Bayesian Estimation of CIR Model

Abstract: This article concerns the Bayesian estimation of interest rate models based on Euler-Maruyama approximation. Assume the short term interest rate follows the CIR model, an iterative method of Bayesian estimation is proposed. Markov Chain Monte Carlo simulation based on Gibbs sampler is used for the posterior estimation of the parameters. The maximum A-posteriori estimation using the genetic algorithm is employed for finding the Bayesian estimates of the parameters. The method and the algorithm are calibrated with the historical data of US Treasury bills

Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration

The Markov chain Monte Carlo method (MCMC), especially the Metropolis-Hastings (MH) algorithm, is a widely used technique for sampling from a target probability distribution $P$ on a state space $\Omega$ and applied to various problems such as estimation of parameters in statistical models in the Bayesian approach. Quantum algorithms for MCMC have been proposed, yielding the quadratic speedup with respect to the spectral gap $\Delta$ compered to classical counterparts. In this paper, we consider the quantum version of the MH algorithm in the case that calculating $P$ is costly because the log-likelihood $L$ for a state $x\in\Omega$ is obtained via computing the sum of many terms $\frac{1}{M}\sum_{i=0}^{M-1} \ell(i,x)$. We propose calculating $L$ by quantum Monte Carlo integration and combine it with the existing method called quantum simulated annealing (QSA) to generate the quantum state that encodes $P$ in amplitudes. We consider not only state generation but also finding a credible interval for a parameter, a common task in Bayesian inference. In the proposed method for credible interval calculation, the number of queries to the quantum circuit to compute $\ell$ scales on $\Delta$, the required accuracy $\epsilon$ and the standard deviation $\sigma$ of $\ell$ as $\tilde{O}(\sigma/\epsilon^2\Delta^{3/2})$, in contrast to $\tilde{O}(M/\epsilon\Delta^{1/2})$ for QSA with $L$ calculated exactly. Therefore, the proposed method is advantageous if $\sigma$ scales on $M$ sublinearly. As one such example, we consider parameter estimation in a gravitational wave experiment, where $\sigma=O(M^{1/2})$

Bayesian optimization for computationally extensive probability distributions

An efficient method for finding a better maximizer of computationally extensive probability distributions is proposed on the basis of a Bayesian optimization technique. A key idea of the proposed method is to use extreme values of acquisition functions by Gaussian processes for the next training phase, which should be located near a local maximum or a global maximum of the probability distribution. Our Bayesian optimization technique is applied to the posterior distribution in the effective physical model estimation, which is a computationally extensive probability distribution. Even when the number of sampling points on the posterior distributions is fixed to be small, the Bayesian optimization provides a better maximizer of the posterior distributions in comparison to those by the random search method, the steepest descent method, or the Monte Carlo method. Furthermore, the Bayesian optimization improves the results efficiently by combining the steepest descent method and thus it is a powerful tool to search for a better maximizer of computationally extensive probability distributions.Comment: 13 pages, 5 figure

Robust Online Hamiltonian Learning

In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.Comment: 24 pages, 12 figures; to appear in New Journal of Physic

Bayesian sequential experimental design for binary response data with application to electromyographic experiments

We develop a sequential Monte Carlo approach for Bayesian analysis of the experimental design for binary response data. Our work is motivated by surface electromyographic (SEMG) experiments, which can be used to provide information about the functionality of subjects' motor units. These experiments involve a series of stimuli being applied to a motor unit, with whether or not the motor unit res for each stimulus being recorded. The aim is to learn about how the probability of ring depends on the applied stimulus (the so-called stimulus response curve); One such excitability parameter is an estimate of the stimulus level for which the motor unit has a 50% chance of ring. Within such an experiment we are able to choose the next stimulus level based on the past observations. We show how sequential Monte Carlo can be used to analyse such data in an online manner. We then use the current estimate of the posterior distribution in order to choose the next stimulus level. The aim is to select a stimulus level that mimimises the expected loss. We will apply this loss function to the estimates of target quantiles from the stimulus-response curve. Through simulation we show that this approach is more ecient than existing sequential design methods for choosing the stimulus values. If applied in practice, it could more than halve the length of SEMG experiments