Normal Likelihood Research Articles

On the Optimal Combination of Elliptically Distributed Biomarkers to Improve Diagnostic Accuracy.

Diagnostic biomarkers play a critical role in biomedical research, particularly for the diagnosis and prediction of diseases, etc. To enhance diagnostic accuracy, extensive research about combining multiple biomarkers has been developed based on the multivariate normality, which is often not true in practice, as most biomarkers follow distributions that deviate from normality. While the likelihood ratio combination is recognized to be the optimal approach, it is complicated to calculate. To achieve a more accurate and effective combination of biomarkers, especially when these biomarkers deviate from normality, we propose using a receiver operating characteristic (ROC) curve methodology based on the optimal combination of elliptically distributed biomarkers. In this paper, we derive the ROC curve function for the elliptical likelihood ratio combination. Further, proceeding from the derived best combinations of biomarkers, we propose an efficient technique via nonparametric maximum likelihood estimate (NPMLE) to build empirical estimation. Simulation results show that the proposed elliptical combination method consistently provided better performance, demonstrating its robustness in handling various distribution types of biomarkers. We apply the proposed method to two real datasets: Autism/autism spectrum disorder (ASD) and neural tube defects (NTD). In both applications, the elliptical likelihood ratio combination improves the AUC value compared to the multivariate normal likelihood ratio combination and the best linear combination.

Network meta-analysis of second line and beyond treatment options in metastatic clear cell renal cell carcinoma

IntroductionDeciding on the optimal second-line (2L) treatment for metastatic clear-cell renal cell carcinoma (ccRCC) remains challenging due to the limited information comparing each of the available options and the influence of the newly expanding first-line (1L) agents. Patients and MethodsWe identified phase II/III randomized controlled trials (RCTs) evaluating 2L treatments in metastatic ccRCC. This Network Meta-analysis (NMA) evaluates the overall survival (OS), progression-free survival (PFS), objective response rate (ORR), and severe adverse events (SAE). We used normal likelihood model to incorporate log hazard ratios (HRs), odds ratios (OR), and 95%-confidence-intervals (CI). Treatment p-scores were used for ranking. Data was analyzed in a fixed-effects model using the netmeta package in R v.1.5-0. ResultsAll therapies demonstrated some benefits over placebo. Lenvatinib + everolimus ranked first for OS (HR = 0.44; 95%CI = 0.24–0.82; p-score = 0.92), PFS (HR = 0.13; 95%CI = 0.07–0.24, p-score = 0.98), and ORR (OR = 35.95; 95%CI = 11.55–111.87; p-score = 0.93) compared to placebo, though with a higher SAE (OR = 5.27; p-score = 0.23). Cabozantinib ranked second for OS (HR = 0.57, p-score = 0.80), PFS (HR = 0.19; p-score = 0.86), and ORR (OR = 27.24, p-score = 0.84). Nivolumab was third for ORR (p-score = 0.79), fourth for OS (p-score = 0.69), fifth for PFS (p-score = 0.61), and last for SAE (p-score = 0.83). Lenvatinib monotherapy ranked worst SAE (OR = 5.89, p-score = 0.17) and third for OS and PFS. The latest drug, tivozanib, was sixth for PFS, OS, and ORR. The NMA matrix revealed no differential OS benefit between cabozantinib, lenvatinib + everolimus, and nivolumab. Other regimens had no significant OS benefit when compared to placebo. ConclusionBased on OS and PFS, the lenvtatinib + everolimus combination yielded superior, followed by cabozantinib and Lenvatinib monotherapies; all were limited by a worse SAE profile. Nivolumab and pazopanib had the lowest odds of SAEs.

