Pseudo-likelihood Function Research Articles

Semiparametric model averaging prediction in nested case-control studies

Survival predictions for patients are becoming increasingly important in clinical practice as they play a crucial role in aiding healthcare professionals to make more informed diagnoses and treatment decisions. The nested case-control designs have been extensively utilized as a cost-effective solution in many large cohort studies across epidemiology and other research fields. To achieve accurate survival predictions of individuals from nested case-control studies, we propose a semiparametric model averaging approach based on the partly linear additive proportional hazards structure to avoid the curse of dimensionality. The inverse probability weighting method is considered to estimate the parameters of submodels used in model averaging. We choose the weights by maximizing the pseudo-likelihood function constructed for the aggregated model and discuss the asymptotic optimality of selected weights. Simulation studies are conducted to assess the performance of our proposed model averaging method in the nested case-control study. Furthermore, we apply the proposed approach to real data to demonstrate its superiority.

Two-stage pseudo maximum likelihood estimation of semiparametric copula-based regression models for semi-competing risks data.

We propose a two-stage estimation procedure for a copula-based model with semi-competing risks data, where the non-terminal event is subject to dependent censoring by the terminal event, and both events are subject to independent censoring. With a copula-based model, the marginal survival functions of individual event times are specified by semiparametric transformation models, and the dependence between the bivariate event times is specified by a parametric copula function. For the estimation procedure, in the first stage, the parameters associated with the marginal of the terminal event are estimated using only the corresponding observed outcomes, and in the second stage, the marginal parameters for the non-terminal event time and the copula parameter are estimated together via maximizing a pseudo-likelihood function based on the joint distribution of the bivariate event times. We derived the asymptotic properties of the proposed estimator and provided an analytic variance estimator for inference. Through simulation studies, we showed that our approach leads to consistent estimates with less computational cost and more robustness than the one-stage procedure developed in Chen YH (Lifetime Data Anal 18:36-57, 2012), where all parameters were estimated simultaneously. In addition, our approach demonstrates more desirable finite-sample performances over another existing two-stage estimation method proposed in Zhu H et al., (Commu Statistics-Theory Methods 51(22):7830-7845, 2021) . An R package PMLE4SCR is developed to implement our proposed method.

Joint Direction of Arrival-Polarization Parameter Tracking Algorithm Based on Multi-Target Multi-Bernoulli Filter

This paper presents a tracking algorithm for joint estimation of direction of arrival (DOA) and polarization parameters, which exhibit dynamic behavior due to the movement of signal source carriers. The proposed algorithm addresses the challenge of real-time estimation in multi-target scenarios with an unknown number. This algorithm is built upon the Multi-target Multi-Bernoulli (MeMBer) filter algorithm, which makes use of a sensor array called Circular Orthogonal Double-Dipole (CODD). The algorithm begins by constructing a Minimum Description Length (MDL) principle, taking advantage of the characteristics of the polarization-sensitive array. This allows for adaptive estimation of the number of signal sources and facilitates the separation of the noise subspace. Subsequently, the joint parameter Multiple Signal Classification (MUSIC) spatial spectrum function is employed as the pseudo-likelihood function, overcoming the limitations imposed by unknown prior information constraints. To approximate the posterior distribution of MeMBer filters, Sequential Monte Carlo (SMC) method is utilized. The simulation results demonstrate that the proposed algorithm achieves excellent tracking accuracy in joint DOA-polarization parameter estimation, whether in scenarios with known or unknown numbers of signal sources. Moreover, the algorithm demonstrates robust tracking convergence even under low Signal-to-Noise Ratio (SNR) conditions.

CONSISTENT NON-GAUSSIAN PSEUDO MAXIMUM LIKELIHOOD ESTIMATORS OF SPATIAL AUTOREGRESSIVE MODELS

This paper studies the non-Gaussian pseudo maximum likelihood (PML) estimation of a spatial autoregressive (SAR) model with SAR disturbances. If the spatial weights matrix $M_{n}$ for the SAR disturbances is normalized to have row sums equal to 1 or the model reduces to a SAR model with no SAR process of disturbances, the non-Gaussian PML estimator (NGPMLE) for model parameters except the intercept term and the variance $\sigma _{0}^{2}$ of independent and identically distributed (i.i.d.) innovations in the model is consistent. Without row normalization of $M_{n}$ , the symmetry of i.i.d. innovations leads to consistent NGPMLE for model parameters except $\sigma _{0}^{2}$ . With neither row normalization of $M_{n}$ nor the symmetry of innovations, a location parameter can be added to the non-Gaussian pseudo likelihood function to achieve consistent estimation of model parameters except $\sigma _{0}^{2}$ . The NGPMLE with no added parameter can have a significant efficiency improvement upon the Gaussian PML estimator and the generalized method of moments estimator based on linear and quadratic moments. We also propose a non-Gaussian score test for spatial dependence, which can be locally more powerful than the Gaussian score test. Monte Carlo results show that our NGPMLE with no added parameter and the score test based on it perform well in finite samples.

Bayesian Predictive Inference Under Nine Methods for Incorporating Survey Weights

Sample surveys play a significant role in obtaining reliable estimators of finite population quantities, and survey weights are used to deal with selection bias and  non-response bias. The main idea of this research is to compare the performance of nine methods with differently constructed survey weights, and we can use these methods for non-probability sampling after weights are estimated (e.g. quasi-randomization). The original survey weights are calibrated to the population size. In particular, the base model does not include survey weights or design weights. We use original survey weights, adjusted survey weights, trimmed survey weights, and adjusted trimmed survey weights into pseudo-likelihood function to build unnormalized or normalized posterior distributions. In this research, we focus on binary data, which occur in many different situations.
 A simulation study is performed and we analyze the simulated data using average posterior mean, average posterior standard deviation, average relative bias, average posterior root mean squared error, and the coverage rate of  95% credible intervals. We also performed an application on body mass index to further understand these nine methods. The results show that methods with trimmed weights are preferred than methods with untrimmed weights, and methods with adjusted weights have higher variability than methods with unadjusted weights.

Robust clustering of COVID-19 cases across U.S. counties using mixtures of asymmetric time series models with time varying and freely indexed covariates

In this paper, we develop a mixture of autoregressive (MoAR) process model with time varying and freely indexed covariates under the flexible class of two–piece distributions using the scale mixtures of normal (TP-SMN) family. This novel family of time series (TP-SMN-MoAR) models was used to examine flexible and robust clustering of reported cases of Covid-19 across 313 counties in the U.S. The TP-SMN distributions allow for symmetrical/ asymmetrical distributions as well as heavy-tailed distributions providing for flexibility to handle outliers and complex data. Developing a suitable hierarchical representation of the TP-SMN family enabled the construction of a pseudo-likelihood function to derive the maximum pseudo-likelihood estimates via an EM-type algorithm.

Weighted functional linear Cox regression model.

The aim of this paper is to develop a weighted functional linear Cox regression model that accounts for the association between a failure time and a set of functional and scalar covariates. We formulate the weighted functional linear Cox regression by incorporating a comprehensive three-stage estimation procedure as a unified methodology. Specifically, the weighted functional linear Cox regression uses a functional principal component analysis to represent the functional covariates and a high-dimensional Cox regression model to capture the joint effects of both scalar and functional covariates on the failure time data. Then, we consider an uncensored probability for each subject by estimating the important parameter of a censoring distribution. Finally, we use such a weight to construct the pseudo-likelihood function and maximize it to acquire an estimator. We also show our estimation and testing procedures through simulations and an analysis of real data from the Alzheimer's Disease Neuroimaging Initiative.

The sufficiency of the evidence, the relevancy of the evidence, and quantifying both with a single number

Consider a data set as a body of evidence that might confirm or disconfirm a hypothesis about a parameter value. If the posterior probability of the hypothesis is high enough, then the truth of the hypothesis is accepted for some purpose such as reporting a new discovery. In that way, the posterior probability measures the sufficiency of the evidence for accepting the hypothesis. It would only follow that the evidence is relevant to the hypothesis if the prior probability were not already high enough for acceptance. A measure of the relevancy of the evidence is the Bayes factor since it is the ratio of the posterior odds to the prior odds. Measures of the sufficiency of the evidence and measures of the relevancy of the evidence are not mutually exclusive. An example falling in both classes is the likelihood ratio statistic, perhaps based on a pseudolikelihood function that eliminates nuisance parameters. There is a sense in which the likelihood ratio statistic measures both the sufficiency of the evidence and its relevancy. That result is established by representing the likelihood ratio statistic in terms of a conditional possibility measure that satisfies logical coherence rather than probabilistic coherence.

Synthetic Microdata for Establishment Surveys Under Informative Sampling

AbstractMany agencies are investigating whether releasing synthetic microdata could be a viable dissemination strategy for highly sensitive data, such as business data, for which disclosure avoidance regulations otherwise prohibit the release of public use microdata. However, existing methods assume that the original data either cover the entire population or comprise a simple random sample, which limits the application of these methods in the context of survey data with unequal weights. This paper discusses synthetic data generation under informative sampling. To utilise design information in survey weights, we rely on the pseudo likelihood approach when building a hierarchical Bayesian model to estimate the distribution of the finite population. Then, synthetic populations are randomly drawn from the estimated finite population density. We present the full conditional distributions of the Markov chain Monte Carlo algorithm for posterior inference with the pseudo likelihood function. Using simulation studies, we show that the suggested synthetic data approach offers high utility for design-based and model-based analyses while offering a high level of disclosure protection. We apply the proposed method to a subset of the 2012 U.S. Economic Census and evaluate results with utility metrics and disclosure avoidance metrics under data attacker scenarios commonly used for business data.

Novel method for a posteriori uncertainty quantification in wildland fire spread simulation

Simulation is used to predict the spread of a wildland fire across land in real-time. Nevertheless, the large uncertainties in these simulations must be quantified in order to provide better information to fire managers. Ensemble forecasts are usually applied for this purpose, with an input parameter distribution that is defined based on expert knowledge. A novel approach is proposed in order to generate calibrated ensembles whose input distribution is defined by a posterior PDF with a pseudo-likelihood function that involves the Wasserstein distance between simulated and observed burned surfaces of several fire cases. Due to the high dimension and the computational requirements of the pseudo-likelihood function, a Gaussian process emulator is built to obtain a sample of the calibrated input distribution with a MCMC algorithm in about one day of computation on 8 computing cores. The calibrated ensembles lead to better overall accuracy than the uncalibrated ensembles. The a posteriori probability distribution of the inputs favors lower values of rate of spread and lower uncertainty in wind direction. This strongly limits overprediction, while keeping the ability of the ensemble to cover the observed burned area.

Time-varying Hazards Model for Incorporating Irregularly Measured, High-Dimensional Biomarkers.

Clinical studies with time-to-event outcomes often collect measurements of a large number of time-varying covariates over time (e.g., clinical assessments or neuroimaging biomarkers) to build time-sensitive prognostic model. An emerging challenge is that due to resource-intensive or invasive (e.g., lumbar puncture) data collection process, biomarkers may be measured infrequently and thus not available at every observed event time point. Lever-aging all available, infrequently measured time-varying biomarkers to improve prognostic model of event occurrence is an important and challenging problem. In this paper, we propose a kernel-smoothing based approach to borrow information across subjects to remedy infrequent and unbalanced biomarker measurements under a time-varying hazards model. A penalized pseudo-likelihood function is proposed for estimation, and an efficient augmented penalization minimization algorithm related to the alternating direction method of multipliers (ADMM) is adopted for computation. Under some regularity conditions to carefully control approximation bias and stochastic variability, we show that even in the presence of ultra-high dimensionality, the proposed method selects important biomarkers with high probability. Through extensive simulation studies, we demonstrate superior performance in terms of estimation and selection performance compared to alternative methods. Finally, we apply the proposed method to analyze a recently completed real world study to model time to disease conversion using longitudinal, whole brain structural magnetic resonance imaging (MRI) biomarkers, and show a substantial improvement in performance over current standards including using baseline measures only.

