ARCH (GARCH) models [see Bollerslev, Chou, and Kroner (1992) for a survey of ARCH modeling], both the mean and log-volatility equations have separate error terms. The ease of evaluating the ARCH likelihood function and the ability of the ARCH specification to accommodate the timevarying volatility

In this paper, Markov chain Monte Carlo sampling methods are exploited to provide a unified, practical likelihood-based framework for the analysis of stochastic volatility models. A highly effective method is developed that samples all the unobserved volatilities at once using an approximating

The output of a classifier should be a calibrated posterior probability to enable post-processing. Standard SVMs do not provide such probabilities. One method to create probabilities is to directly train a kernel classifier with a logit link function and a regularized maximum likelihood score

penalized likelihood functions. The proposed ideas are widely applicable. They are readily applied to a variety of parametric models such as generalized linear models and robust regression models. They can also be applied easily to nonparametric modeling by using wavelets and splines. Rates of convergence

for additive expansions based on any tting criterion. Specic algorithms are presented for least{squares, least{absolute{deviation, and Huber{M loss functions for regression, and multi{class logistic likelihood for classication. Special enhancements are derived for the particular case where the individual

A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class

of parameters in a latent variable model closely related to factor analysis. We consider the properties of the associated likelihood function, giving an EM algorithm for estimating the principal subspace iteratively, and discuss, with illustrative examples, the advantages conveyed by this probabilistic approach

but effective GMMs for likelihood functions, a universal background model (UBM) for alternative speaker representation, and a form of Bayesian adaptation to derive speaker models from the UBM. The development and use of a handset detector and score normalization to greatly improve verification performance

. The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect

We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters