A novel sequential inferential method for Bayesian dynamic generalised linear models is presented, addressing both univariate and multivariate $k$-parametric exponential families. It efficiently handles diverse responses, including multinomial, gamma, normal, and Poisson distributed outcomes, by leveraging the conjugate and predictive structure of the exponential family. The approach integrates information geometry concepts, such as the projection theorem and Kullback-Leibler divergence, and aligns with recent advances in variational inference. Applications to both synthetic and real datasets highlight its computational efficiency and scalability, surpassing alternative methods. The approach supports the strategic integration of new information, facilitating monitoring, intervention, and the application of discount factors, which are typical in sequential analyses. The R package kDGLM is available for direct use by applied researchers, facilitating the implementation of the method for specific k-parametric dynamic generalised models.
In health-related experiments, treatment effects can be identified using paired data that consist of pre- and post-treatment measurements. In this framework, sequential testing strategies are widely accepted statistical tools in practice. Since performances of parametric sequential testing procedures vitally depend on the validity of the parametric assumptions regarding underlying data distributions, we focus on distribution-free mechanisms for sequentially evaluating treatment effects. In fixed sample size designs, the density-based empirical likelihood (DBEL) methods provide powerful nonpara…
It is commonly required to detect change points in sequences of random variables. In the most difficult setting of this problem, change detection must be performed sequentially with new observations being constantly received over time. Further, the parameters of both the pre- and post- change distributions may be unknown. In a recent paper by Hawkins and Zamba (2005), the sequential generalised likelihood ratio test was introduced for detecting changes in this context, under the assumption that the observations follow a Gaussian distribution. However, we show that the asymptotic approximation …
Given additional distributional information in the form of moment restrictions, kernel density and distribution function estimators with implied generalised empirical likelihood probabilities as weights achieve a reduction in variance due to the systematic use of this extra information. The particular interest here is the estimation of densities or distributions of (generalised) residuals in semi-parametric models defined by a finite number of moment restrictions. Such estimates are of great practical interest, being potentially of use for diagnostic purposes, including tests of parametric ass…
In classical newsvendor model, piece-wise linear shortage and excess costs are balanced out to determine the optimal order quantity. However, for critical perishable commodities, severity of the costs may be much more than linear. In this paper we discuss a generalisation of the newsvendor model with piece-wise polynomial cost functions to accommodate their severity. In addition, the stochastic demand has been assumed to follow a completely unknown probability distribution. Subsequently, non-parametric estimator of the optimal order quantity has been developed from a random polynomial type est…
Regularly varying stochastic processes model extreme dependence between process values at different locations and/or time points. For such processes we propose a two-step parameter estimation of the extremogram, when some part of the domain of interest is fixed and another increasing. We provide conditions for consistency and asymptotic normality of the empirical extremogram centred by a pre-asymptotic version for such observation schemes. For max-stable processes with Fr{é}chet margins we provide conditions, such that the empirical extremogram (or a bias-corrected version) centred by its true…
It is well-known that value added per worker is extremely heterogeneous among firms, but relatively little has been done to characterize this heterogeneity more precisely. Here we show that the distribution of value-added per worker exhibits heavy tails, a very large support, and consistently features a proportion of negative values, which prevents log transformation. We propose to model the distribution of value added per worker using the four parameter Lévy stable distribution, a natural candidate deriving from the Generalised Central Limit Theorem, and we show that it is a better fit than k…
We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm…
Group sequential design (GSD) is widely used in clinical trials in which correlated tests of multiple hypotheses are used. Multiple primary objectives resulting in tests with known correlations include evaluating 1) multiple experimental treatment arms, 2) multiple populations, 3) the combination of multiple arms and multiple populations, or 4) any asymptotically multivariate normal tests. In this paper, we focus on the first 3 of these and extend the framework of the weighted parametric multiple test procedure from fixed designs with a single analysis per objective to a GSD setting where diff…
We consider sequential or active ranking of a set of n items based on noisy pairwise comparisons. Items are ranked according to the probability that a given item beats a randomly chosen item, and ranking refers to partitioning the items into sets of pre-specified sizes according to their scores. This notion of ranking includes as special cases the identification of the top-k items and the total ordering of the items. We first analyze a sequential ranking algorithm that counts the number of comparisons won, and uses these counts to decide whether to stop, or to compare another pair of items, ch…
Product models of low dimensional experts are a powerful way to avoid the curse of dimensionality. We present the ``under-complete product of experts' (UPoE), where each expert models a one dimensional projection of the data. The UPoE is fully tractable and may be interpreted as a parametric probabilistic model for projection pursuit. Its ML learning rules are identical to the approximate learning rules proposed before for under-complete ICA. We also derive an efficient sequential learning algorithm and discuss its relationship to projection pursuit density estimation and feature induction alg…
