Estimating Signals with Finite Rate of Innovation from Noisy Samples: A Stochastic Algorithm

As an example of the recently-introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples. Even though variants of this problem has been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example kernels which satisfy the Strang-Fix condition. In this paper, we consider the infinite-support Gaussian kernel, which does not satisfy the Strang-Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo (MCMC) method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.Comment: Submitted to IEEE Transactions on Signal Processin

Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials

In 1951 Robbins and Monro published the seminal article on stochastic approximation and made a specific reference to its application to the "estimation of a quantal using response, nonresponse data." Since the 1990s, statistical methodology for dose-finding studies has grown into an active area of research. The dose-finding problem is at its core a percentile estimation problem and is in line with what the Robbins--Monro method sets out to solve. In this light, it is quite surprising that the dose-finding literature has developed rather independently of the older stochastic approximation literature. The fact that stochastic approximation has seldom been used in actual clinical studies stands in stark contrast with its constant application in engineering and finance. In this article, I explore similarities and differences between the dose-finding and the stochastic approximation literatures. This review also sheds light on the present and future relevance of stochastic approximation to dose-finding clinical trials. Such connections will in turn steer dose-finding methodology on a rigorous course and extend its ability to handle increasingly complex clinical situations.Comment: Published in at http://dx.doi.org/10.1214/10-STS334 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantum Inference on Bayesian Networks

Performing exact inference on Bayesian networks is known to be #P-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values $e$ of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on $n$ variables with at most $m$ parents per node in time $\mathcal{O}(nmP(e)^{-1})$, depending critically on $P(e)$, the probability the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking $\mathcal{O}(n2^mP(e)^{-\frac12})$ time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized -- we count primitive operations and require no blackbox oracle queries.Comment: 8 pages, 3 figures. Submitted to PR

Introducing shrinkage in heavy-tailed state space models to predict equity excess returns

We forecast S&P 500 excess returns using a flexible Bayesian econometric state space model with non-Gaussian features at several levels. More precisely, we control for overparameterization via novel global-local shrinkage priors on the state innovation variances as well as the time-invariant part of the state space model. The shrinkage priors are complemented by heavy tailed state innovations that cater for potential large breaks in the latent states. Moreover, we allow for leptokurtic stochastic volatility in the observation equation. The empirical findings indicate that several variants of the proposed approach outperform typical competitors frequently used in the literature, both in terms of point and density forecasts