High-dimensional structure learning of binary pairwise Markov networks: A comparative numerical study

Learning the undirected graph structure of a Markov network from data is a problem that has received a lot of attention during the last few decades. As a result of the general applicability of the model class, a myriad of methods have been developed in parallel in several research fields. Recently, as the size of the considered systems has increased, the focus of new methods has been shifted towards the high-dimensional domain. In particular, introduction of the pseudo-likelihood function has pushed the limits of score-based methods which were originally based on the likelihood function. At the same time, methods based on simple pairwise tests have been developed to meet the challenges arising from increasingly large data sets in computational biology. Apart from being applicable to high-dimensional problems, methods based on the pseudo-likelihood and pairwise tests are fundamentally very different. To compare the accuracy of the different types of methods, an extensive numerical study is performed on data generated by binary pairwise Markov networks. A parallelizable Gibbs sampler, based on restricted Boltzmann machines, is proposed as a tool to efficiently sample from sparse high-dimensional networks. The results of the study show that pairwise methods can be more accurate than pseudo-likelihood methods in settings often encountered in high-dimensional structure learning applications.

Simplex regression models with measurement error

This paper considers the simplex regression model when there is measurement error in the covariate. We consider a structural approach where the measurement error follows a normal or gamma distribution. We apply a Monte Carlo EM algorithm to estimate the parameters using a pseudo-likelihood function. A simulation study is used to investigate the impact of ignoring the measurement error. Finally, the results are illustrated with a data set.

Automated Harmonization of Bass Lines from Bach Chorales: A Hybrid Approach

Abstract In this article, a combination of two novel approaches to the harmonization of chorales in the style of J. S. Bach is proposed, implemented, and profiled. The first is the use of the bass line, as opposed to the melody, as the primary input into a chorale-harmonization algorithm. The second is a compromise between methods guided by music knowledge and by machine-learning techniques, designed to mimic the way a music student learns. Specifically, our approach involves learning harmonic structure through a hidden Markov model, and determining individual voice lines by optimizing a Boltzmann pseudolikelihood function incorporating musical constraints through a weighted linear combination of constraint indicators. Although previous generative models have focused only on codifying musical rules or on machine learning without any rule specification, by using a combination of musicologically sound constraints with weights estimated from chorales composed by Bach, we were able to produce musical output in a style that closely resembles Bach's chorale harmonizations. A group of test subjects was able to distinguish which chorales were computer generated only 51.3% of the time, a rate not significantly different from guessing.

On semiparametric transformation model with LTRC data: pseudo likelihood approach

When the distribution of the truncation time is known up to a finite-dimensional parameter vector, many researches have been conducted with the objective to improve the efficiency of estimation for nonparametric or semiparametric model with left-truncated and right-censored (LTRC) data. When the distribution of truncation times is unspecified, one approach is to use the conditional maximum likelihood estimators (cMLE) (Chen and Shen in Lifetime Data Anal https://doi.org/10.1007/s10985-016-9385-9, 2017). Although the cMLE has nice asymptotic properties, it is not efficient since the conditional likelihood function does not incorporate information on the distribution of truncation time. In this article, we aim to develop a more efficient estimator by considering the full likelihood function. Following Turnbull (J R Stat Soc B 38:290–295, 1976) and Qin et al. (J Am Stat Assoc 106:1434–1449, 2011), we treat the unobserved (left-truncated) subpopulation as missing data and propose a two-stage approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first stage, the distribution of left truncation time is estimated by the inverse-probability-weighted (IPW) estimator (Wang in J Am Stat Assoc 86:130–143, 1991). In the second stage, we obtain the pseudo complete-data likelihood function by replacing the distribution of truncation time with the IPW estimator in the full likelihood. We propose an expectation–maximization algorithm for obtaining the PMLE and establish the consistency of the PMLE. Simulation results show that the PMLE outperforms the cMLE in terms of mean squared error. The PMLE can also be used to analyze the length-biased data, where the truncation time is uniformly distributed. We demonstrate that the PMLE works more robust against the support assumption of truncation time for length-biased data compared with the MLE proposed by Qin et al. (2011). We apply our proposed method to the channing house data. While the PMLE is quite appealing under specific cases with independent censoring and time-invariant covariates, its applicability, as shown in simulation study, can be rather restricted for more general settings.