Suppose that a sequence of treatments are assigned to influence an outcome of interest that occurs after the last treatment. Between treatments there exist time-dependent covariates that may be posttreatment variables of the earlier treatments and confounders of the subsequent treatments. In this article, we develop a parametric approach to inference of the causal effect of the treatment sequence on the outcome called the sequential causal effect. We construct a point parametrization for the conditional distribution of an outcome given all treatments and time-dependent covariates, in which the…
In group sequential analysis, data is collected and analyzed in batches until pre-defined stopping criteria are met. Inference in the parametric setup typically relies on the limiting asymptotic multivariate normality of the repeatedly computed maximum likelihood estimators (MLEs), a result first rigorously proved by Jennison and Turbull (1997) under general regularity conditions. In this work, using Stein's method we provide optimal order, non-asymptotic bounds on the distance for smooth test functions between the joint group sequential MLEs and the appropriate normal distribution under the s…
The generalised linear model (GLM) is a very important tool for analysing real data in biology, sociology, agriculture, engineering and many other application domain where the relationship between the response and explanatory variables may not be linear or the distributions may not be normal in all the cases. However, quite often such real data contain a significant number of outliers in relation to the standard parametric model used in the analysis; in such cases the classical maximum likelihood estimator may fail to produce reasonable estimators and related inference could be unreliable. In …
We study the problem of sequential experimental design to estimate the parametric component of a partially linear model with a Gaussian process prior. We consider an active learning setting where an experimenter adaptively decides which data to collect to achieve their goal efficiently. The experimenter's goals may vary, such as reducing the classification error probability or improving the accuracy of estimating the parameters of the data generating process. This study aims to improve the accuracy of estimating the parametric component of a partially linear model. Under some assumptions, the …
An algorithm is proposed that enables the imposition of shape constraints on regression curves, without requiring the constraints to be written as closed-form expressions, nor assuming the functional form of the loss function. This algorithm is based on Sequential Monte Carlo-Simulated Annealing and only relies on an indicator function that assesses whether or not the constraints are fulfilled, thus allowing the enforcement of various complex constraints by specifying an appropriate indicator function without altering other parts of the algorithm. The algorithm is illustrated by fitting ration…
Existing methods in estimating the mean outcome under a given dynamic treatment regime rely on intention-to-treat analyses which estimate the effect of following a certain dynamic treatment regime regardless of compliance behavior of patients. There are two major concerns with intention-to-treat analyses: (1) the estimated effects are often biased toward the null effect; (2) the results are not generalizable and reproducible due to the potential differential compliance behavior. These are particularly problematic in settings with high level of non-compliance such as substance use disorder trea…
Survival data is encountered in a range of disciplines, most notably health and medical research. Although Bayesian approaches to the analysis of survival data can provide a number of benefits, they are less widely used than classical (e.g. likelihood-based) approaches. This may be in part due to a relative absence of user-friendly implementations of Bayesian survival models. In this article we describe how the rstanarm R package can be used to fit a wide range of Bayesian survival models. The rstanarm package facilitates Bayesian regression modelling by providing a user-friendly interface (us…
Information of interest can often only be extracted from data by model fitting. When the functional form of such a model can not be deduced from first principles, one has to make a choice between different possible models. A common approach in such cases is to minimise the information loss in the model by trying to reduce the number of fit variables (or the model flexibility, respectively) as much as possible while still yielding an acceptable fit to the data. Model selection via the Akaike Information Criterion (AIC) provides such an implementation of Occam's razor. We argue that the same pri…
A Monte Carlo algorithm is said to be adaptive if it automatically calibrates its current proposal distribution using past simulations. The choice of the parametric family that defines the set of proposal distributions is critical for good performance. In this paper, we present such a parametric family for adaptive sampling on high-dimensional binary spaces. A practical motivation for this problem is variable selection in a linear regression context. We want to sample from a Bayesian posterior distribution on the model space using an appropriate version of Sequential Monte Carlo. Raw versions …
We overview Bayesian estimation, hypothesis testing, and model-averaging and illustrate how they benefit parametric survival analysis. We contrast the Bayesian framework to the currently dominant frequentist approach and highlight advantages, such as seamless incorporation of historical data, continuous monitoring of evidence, and incorporating uncertainty about the true data generating process. We illustrate the application of the Bayesian approaches on an example data set from a colon cancer trial. We compare the Bayesian parametric survival analysis and frequentist models with AIC/BIC model…
We model the spatial dynamics of a forest stand by using a special class of spatio-temporal point processes, the sequential spatial point process, where the spatial dimension is parameterized and the time component is atomic. The sequential spatial point processes differ from spatial point processes in the sense that the realizations are ordered sequences of spatial locations and the order of points allows us to approximate the spatial evolutionary dynamics of the process. This feature shall be useful to interpret the long-term dependence and the memory formed by the spatial history of the pro…
Sequential Permutation Testing of Random Forest Variable Importance Measures
Institute of General Practice and Health Services Research（全科医学与卫生服务研究所）；Institute of AI and Informatics in Medicine（医学人工智能与信息学研究所）；Technical University of Munich（慕尼黑工业大学）；Institute for Medical Information Processing, Biometry and Epidemiology（医学信息处理、生物统计学与流行病学研究所）；University of Munich（慕尼黑大学）
