Consider a set of categorical variables where at least one of them is binary. The log-linear model that describes the counts in the resulting contingency table implies a specific logistic regression model, with the binary variable as the outcome. Within the Bayesian framework, the $g$-prior and mixtures of $g$-priors are commonly assigned to the parameters of a generalized linear model. We prove that assigning a $g$-prior (or a mixture of $g$-priors) to the parameters of a certain log-linear model designates a $g$-prior (or a mixture of $g$-priors) on the parameters of the corresponding logistic regression. By deriving an asymptotic result, and with numerical illustrations, we demonstrate that when a $g$-prior is adopted, this correspondence extends to the posterior distribution of the model parameters. Thus, it is valid to translate inferences from fitting a log-linear model to inferences within the logistic regression framework, with regard to the presence of main effects and interaction terms.
We aim at modelling the appearance of distinct tags in a sequence of labelled objects. Common examples of this type of data include words in a corpus or distinct species in a sample. These sequential discoveries are often summarised via accumulation curves, which count the number of distinct entities observed in an increasingly large set of objects. We propose a novel Bayesian nonparametric method for species sampling modelling by directly specifying the probability of a new discovery, therefore allowing for flexible specifications. The asymptotic behavior and finite sample properties of such …
We introduce an extended generalised logistic growth model for discrete outcomes, in which a network structure can be specified to deal with spatial dependence and time dependence is dealt with using an Auto-Regressive approach. A major challenge concerns the specification of the network structure, crucial to consistently estimate the canonical parameters of the generalised logistic curve, e.g. peak time and height. Parameters are estimated under the Bayesian framework, using the {\texttt{ Stan}} probabilistic programming language. The proposed approach is motivated by the analysis of the firs…
Family planning has been characterized by highly different strategic programs in India, including method-specific contraceptive targets, coercive sterilization, and more recent target-free approaches. These major changes in family planning policies over time have motivated a considerable interest towards assessing the effectiveness of the different programs, while understanding which subsets of the population have not been properly addressed. Current studies consider specific aspects of the above policies, including, for example, the factors associated with the choice of alternative contracept…
We present a new method of modelling numerical systems where there are two distinct output solution classes, for example tipping points or bifurcations. Gaussian process emulation is a useful tool in understanding these complex systems and provides estimates of uncertainty, but we aim to include systems where there are discontinuities between the two output solutions. Due to continuity assumptions, we consider current methods of classification to split our input space into two output regions. Classification and logistic regression methods currently rely on drawing from an independent Bernoulli…
Consider a set of categorical variables $\mathcal{P}$ where at least one, denoted by $Y$, is binary. The log-linear model that describes the counts in the resulting contingency table implies a specific logistic regression model, with the binary variable as the outcome. Extending results in Christensen (1997), by also considering the case where factors present in the contingency table disappear from the logistic regression model, we prove that the Maximum Likelihood Estimate (MLE) for the parameters of the logistic regression equals the MLE for the corresponding parameters of the log-linear mod…
Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximate Bayesian inference for LGP density estimation in a grid using Laplace's method to integrate over the non-Gaussian posterior distribution of latent function values and to determine the covariance function parameters with type-II maximum a posteriori (MAP) estimation. We demonstr…
Risk prediction models are a crucial tool in healthcare. Risk prediction models with a binary outcome (i.e., binary classification models) are often constructed using methodology which assumes the costs of different classification errors are equal. In many healthcare applications this assumption is not valid, and the differences between misclassification costs can be quite large. For instance, in a diagnostic setting, the cost of misdiagnosing a person with a life-threatening disease as healthy may be larger than the cost of misdiagnosing a healthy person as a patient. In this work, we present…
One difficulty for real-time tracking of epidemics is related to reporting delay. The reporting delay may be due to laboratory confirmation, logistic problems, infrastructure difficulties and so on. The ability to correct the available information as quickly as possible is crucial, in terms of decision making such as issuing warnings to the public and local authorities. A Bayesian hierarchical modelling approach is proposed as a flexible way of correcting the reporting delays and to quantify the associated uncertainty. Implementation of the model is fast, due to the use of the integrated neste…
