scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

Finite-dimensional models for hidden Markov chains

1995 ◽  
Vol 27 (01) ◽  
pp. 146-160
Author(s):  
Lakhdar Aggoun ◽  
Robert J. Elliott
Keyword(s):  
Markov Chains ◽  
Expectation Maximization ◽  
Continuous Time ◽  
Hidden Markov ◽  
Expectation Maximization Algorithm ◽  
Linear Filtering ◽  
Occupation Times ◽  
Finite Dimensional ◽  
Self Tuning ◽  
Filtering Problem

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.

scholarly journals Unbiased estimation of the gradient of the log-likelihood in inverse problems

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Ajay Jasra ◽  
Kody J. H. Law ◽  
Deng Lu
Keyword(s):  
Numerical Approximation ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Multilevel Monte Carlo ◽  
Gradient Algorithms ◽  
Log Likelihood ◽  
Bayesian Inverse Problem ◽  
Unbiased Estimates ◽  
The Cost ◽  
Single Processor

AbstractWe consider the problem of estimating a parameter $$\theta \in \Theta \subseteq {\mathbb {R}}^{d_{\theta }}$$ θ ∈ Θ ⊆ R d θ associated with a Bayesian inverse problem. Typically one must resort to a numerical approximation of gradient of the log-likelihood and also adopt a discretization of the problem in space and/or time. We develop a new methodology to unbiasedly estimate the gradient of the log-likelihood with respect to the unknown parameter, i.e. the expectation of the estimate has no discretization bias. Such a property is not only useful for estimation in terms of the original stochastic model of interest, but can be used in stochastic gradient algorithms which benefit from unbiased estimates. Under appropriate assumptions, we prove that our estimator is not only unbiased but of finite variance. In addition, when implemented on a single processor, we show that the cost to achieve a given level of error is comparable to multilevel Monte Carlo methods, both practically and theoretically. However, the new algorithm is highly amenable to parallel computation.

Finite-dimensional models for hidden Markov chains

1995 ◽  
Vol 27 (1) ◽  
pp. 146-160 ◽  
Author(s):  
Lakhdar Aggoun ◽  
Robert J. Elliott
Keyword(s):  
Markov Chains ◽  
Expectation Maximization ◽  
Continuous Time ◽  
Hidden Markov ◽  
Expectation Maximization Algorithm ◽  
Linear Filtering ◽  
Occupation Times ◽  
Finite Dimensional ◽  
Self Tuning ◽  
Filtering Problem

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.

Multilevel Monte Carlo by using the Halton sequence

2020 ◽  
Vol 26 (3) ◽  
pp. 193-203
Author(s):  
Shady Ahmed Nagy ◽  
Mohamed A. El-Beltagy ◽  
Mohamed Wafa
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Computational Cost ◽  
Random Numbers ◽  
Multilevel Monte Carlo ◽  
Halton Sequence ◽  
Random Generator ◽  
Mc Simulation ◽  
Multilevel Monte Carlo Method

AbstractMonte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

An Exact Discrete Analog to a Closed Linear First-Order Continuous-Time System with Mixed Sample

1987 ◽  
Vol 3 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Terence D. Agbeyegbe
Keyword(s):  
Continuous Time ◽  
Discrete Model ◽  
Moving Average ◽  
Discrete Analog ◽  
Time System ◽  
First Order ◽  
Exact Discrete Model ◽  
Asymptotically Efficient ◽  
Order Continuous ◽  
Continuous Time System

This article deals with the derivation of the exact discrete model that corresponds to a closed linear first-order continuous-time system with mixed stock and flow data. This exact discrete model is (under appropriate additional conditions) a stationary autoregressive moving average time series model and may allow one to obtain asymptotically efficient estimators of the parameters describing the continuous-time system.

scholarly journals Unbiased estimation of linkage disequilibrium from unphased data

2019 ◽  
Author(s):  
Aaron P. Ragsdale ◽  
Simon Gravel
Keyword(s):  
Linkage Disequilibrium ◽  
Population Size ◽  
Evolutionary History ◽  
Unbiased Estimation ◽  
Sequencing Data ◽  
Unbiased Estimators ◽  
Population Size Estimates ◽  
Wide Range ◽  
The Common ◽  
Size Estimates

AbstractLinkage disequilibrium is used to infer evolutionary history and to identify regions under selection or associated with a given trait. In each case, we require accurate estimates of linkage disequilibrium from sequencing data. Unphased data presents a challenge because the co-occurrence of alleles at different loci is ambiguous. Commonly used estimators for the common statistics r2 and D2 exhibit large and variable upward biases that complicate interpretation and comparison across cohorts. Here, we show how to find unbiased estimators for a wide range of two-locus statistics, including D2, for both single and multiple randomly mating populations. These provide accurate estimates over three orders of magnitude in LD. We also use these estimators to construct an estimator for r2 that is less biased than commonly used estimators, but nevertheless argue for using rather than r2 for population size estimates.

Proof of the antithetic-variates theorem for unbounded functions

1979 ◽  
Vol 86 (3) ◽  
pp. 477-479 ◽  
Author(s):  
James R. Wilson
Keyword(s):  
Marginal Distribution ◽  
Functional Dependence ◽  
Finite Variance ◽  
Response Functions ◽  
Unbiased Estimators ◽  
Monte Carlo Techniques ◽  
Joint Distributions ◽  
Antithetic Variates ◽  
Borel Functions ◽  
Image Position

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the formwhere each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.

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