Bayesian boundary trend filtering

Estimating boundary curves has many applications such as economics, climate science, and medicine. Bayesian trend filtering has been developed as one of locally adaptive smoothing methods to estimate the non-stationary trend of data. This paper develops a Bayesian trend filtering for estimating the boundary trend. To this end, the truncated multivariate normal working likelihood and global-local shrinkage priors based on the scale mixtures of normal distribution are introduced. In particular, well-known horseshoe prior for difference leads to locally adaptive shrinkage estimation for boundary trend. However, the full conditional distributions of the Gibbs sampler involve high-dimensional truncated multivariate normal distribution. To overcome the difficulty of sampling, an approximation of truncated multivariate normal distribution is employed. Using the approximation, the proposed models lead to an efficient Gibbs sampling algorithm via the Pólya-Gamma data augmentation. The proposed method is also extended by considering a nearly isotonic constraint. The performance of the proposed method is illustrated through some numerical experiments and real data examples.

Bayesian approach for a 2 x 2 crossover design with repeated measures: a simulation study

In crossover designs, the subjects receive all treatments, according to the groups of sequences formed. Therefore, if carryover effects are present in the model, inferences about the treatments effects become difficult. Furthermore, repeated measures of the response variable can be taken over time in the same experimental unit; however, these measures may be correlated. In this way, we aimed to analyze a 2 x 2 crossover design with repeated measures within the treatment period, using a Bayesian approach. A simulation study was performed to evaluate the performance. The posterior estimates of the model parameters were obtained under non-informative prior distributions and the normal likelihood function. The model performed well with a sample size of 20 subjects, showing even better results with samples of 100 subjects. With larger samples, exact tests for the differences in carryover effects and time effects were obtained. However, the test of time effect proved to be powerful even with small samples. In turn, considering carryover effects different from zero did not influence the estimates of treatment differences, although biased estimates of the period effect were obtained.

Fast Methods for Posterior Inference of Two-Group Normal-Normal Models

We describe a class of algorithms for evaluating posterior moments of certain Bayesian linear regression models with a normal likelihood and a normal prior on the regression coefficients. The proposed methods can be used for hierarchical mixed effects models with partial pooling over one group of predictors, as well as random effects models with partial pooling over two groups of predictors. We demonstrate the performance of the methods on two applications, one involving U.S. opinion polls and one involving the modeling of COVID-19 outbreaks in Israel using survey data. The algorithms involve analytical marginalization of regression coefficients followed by numerical integration of the remaining low-dimensional density. The dominant cost of the algorithms is an eigendecomposition computed once for each value of the outside parameter of integration. Our approach drastically reduces run times compared to state-of-the-art Markov chain Monte Carlo (MCMC) algorithms. The latter, in addition to being computationally expensive, can also be difficult to tune when applied to hierarchical models.

Bayesian model updating utilizing scaled likelihood ratio and BCT-PCA with frequency response function

Bayesian inference has now been widely practiced for model updating. In this study, the discussion is conducted about two issues facing the implementation of Bayesian model updating when the Gaussian distribution is assumed. To begin with, the problem of the normal likelihood function used in Markov chain Monte Carlo (MCMC) is demonstrated and presented with an alternative approach by defining the scaled likelihood ratio (SLR). Then, a data preprocessing technique is proposed by introducing the Box-Cox transformation and principal component analysis (BCT-PCA) to address the obstacles in the practice of frequency response function (FRF)-based Bayesian model updating. The differential evolution adaptive metropolis (DREAM) is applied to explore the posterior distribution of the parameters. In addition, a novel procedure for FRF-based Bayesian model updating is presented. The effectiveness of the proposed methodology is verified in both numerical and experimental cases of dynamical model updating. In the numerical case, the results of eigenfrequency-based model updating show that the SLR is advantageous over the normal likelihood ratio (NLR) in estimating the posterior distribution of parameters. The results of FRF-based model updating show that the consistency and robustness of model updating performance as achieved through the BCT-PCA method are improved when different schemes of frequency selection are implemented. The results of the experimental case also show the superiority of the proposed method.