In Oncology, trials evaluating drug combinations are becoming more common. While combination therapies bring the potential for greater efficacy, they also create unique challenges for ensuring drug safety. In Phase-I dose escalation trials of drug combinations, model-based approaches enable efficient use of information gathered, but the models need to account for trial complexities: appropriate modeling of interactions becomes increasingly important with growing numbers of drugs being tested simultaneously in a given trial. In principle, we can use data from multiple arms testing varying combi…
Balancing a rare and serious possibility against a more common and less serious one is a familiar problem in many situations, such as the prediction of rare diseases. The relative costs of forecasting errors can be used for any prediction method that gives an estimated probability of a future event. The probability at which the likely cost (defined as cost x probability) of a possible false negative is exactly equal to that of a possible false positive gives the relevant cutpoint and all subjects with probability of disease greater than this have a positive test result. All standard methods of…
Classification is a vital tool that is important for modelling many complex numerical models. A model or system may be such that, for certain areas of input space, the output either does not exist, or is not in a quantifiable form. Here, we present a new method for classification where the model outputs are given distinct classifying labels, which we model using a latent Gaussian process (GP). The latent variable is estimated using MCMC sampling, a unique likelihood and distinct prior specifications. Our classifier is then verified by calculating a misclassification rate across the input space…
Multinomial logistic regression is one of the most popular models for modelling the effect of explanatory variables on a subject choice between a set of specified options. This model has found numerous applications in machine learning, psychology or economy. Bayesian inference in this model is non trivial and requires, either to resort to a MetropolisHastings algorithm, or rejection sampling within a Gibbs sampler. In this paper, we propose an alternative model to multinomial logistic regression. The model builds on the Plackett-Luce model, a popular model for multiple comparisons. We show tha…
In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, a…
In this paper, we introduce and evaluate a data-driven staged mixture modeling technique for building density, regression, and classification models. Our basic approach is to sequentially add components to a finite mixture model using the structural expectation maximization (SEM) algorithm. We show that our technique is qualitatively similar to boosting. This correspondence is a natural byproduct of the fact that we use the SEM algorithm to sequentially fit the mixture model. Finally, in our experimental evaluation, we demonstrate the effectiveness of our approach on a variety of prediction an…
Bivariate observations of binary and ordinal data arise frequently and require a bivariate modelling approach in cases where one is interested in aspects of the marginal distributions as separate outcomes along with the association between the two. We consider methods for constructing such bivariate models with logistic marginals and propose a model based on the Ali-Mikhail-Haq bivariate logistic distribution. We motivate the model as an extension of that based on the Gumbel type 2 distribution as considered by other authors and as a bivariate extension of the logistic distribution which prese…
Dropout has recently emerged as a powerful and simple method for training neural networks preventing co-adaptation by stochastically omitting neurons. Dropout is currently not grounded in explicit modelling assumptions which so far has precluded its adoption in Bayesian modelling. Using Bayesian entropic reasoning we show that dropout can be interpreted as optimal inference under constraints. We demonstrate this on an analytically tractable regression model providing a Bayesian interpretation of its mechanism for regularizing and preventing co-adaptation as well as its connection to other Baye…
Identifying genetic interactions for a given microorganism such as yeast is difficult. Quantitative Fitness Analysis (QFA) is a high-throughput experimental and computational methodology for quantifying the fitness of microbial cultures. QFA can be used to compare between fitness observations for different genotypes and thereby infer genetic interaction strengths. Current "naive" frequentist statistical approaches used in QFA do not model between-genotype variation or difference in genotype variation under different conditions. In this thesis, a Bayesian approach is introduced to evaluate hier…
We propose a multiple imputation method to deal with incomplete categorical data. This method imputes the missing entries using the principal components method dedicated to categorical data: multiple correspondence analysis (MCA). The uncertainty concerning the parameters of the imputation model is reflected using a non-parametric bootstrap. Multiple imputation using MCA (MIMCA) requires estimating a small number of parameters due to the dimensionality reduction property of MCA. It allows the user to impute a large range of data sets. In particular, a high number of categories per variable, a …
