658 research outputs found
Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities
Information-theoretic measures such as the entropy, cross-entropy and the
Kullback-Leibler divergence between two mixture models is a core primitive in
many signal processing tasks. Since the Kullback-Leibler divergence of mixtures
provably does not admit a closed-form formula, it is in practice either
estimated using costly Monte-Carlo stochastic integration, approximated, or
bounded using various techniques. We present a fast and generic method that
builds algorithmically closed-form lower and upper bounds on the entropy, the
cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate
the versatile method by reporting on our experiments for approximating the
Kullback-Leibler divergence between univariate exponential mixtures, Gaussian
mixtures, Rayleigh mixtures, and Gamma mixtures.Comment: 20 pages, 3 figure
Closed-Form Bayesian Inferences for the Logit Model via Polynomial Expansions
Articles in Marketing and choice literatures have demonstrated the need for
incorporating person-level heterogeneity into behavioral models (e.g., logit
models for multiple binary outcomes as studied here). However, the logit
likelihood extended with a population distribution of heterogeneity doesn't
yield closed-form inferences, and therefore numerical integration techniques
are relied upon (e.g., MCMC methods).
We present here an alternative, closed-form Bayesian inferences for the logit
model, which we obtain by approximating the logit likelihood via a polynomial
expansion, and then positing a distribution of heterogeneity from a flexible
family that is now conjugate and integrable. For problems where the response
coefficients are independent, choosing the Gamma distribution leads to rapidly
convergent closed-form expansions; if there are correlations among the
coefficients one can still obtain rapidly convergent closed-form expansions by
positing a distribution of heterogeneity from a Multivariate Gamma
distribution. The solution then comes from the moment generating function of
the Multivariate Gamma distribution or in general from the multivariate
heterogeneity distribution assumed.
Closed-form Bayesian inferences, derivatives (useful for elasticity
calculations), population distribution parameter estimates (useful for
summarization) and starting values (useful for complicated algorithms) are
hence directly available. Two simulation studies demonstrate the efficacy of
our approach.Comment: 30 pages, 2 figures, corrected some typos. Appears in Quantitative
Marketing and Economics vol 4 (2006), no. 2, 173--20
Nonparametric Hierarchical Clustering of Functional Data
In this paper, we deal with the problem of curves clustering. We propose a
nonparametric method which partitions the curves into clusters and discretizes
the dimensions of the curve points into intervals. The cross-product of these
partitions forms a data-grid which is obtained using a Bayesian model selection
approach while making no assumptions regarding the curves. Finally, a
post-processing technique, aiming at reducing the number of clusters in order
to improve the interpretability of the clustering, is proposed. It consists in
optimally merging the clusters step by step, which corresponds to an
agglomerative hierarchical classification whose dissimilarity measure is the
variation of the criterion. Interestingly this measure is none other than the
sum of the Kullback-Leibler divergences between clusters distributions before
and after the merges. The practical interest of the approach for functional
data exploratory analysis is presented and compared with an alternative
approach on an artificial and a real world data set
Distributionally robust optimization through the lens of submodularity
Distributionally robust optimization is used to solve decision making
problems under adversarial uncertainty where the distribution of the
uncertainty is itself ambiguous. In this paper, we identify a class of these
instances that is solvable in polynomial time by viewing it through the lens of
submodularity. We show that the sharpest upper bound on the expectation of the
maximum of affine functions of a random vector is computable in polynomial time
if each random variable is discrete with finite support and upper bounds
(respectively lower bounds) on the expected values of a finite set of
submodular (respectively supermodular) functions of the random vector are
specified. This adds to known polynomial time solvable instances of the
multimarginal optimal transport problem and the generalized moment problem by
bridging ideas from convexity in continuous optimization to submodularity in
discrete optimization. In turn, we show that a class of distributionally robust
optimization problems with discrete random variables is solvable in polynomial
time using the ellipsoid method. When the submodular (respectively
supermodular) functions are structured, the sharp bound is computable by
solving a compact linear program. We illustrate this in two cases. The first is
a multimarginal optimal transport problem with given univariate marginal
distributions and bivariate marginals satisfying specific positive dependence
orders along with an extension to incorporate higher order marginal
information. The second is a discrete moment problem where a set of marginal
moments of the random variables are given along with lower bounds on the cross
moments of pairs of random variables. Numerical experiments show that the
bounds improve by 2 to 8 percent over bounds that use only univariate
information in the first case, and by 8 to 15 percent over bounds that use the
first moment in the second case.Comment: 36 Pages, 6 Figure
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