Multivariate functional clustering and its application to typhoon data

We propose a multivariate nonlinear mixed effects model for clustering multiple longitudinal data. Advantages of the nonlinear mixed effects model are that it is easy to handle unbalanced data which highly occur in the longitudinal study, and it can take into account associations among longitudinal variables at a given time point. The joint modeling for multivariate longitudinal data, however, requires a high computational cost because numerous parameters are included in the model. To overcome this issue, we perform a pairwise fitting procedure based on a pseudo-likelihood function. Unknown parameters included in each bivariate model are estimated by the maximum likelihood method along with the EM algorithm, and then the number of basis functions included in the model is selected by model selection criteria. After estimating the model, a non-hierarchical clustering algorithm by self-organizing maps is applied to the predicted coefficient vectors of individual specific random effect functions. We present the results of the application of the proposed method to the analysis of data of typhoons that occurred between 2000 and 2017 in Asia.

Improving conditional random field model for prediction of protein‐RNA residue‐base contacts

BackgroundFor understanding biological cellular systems, it is important to analyze interactions between protein residues and RNA bases. A method based on conditional random fields (CRFs) was developed for predicting contacts between residues and bases, which receives multiple sequence alignments for given protein and RNA sequences, respectively, and learns the model with many parameters involved in relationships between neighboring residue‐base pairs by maximizing the pseudo likelihood function.MethodsIn this paper, we proposed a novel CRF‐based model with more complicated dependency relationships between random variables than the previous model, but which takes less parameters for the sake of avoidance of overfitting to training data.ResultsWe performed cross‐validation experiments for evaluating the proposed model, and took the average of AUC (area under receiver operating characteristic curve) scores. The result suggests that the proposed CRF‐based model without using L1‐norm regularization (lasso) outperforms the existing model with and without the lasso under several input observations to CRFs.ConclusionsWe proposed a novel stochastic model for predicting protein‐RNA residue‐base contacts, and improved the prediction accuracy in terms of the AUC score. It implies that more dependency relationships in a CRF could be controlled by less parameters.

Pseudo maximum likelihood estimation of the univariate GARCH (2,2) and asymptotic normality under dependent innovations

ABSTRACTIn this paper, we first consider the pseudo maximum likelihood estimation of the univariate GARCH (2,2) model and derive the underlying estimator. Then, we make use of the technique of martingales to establish the asymptotic normality of the pseudo-maximum likelihood estimator (PMLE) of the univariate GARCH (2,2) model. Contrary to previous approaches encountered in the statistical literature, the pseudo-likelihood function uses the general form of the density laws of the quadratic exponential family.

Model-free scoring system for risk prediction with application to hepatocellular carcinoma study.

There is an increasing need to construct a risk-prediction scoring system for survival data and identify important risk factors (e.g., biomarkers) for patient screening and treatment recommendation. However, most existing methodologies either rely on strong model assumptions (e.g., proportional hazards) or only handle binary outcomes. In this article, we propose a flexible method that simultaneously selects important risk factors and identifies the optimal linear combination of risk factors by maximizing a pseudo-likelihood function based on the time-dependent area under the receiver operating characteristic curve. Our method is particularly useful for risk evaluation and recommendation of optimal subsequent treatments. We show that the proposed method has desirable theoretical properties, including asymptotic normality and the oracle property after variable selection. Numerical performance is evaluated on several simulation data sets and an application to hepatocellular carcinoma data.

Efficient Bayesian inference for exponential random graph models by correcting the pseudo-posterior distribution

Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves the calculation of an intractable normalizing constant. This barrier motivates the consideration of tractable approximations to the likelihood function, such as the pseudolikelihood function, which offers an approach to constructing such an approximation. Naive implementation of what we term a pseudo-posterior resulting from replacing the likelihood function in the posterior distribution by the pseudolikelihood is likely to give misleading inferences. We provide practical guidelines to correct a sample from such a pseudo-posterior distribution so that it is approximately distributed from the target posterior distribution and discuss the computational and statistical efficiency that result from this approach. We illustrate our methodology through the analysis of real-world graphs. Comparisons against the approximate exchange algorithm of Caimo and Friel (2011) are provided, followed by concluding remarks.