The gapped local alignment score of two random sequences follows a Gumbel distribution. If computers could estimate the parameters of the Gumbel distribution within one second, the use of arbitrary alignment scoring schemes could increase the sensitivity of searching biological sequence databases over the web. Accordingly, this article gives a novel equation for the scale parameter of the relevant Gumbel distribution. We speculate that the equation is exact, although present numerical evidence is limited. The equation involves ascending ladder variates in the global alignment of random sequences. In global alignment simulations, the ladder variates yield stopping times specifying random sequence lengths. Because of the random lengths, and because our trial distribution for importance sampling occurs on a different sample space from our target distribution, our study led to a mapping theorem, which led naturally in turn to an efficient dynamic programming algorithm for the importance sampling weights. Numerical studies using several popular alignment scoring schemes then examined the efficiency and accuracy of the resulting simulations.
Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that…
We propose a new unbiased estimator for estimating the utility of the optimal stopping problem. The MUSE, short for Multilevel Unbiased Stopping Estimator, constructs the unbiased Multilevel Monte Carlo (MLMC) estimator at every stage of the optimal stopping problem in a backward recursive way. In contrast to traditional sequential methods, the MUSE can be implemented in parallel. We prove the MUSE has finite variance, finite computational complexity, and achieves $ε$-accuracy with $O(1/ε^2)$ computational cost under mild conditions. We demonstrate MUSE empirically in an option pricing problem…
Some classical uncertainty quantification problems require the estimation of multiple expectations. Estimating all of them accurately is crucial and can have a major impact on the analysis to perform, and standard existing Monte Carlo methods can be costly to do so. We propose here a new procedure based on importance sampling and control variates for estimating more efficiently multiple expectations with the same sample. We first show that there exists a family of optimal estimators combining both importance sampling and control variates, which however cannot be used in practice because they r…
In many applications we seek to maximize an expectation with respect to a distribution over discrete variables. Estimating gradients of such objectives with respect to the distribution parameters is a challenging problem. We analyze existing solutions including finite-difference (FD) estimators and continuous relaxation (CR) estimators in terms of bias and variance. We show that the commonly used Gumbel-Softmax estimator is biased and propose a simple method to reduce it. We also derive a simpler piece-wise linear continuous relaxation that also possesses reduced bias. We demonstrate empirical…
We suppose that a Lévy process is observed at discrete time points. Starting from an asymptotically minimax family of estimators for the continuous part of the Lévy Khinchine characteristics, i.e., the covariance, we derive a data-driven parameter choice for the frequency of estimating the covariance. We investigate a Lepskiĭ-type stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible data-driven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented.
Markov chain Monte Carlo (MCMC) is a sampling-based method for estimating features of probability distributions. MCMC methods produce a serially correlated, yet representative, sample from the desired distribution. As such it can be difficult to know when the MCMC method is producing reliable results. We introduce some fundamental methods for ensuring a trustworthy simulation experiment. In particular, we present a workflow for output analysis in MCMC providing estimators, approximate sampling distributions, stopping rules, and visualization tools.
Markov chain Monte Carlo (MCMC) algorithms provide a very general recipe for estimating properties of complicated distributions. While their use has become commonplace and there is a large literature on MCMC theory and practice, MCMC users still have to contend with several challenges with each implementation of the algorithm. These challenges include determining how to construct an efficient algorithm, finding reasonable starting values, deciding whether the sample-based estimates are accurate, and determining an appropriate length (stopping rule) for the Markov chain. We describe an approach…
Log-linear models are arguably the most successful class of graphical models for large-scale applications because of their simplicity and tractability. Learning and inference with these models require calculating the partition function, which is a major bottleneck and intractable for large state spaces. Importance Sampling (IS) and MCMC-based approaches are lucrative. However, the condition of having a "good" proposal distribution is often not satisfied in practice. In this paper, we add a new dimension to efficient estimation via sampling. We propose a new sampling scheme and an unbiased esti…
For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) proce…
We study the problem of estimating the source of a network cascade. The cascade starts from a single vertex at time 0 and spreads over time, but only a noisy version of the propagation is observable. The goal is then to design a stopping time and estimator that will estimate the source well while ensuring the cost of the cascade to the system is not too large. We rigorously formulate a Bayesian approach to the problem. If vertices can be labelled by vectors in Euclidean space (which is natural in geo-spatial networks), the optimal estimator is the conditional mean estimator, and we derive an e…
Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time $τ_\ell$ of a given level $\ell$ with a stopping time $η$ defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation $\inf_η\ex|η-τ_\ell|$ which become tight as $\ell\to\infty$. Interestingly, in this regime the estimation error does not get smaller if we allow $ η$ to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particul…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Consider $K$ processes, each generating a sequence of identical and independent random variables. The probability measures of these processes have random parameters that must be estimated. Specifically, they share a parameter $θ$ common to all probability measures. Additionally, each process $i\in\{1, \dots, K\}$ has a private parameter $α_i$. The objective is to design an active sampling algorithm for sequentially estimating these parameters in order to form reliable estimates for all shared and private parameters with the fewest number of samples. This sampling algorithm has three key compon…
An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up to a bounded factor in $O\left(t_{\mathrm{rel}}^{3/4}\sqrt{m/d}\right)$ steps, where $t_{\mathrm{rel}}$ is the relaxation time of the lazy random walk on $G$ and $d$ is the minimum degree in $G$. Alternatively, $m$ can be estimated in $O\left(t_{\mathrm{unif}} +t_{\mathrm{rel}}^{5/6}\sqrt{n}\right)$, where $n$ is the number of vertices and $t_{\mathrm{unif}…
A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function $f$ quantifies specificity for each possible outcome $x$ of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target than by chance. The distr…
