Localized Schrödinger bridge sampler

Many approaches for conducting Bayesian inference on discretely observed diffusions involve imputing diffusion bridges between observations. This can be computationally challenging in settings in which the temporal horizon between subsequent observations is large, due to the poor scaling of algorithms for simulating bridges as observation distance increases. It is common in practical settings to use a blocking scheme, in which the path is split into a (user-specified) number of overlapping segments and a Gibbs sampler is employed to update segments in turn. Substituting the independent simulation of diffusion bridges for one obtained using blocking introduces an inherent trade-off: we are now imputing shorter bridges at the cost of introducing a dependency between subsequent iterations of the bridge sampler. This is further complicated by the fact that there are a number of possible ways to implement the blocking scheme, each of which introduces a different dependency structure between iterations. Although blocking schemes have had considerable empirical success in practice, there has been no analysis of this trade-off nor guidance to practitioners on the particular specifications that should be used to obtain a computationally efficient implementation. In this article we conduct this analysis and demonstrate that the expected computational cost of a blocked path-space rejection sampler applied to Brownian bridges scales asymptotically at a cubic rate with respect to the observation distance and that this rate is linear in the case of the Ornstein–Uhlenbeck process. Numerical experiments suggest applicability both of the results of our paper and of the guidance we provide beyond the class of linear diffusions considered.

Bayes analysis of some important lifetime models using MCMC based approaches when the observations are left truncated and right censored

Computation of normalizing constants is a fundamental mathematical problem in various disciplines, particularly in Bayesian model selection problems. A sampling-based technique known as bridge sampling (Meng and Wong in Stat Sin 6(4):831–860, 1996) has been found to produce accurate estimates of normalizing constants and is shown to possess good asymptotic properties. For small to moderate sample sizes (as in situations with limited computational resources), we demonstrate that the (optimal) bridge sampler produces biased estimates. Specifically, when one density (we denote as $$p_2$$) is constructed to be close to the target density (we denote as $$p_1$$) using method of moments, our simulation-based results indicate that the correlation-induced bias through the moment-matching procedure is non-negligible. More crucially, the bias amplifies as the dimensionality of the problem increases. Thus, a series of theoretical as well as empirical investigations is carried out to identify the nature and origin of the bias. We then examine the effect of sample size allocation on the accuracy of bridge sampling estimates and discovered that one possibility of reducing both the bias and standard error with a small increase in computational effort is by drawing extra samples from the moment-matched density $$p_2$$ (which we assume easy to sample from), provided that the evaluation of $$p_1$$ is not too expensive. We proceed to show how the simple adaptive approach we termed “splitting” manages to alleviate the correlation-induced bias at the expense of a higher standard error, irrespective of the dimensionality involved. We also slightly modified the strategy suggested by Wang et al. (Warp bridge sampling: the next generation, Preprint, 2019. arXiv:1609.07690) to address the issue of the increase in standard error due to splitting, which is later generalized to further improve the efficiency. We conclude the paper by offering our insights of the application of a combination of these adaptive methods to improve the accuracy of bridge sampling estimates in Bayesian applications (where posterior samples are typically expensive to generate) based on the preceding investigations, with an application to a practical example.

Non-Gaussian Bridge Sampling with an Application

Ambulatory cardiovascular (CV) measurements provide valuable insights into individuals' health conditions in "real-life," everyday settings. Current methods of modeling ambulatory CV data do not consider the dynamic characteristics of the full data set and their relationships with covariates such as caffeine use and stress. We propose a stochastic differential equation (SDE) in the form of a dual nonlinear Ornstein-Uhlenbeck (OU) model with person-specific covariates to capture the morning surge and nighttime dipping dynamics of ambulatory CV data. To circumvent the data analytic constraint that empirical measurements are typically collected at irregular and much larger time intervals than those evaluated in simulation studies of SDEs, we adopt a Bayesian approach with a regularized Brownian Bridge sampler (RBBS) and an efficient multiresolution (MR) algorithm to fit the proposed SDE. The MR algorithm can produce more efficient MCMC samples that is crucial for valid parameter estimation and inference. Using this model and algorithm to data from the Duke Behavioral Investigation of Hypertension Study, results indicate that age, caffeine intake, gender and race have effects on distinct dynamic characteristics of the participants' CV trajectories.