Predictive and Regenerative Medicine for Humans

In human history, finding a cure for the devastating disease of cancer has been a long, ongoing battle. Recently, scientific research has shown several feasible bio-medicinal ways for the treatment of cancer. The first method is through predictive regenerative medicine, which converts (breast) cancer cells back into phenotypically benign cells using continually updated predictions from Bayesian inference models. Another common method is the rewriting of damaged coordinating genes, but this has underlying moral issues — who can authorize the right to alter human genes? To investigate these methods, this paper will discuss elementary point-set topologies, physical gene mapping, and gene sequencing. The main purpose of this paper is to establish a statistical method with the help of mathematics in our biological life science. The proposed method can be applied in the case of manipulating genetic transcription and expression with a suitable Bayesian inference model (for genetic reprogramming). The peak value of cancer cell conversion is its Bayesian regression (based on the model’s parameters) from the corresponding Bayesian inference (for the normal posterior distribution, prior and likelihood). The relevant predictive conditions can be determined from the respective model, which could be used to reveal elements that help cancer cell conversion maximization/minimization. Practically, we may apply both of the Bayesian optimization and the related attribute adjustment (to the Oct4, SoX2, Klf4 and c-Myc transcription factors) that can induce the reprogramming of cancer cells. The expected outcome is that we can biomedical-engine the cancer cells conversion. To go a further step, one may apply my proposed Net-Seizing Theory to capture the genetic mutations that underly cancer disease. It is hope that we can design the corresponding universal strategy to heal cancer. Finally, one may establish an intelligent warning mechanism for those high risked cancer candidates but NOT for the thinking control in most of the dictatorship countries.

The Effect of Prior Parameters in a Bayesian Approach to Inferring Material Properties from Experimental Measurements

A Bayesian formulation for inference of material properties from experimental signals is proposed with a normal likelihood quantifying noise and an inverse gamma prior expressing uncertainty about noise variance. Asymptotic analysis shows that, for large samples, the posterior distribution is maximized at the database signal that minimizes an L2 norm between the experimental data and database entries. The largest posterior probability tends to 1 as the number of data points on the signals tends to ∞. There exists a critical ratio between the mode of the inverse gamma prior and the true variance of the normally distributed noise. This ratio determines how the posterior probability varies with the shape parameter of the inverse gamma prior. Numerical results based on an analytical expression and simulations confirm the conclusions from the asymptotic analysis. These results show that posterior probabilities are insensitive to the choice of prior mode when the prior’s shape parameter is taken to be small. In spite of the use of interpolation in database construction, the approach is limited to cases where a small number of parameters is to be identified. When multiple signal types are analyzed simultaneously, a method using a weighted average of posterior distributions from the different signal types is proposed. This method is compared with a classical Bayesian approach that uses a joint likelihood. Effective choices of prior parameters are recommended.

Low-Rank Covariance Matrix Estimation for Factor Analysis in Anisotropic Noise: Application to Array Processing and Portfolio Selection

Factor analysis (FA) or principal component analysis (PCA) models the covariance matrix of the observed data as R = SS' + {\Sigma}, where SS' is the low-rank covariance matrix of the factors (aka latent variables) and {\Sigma} is the diagonal matrix of the noise. When the noise is anisotropic (aka nonuniform in the signal processing literature and heteroscedastic in the statistical literature), the diagonal elements of {\Sigma} cannot be assumed to be identical and they must be estimated jointly with the elements of SS'. The problem of estimating SS' and {\Sigma} in the above covariance model is the central theme of the present paper. After stating this problem in a more formal way, we review the main existing algorithms for solving it. We then go on to show that these algorithms have reliability issues (such as lack of convergence or convergence to infeasible solutions) and therefore they may not be the best possible choice for practical applications. Next we explain how to modify one of these algorithms to improve its convergence properties and we also introduce a new method that we call FAAN (Factor Analysis for Anisotropic Noise). FAAN is a coordinate descent algorithm that iteratively maximizes the normal likelihood function, which is easy to implement in a numerically efficient manner and has excellent convergence properties as illustrated by the numerical examples presented in the paper. Out of the many possible applications of FAAN we focus on the following two: direction-of-arrival (DOA) estimation using array signal processing techniques and portfolio selection for financial asset management.

