series of length 25 per class, creating a challenging classification problem. For comparison, maximum likelihood models for LDSs and SLDSs were fit to the data and used forclassification as well as SVMs trained using Gaussian RBF kernels. All testing was performed using leave one out cross validation. The SLDS kernel was approximated using sampling (50 samples)and the probability product kernels were normalized to improve performance.

The maximum likelihood LDS had an error rate of .35, the SLDS .30, and the Gaussian RBF.34. Exploring the setting of the parameters r and T for the LDS and SLDS kernels, we were able to outperform the maximum likelihood methods and the Gaussian RBF kernel with an optimal errorfor the LDS and SLDS kernel of .28 and .25 respectively. Figure 10 (a) and (b) show the error rate versus T and r respectively. It can be seen that increasing T generally improves performancewhich is to be expected. Although it does appear that T can be chosen too large. Meanwhile, r, generally appears to perform better when it is smaller (a counter example is the LDS kernel at thesetting T

= 5). Overall, the kernels performed comparably to or better than standard methods formost settings of r and T with the exception of extreme settings.

9. Conclusions Discriminative learning, statistical learning theory and kernel-based algorithms have brought math-ematical rigor and practical success to the field making it is easy to overlook the advantages of

generative probabilistic modeling. Yet generative models are intuitive and offer flexibility for insert-ing structure, handling unusual input spaces, and accommodating prior knowledge. By proposing a kernel between the models themselves, this paper provides a bridge between the two schools ofthought.

The form of the kernel is almost as simple as possible, yet it still gives rise to interesting nonlin-earities and properties when applied to different generative models. It can be computed explicitly for some of the most commonly used parametric distributions in statistics such as the exponentialfamily. For more elaborate models (graphical models, hidden Markov models, linear dynamical systems, etc.), computing the kernel reduces to using standard algorithms in the field. Experiments

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JEBARA, KONDOR AND HOWARD 5 10 15 20 250.25 0.3 0.35

0.4 0.45

Time Steps T Error

LDS rho=1/10LDS rho=1/2 LDS rho=1SLDS rho=1/10 SLDS rho=1/2SLDS rho=1

0.1 0.25 0.5 1 20.25 0.3 0.35

0.4 0.45

0.5

Rho Error

LDS T=5LDS T=15 LDS T=25SLDS T=5 SLDS T=15SLDS T=25

(a) (b) Figure 10: A comparison of the choice of parameters (a) T and (b) r. The x-axis is the parameterand the y-axis is the error rate.

show that probability product kernels hold promise in practical domains. Ultimately, engineeringkernels between structured, discrete or unusual objects is an area of active research and, via generative modeling, probability product kernels can bring additional flexibility, formalism and potential.

Acknowledgments The authors thank the anonymous reviewers for their helpful comments. This work was funded inpart by the National Science Foundation (grants IIS-0347499 and CCR-0312690).

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