This paper proposes a method for the maximum likelihood estimation of continuous time stochastic volatility models. The key step is to introduce approximating GARCH processes that have higher frequencies of construction but are observed at lower frequencies. The latency of the volatility process is retained by augmenting data points between price observations. The convergence of the likelihood function can be obtained with mild regularity conditions. Such an approach reconciles discrete and continuous time models, and it can be implemented easily under the context of the simulated maximum likelihood. As an extension to the commonly used modified Brownian bridge sampler, we propose generating paths with skewed density to match the dynamics of the volatilities.

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, sayb. In great generality such a probability converges to 0 exponentially fast in a power ofb. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (inb) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hittingbin finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number,n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations asn↗ ∞. A remarkable feature of TBS is that it isnotbased on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, sayb. In great generality such a probability converges to 0 exponentially fast in a power ofb. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (inb) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hittingbin finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number,n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations asn↗ ∞. A remarkable feature of TBS is that it isnotbased on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

Sparsely sampled diffusion processes, in this paper interpreted as data sampled sparsely in time relative to the time constant, is a challenging statistical problem. Most approximations of the transition kernel are derived under the assumption that data is frequently sampled and these approximations are often severely biased for sparsely sampled data. Monte Carlo methods can be used for this problem as the transition density can be estimated with arbitrary accuracy regardless of the sampling frequency, but this is computationally expensive or even prohibited unless effective variance reduction is applied. The state of art variance reduction technique for diffusion processes is the Durham-Gallant (modified) bridge sampler. Their importance sampler is derived using a linearized, Gaussian approximation of the dynamics, and have proved successful for frequently sampled data. However, the approximation is often not valid for sparsely sampled data. We present a flexible, alternative derivation of the modified bridge sampler for multivariate, discretely observed diffusion models and modify it by taking uncertainty into account. The resulting sampler can be viewed as a combination of the basic sampler and the Durham-Gallant sampler, using the sampler that is most appropriate for the given problem, while still being computationally efficient. Our sampler is providing effective variance reduction for frequently and sparsely sampled data.

This article focuses on two methods to approximate the log-likelihood of discretely observed univariate diffusions: (1) the simulation approach using a modified Brownian bridge as the importance sampler, and (2) the closed-form approximation approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating the numerical variance stabilizing transformation, computing derivatives of the simulated log-likelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications under a benchmark model which has an unknown transition density and has no analytical variance stabilizing transformation. The closed-form approximation, particularly the second-order closed-form, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tails of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closed-form approximation when the observation frequency is lower. Both methods perform better when the volatility level is lower, but the simulation method is more robust to the volatility level. When applied to two well-known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different log-likelihoods in the case of higher volatility.

We develop and study efficient Monte Carlo algorithms for pricing path-dependent options with the variance gamma model. The key ingredient is difference-of-gamma bridge sampling, based on the representation of a variance gamma process as the difference of two increasing gamma processes. For typical payoffs, we obtain a pair of estimators (named low and high) with expectations that (1) are monotone along any such bridge sampler, and (2) contain the continuous-time price. These estimators provide pathwise bounds on unbiased estimators that would be more expensive (infinitely expensive in some situations) to compute. By using these bounds with extrapolation techniques, we obtain significant efficiency improvements by work reduction. We then combine the gamma bridge sampling with randomized quasi–Monte Carlo to reduce the variance and thus further improve the efficiency. We illustrate the large efficiency improvements on numerical examples for Asian, lookback, and barrier options.