Related papers: Laplace and Saddlepoint Approximations in High Dim…
The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic classical…
We study the saddlepoint approximation (SPA) for sums of $n$ i.i.d. random vectors $X_i\in\mathbb R^d$ in growing dimensions. SPA provides highly accurate approximations to probability densities and distribution functions via the moment…
We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail…
Laplace approximations are commonly used to approximate high-dimensional integrals in statistical applications, but the quality of such approximations as the dimension of the integral grows is not well understood. In this paper, we prove a…
Laplace approximation is a very useful tool in Bayesian inference and it claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022} established some rather accurate finite sample results about the quality of Laplace…
In Bayesian inference, making deductions about a parameter of interest requires one to sample from or compute an integral against a posterior distribution. A popular method to make these computations cheaper in high-dimensional settings is…
We consider Bayesian variable selection in sparse high-dimensional regression, where the number of covariates $p$ may be large relative to the samples size $n$, but at most a moderate number $q$ of covariates are active. Specifically, we…
We obtain two theorems extending the use of a saddlepoint approximation to multiparameter problems for likelihood ratio-like statistics which allow their use in permutation and rank tests and could be used in bootstrap approximations. In…
The saddlepoint approximation to the likelihood, and its corresponding maximum likelihood estimate (MLE), offer an alternative estimation method when the true likelihood is intractable or computationally expensive. However, maximizing this…
Posterior inference for Dirichlet process mixture models is analytically intractable and typically relies on Markov chain Monte Carlo methods, which can become computationally prohibitive at moderate to large sample sizes. In this work, we…
A saddlepoint approximation of the Student's t-statistic was derived by Daniels and Young [Biometrika 78 (1991) 169-179] under the very stringent exponential moment condition that requires that the underlying density function go down at…
Laplace's method is used to approximate intractable integrals in a statistical problems. The relative error rate of the approximation is not worse than $O_p(n^{-1})$. We provide the first statistical lower bounds showing that the $n^{-1}$…
Statistical applications often involve the calculation of intractable multidimensional integrals. The Laplace formula is widely used to approximate such integrals. However, in high-dimensional or small sample size problems, the shape of the…
In this paper we investigate the convergence of a recently popular class of first-order primal-dual algorithms for saddle point problems under the presence of errors occurring in the proximal maps and gradients. We study several types of…
Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The…
We study high-dimensional Laplace-type integrals $I(\lambda):=(\lambda/2\pi)^{d/2}\int_{\mathbb R^d} g(x)e^{-\lambda f(x)}dx$ in the regime where both $d$ and $\lambda$ are large. Existing rigorous Laplace-expansion results in growing…
Stochastic saddle point (SSP) problems are, in general, less studied compared to stochastic minimization problems. However, SSP problems emerge from machine learning (adversarial training, e.g., GAN, AUC maximization), statistics (robust…
The intensity of a Gibbs point process is usually an intractable function of the model parameters. For repulsive pairwise interaction point processes, this intensity can be expressed as the Laplace transform of some particular function.…
We introduce $\infty$-dimensional versions of three common models of random hetero-polymers, in which both the polymer density and the density of the polymer-solvent mixture are finite. These solvable models give valuable insight into the…
The Laplace approximation is an old, but frequently used method to approximate integrals for Bayesian calculations. In this paper we develop an extension of the Laplace approximation, by applying it iteratively to the residual, i.e., the…