This paper is motivated by the panel surveys, called Statistics on Income and Living Conditions (SILC), conducted annually on (randomly selected) country-representative households to monitor EU 2020 aims on poverty reduction. We particularly consider the surveys conducted in Turkey, within the scope of integration to the EU, between 2010 and 2013. Our main interests are on health aspects of economic and living conditions. The outcome is {\it self-reported health} that is clustered longitudinal ordinal, since repeated measures of it are nested within individuals and individuals are nested withi…
This study is concerned with the analysis of three-level ordinal outcome data with polytomous logistic regression in the presence of random-effects. It is assumed that the random-effects follow a Bridge distribution for the logit link, which allows one to obtain marginal interpretations of the regression coefficients. The data are obtained from the Turkish Income and Living Conditions Study, where the outcome variable is self-rated health (SRH), which is ordinal in nature. The analysis of these data is to compare covariate sub-groups and draw region- and family-level inferences in terms of SRH…
It is of particular interests in many application fields to draw doubly robust inference of a logistic partially linear model with the predictor specified as combination of a targeted low dimensional linear parametric function and a nuisance nonparametric function. In recent, Tan (2019) proposed a simple and flexible doubly robust estimator for this purpose. They introduced the two nuisance models, i.e. nonparametric component in the logistic model and conditional mean of the exposure covariates given the other covariates and fixed response, and specified them as fixed dimensional parametric m…
This thesis responds to the challenges of using a large number, such as thousands, of features in regression and classification problems. There are two situations where such high dimensional features arise. One is when high dimensional measurements are available, for example, gene expression data produced by microarray techniques. For computational or other reasons, people may select only a small subset of features when modelling such data, by looking at how relevant the features are to predicting the response, based on some measure such as correlation with the response in the training data. A…
merlin can do a lot of things. From simple stuff, like fitting a linear regression or a Weibull survival model, to a three-level logistic mixed effects model, or a multivariate joint model of multiple longitudinal outcomes (of different types) and a recurrent event and survival with non-linear effects...the list is rather endless. merlin can do things I haven't even thought of yet. I'll take a single dataset, and attempt to show you the full range of capabilities of merlin, and discuss some future directions for the implementation in Stata.
We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards…
Control for confounders in observational studies was generally handled through stratification and standardization until the 1960s. Standardization typically reweights the stratum-specific rates so that exposure categories become comparable. With the development first of loglinear models, soon also of nonlinear regression techniques (logistic regression, failure time regression) that the emerging computers could handle, regression modelling became the preferred approach, just as was already the case with multiple regression analysis for continuous outcomes. Since the mid 1990s it has become inc…
Wildlife monitoring for open populations can be performed using a number of different survey methods. Each survey method gives rise to a type of data and, in the last five decades, a large number of associated statistical models have been developed for analysing these data. Although these models have been parameterised and fitted using different approaches, they have all been designed to model the pattern with which individuals enter and exit the population and to estimate the population size. However, existing approaches rely on a predefined model structure and complexity, either by assuming …
We propose a method for estimation in high-dimensional linear models with nominal categorical data. Our estimator, called SCOPE, fuses levels together by making their corresponding coefficients exactly equal. This is achieved using the minimax concave penalty on differences between the order statistics of the coefficients for a categorical variable, thereby clustering the coefficients. We provide an algorithm for exact and efficient computation of the global minimum of the resulting nonconvex objective in the case with a single variable with potentially many levels, and use this within a block…
Numerical weather forecasts can exhibit systematic errors due to simplifying model assumptions and computational approximations. Statistical postprocessing is a statistical approach to correcting such biases. A statistical postprocessing model takes input data from a numerical forecast model, and outputs a parametric predictive distribution of a real-world observation, with model parameters learned from past forecast-observation pairs. In this paper we develop and discuss methods for postprocessing of gridded data. We show that estimates of postprocessing parameters on a spatial grid can be im…
