اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn rates of convergence for posterior distributions in infinite-dimensional models
228
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular, we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model.
قيم البحث
اقرأ أيضاً
We show that diffusion processes can be exploited to study the posterior contraction rates of parameters in Bayesian models. By treating the posterior distribution as a stationary distribution of a stochastic differential equation (SDE), posterior co
nvergence rates can be established via control of the moments of the corresponding SDE. Our results depend on the structure of the population log-likelihood function, obtained in the limit of an infinite sample sample size, and stochastic perturbation bounds between the population and sample log-likelihood functions. When the population log-likelihood is strongly concave, we establish posterior convergence of a $d$-dimensional parameter at the optimal rate $(d/n)^{1/ 2}$. In the weakly concave setting, we show that the convergence rate is determined by the unique solution of a non-linear equation that arises from the interplay between the degree of weak concavity and the stochastic perturbation bounds. We illustrate this general theory by deriving posterior convergence rates for three concrete examples: Bayesian logistic regression models, Bayesian single index models, and over-specified Bayesian mixture models.
We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for distributions
with $beta$-Holder smooth densities where $beta in (0, 2)$, we observe polynomially faster minimax rates of estimation when, additionally, the total positivity condition is imposed. Moreover, we demonstrate fast algorithms to compute the proposed estimators and corroborate the theoretical rates of estimation by simulation studies.
We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression and its plug-in property for differential operators. For a general class of kernels, we establish convergence rates of the posterio
r measure of the regression function and its derivatives, which are both minimax optimal up to a logarithmic factor for functions in certain classes. Our calculation shows that the rate-optimal estimation of the regression function and its derivatives share the same choice of hyperparameter, indicating that the Bayes procedure remarkably adapts to the order of derivatives and enjoys a generalized plug-in property that extends real-valued functionals to function-valued functionals. This leads to a practically simple method for estimating the regression function and its derivatives, whose finite sample performance is assessed using simulations. Our proof shows that, under certain conditions, to any convergence rate of Bayes estimators there corresponds the same convergence rate of the posterior distributions (i.e., posterior contraction rate), and vice versa. This equivalence holds for a general class of Gaussian processes and covers the regression function and its derivative functionals, under both the $L_2$ and $L_{infty}$ norms. In addition to connecting these two fundamental large sample properties in Bayesian and non-Bayesian regimes, such equivalence enables a new routine to establish posterior contraction rates by calculating convergence rates of nonparametric point estimators. At the core of our argument is an operator-theoretic framework for kernel ridge regression and equivalent kernel techniques. We derive a range of sharp non-asymptotic bounds that are pivotal in establishing convergence rates of nonparametric point estimators and the equivalence theory, which may be of independent interest.
Kernel ridge regression is an important nonparametric method for estimating smooth functions. We introduce a new set of conditions, under which the actual rates of convergence of the kernel ridge regression estimator under both the L_2 norm and the n
orm of the reproducing kernel Hilbert space exceed the standard minimax rates. An application of this theory leads to a new understanding of the Kennedy-OHagan approach for calibrating model parameters of computer simulation. We prove that, under certain conditions, the Kennedy-OHagan calibration estimator with a known covariance function converges to the minimizer of the norm of the residual function in the reproducing kernel Hilbert space.
In functional linear regression, the slope ``parameter is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a ran
ge of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigenfunctions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