Bayesian evidence-driven likelihood selection for sky-averaged 21-cm signal extraction

AbstractWe demonstrate that the Bayesian evidence can be used to find a good approximation of the ground truth likelihood function of a dataset, a goal of the likelihood-free inference (LFI) paradigm. As a concrete example, we use forward modelled sky-averaged 21-cm signal antenna temperature datasets where we artificially inject noise structures of various physically motivated forms. We find that the Gaussian likelihood performs poorly when the noise distribution deviates from the Gaussian case, for example, heteroscedastic radiometric or heavy-tailed noise. For these non-Gaussian noise structures, we show that the generalised normal likelihood is on a similar Bayesian evidence scale with comparable sky-averaged 21-cm signal recovery as the ground truth likelihood function of our injected noise. We therefore propose the generalised normal likelihood function as a good approximation of the true likelihood function if the noise structure is a priori unknown.

Adaptive Bayesian variable clustering via structural learning of breast cancer data.

The clustering of proteins is of interest in cancer cell biology. This article proposes a hierarchical Bayesian model for protein (variable) clustering hinging on correlation structure. Starting from a multivariate normal likelihood, we enforce the clustering through prior modeling using angle-based unconstrained reparameterization of correlations and assume a truncated Poisson distribution (to penalize a large number of clusters) as prior on the number of clusters. The posterior distributions of the parameters are not in explicit form and we use a reversible jump Markov chain Monte Carlobased technique is used to simulate the parameters from the posteriors. The end products of the proposed method are estimated cluster configuration of the proteins (variables) along with the number of clusters. The Bayesian method is flexible enough to cluster the proteins as well as estimate the number of clusters. The performance of the proposed method has been substantiated with extensive simulation studies and one protein expression data with a hereditary disposition in breast cancer where the proteins are coming from different pathways.

On the use of distribution-adaptive likelihood functions: Generalized and universal likelihood functions, scoring rules and multi-criteria ranking

This paper is concerned with the formulation of an adequate likelihood function in the application of Bayesian epistemology to uncertainty quantification of hydrologic models. We focus our attention on a special class of likelihood functions (hereinafter referred to as distribution-adaptive likelihood functions), which do not require prior assumptions about the expected distribution of the residuals, rather inference takes place over the hypotheses (model parameters) and space of distribution functions. Our goals are threefold. First, we present theory of a revised implementation of the generalized likelihood (GL) function of Schoups and Vrugt (2010) wherein residual standardization precedes the treatment of serial correlation. This so-called GL+ function, enjoys a solid statistical underpinning and guarantees a more robust joint inference of the autoregressive coefficients and residual properties. Then, as secondary goal, we present a further generalization of the GL+ function, coined the universal likelihood (UL) function, which extends applicability to highly asymmetrical lepto- and platy-kurtic residual distributions. The UL function builds on the 5-parameter skewed generalized Student’s t distribution of Theodossiou (2015) which makes up a large family of continuous probability distributions including (but not limited to) symmetric and skewed forms of the generalized normal, generalized t, Laplace, normal, Student’s t, and Cauchy-Lorentz distributions. As our third and last goal, we present the use of strictly proper scoring rules to evaluate, compare and rank likelihood functions. These scoring rules condense the accuracy of a distribution forecast to a single value while retaining attractive statistical properties. The GL+ and UL functions are illustrated using data of a simple autoregressive scheme and benchmarked against the GL function, Student t likelihood (SL) of Scharnagl et al. (2015) and normal likelihood (NL) for a conceptual hydrologic model using measured streamflow data. Our results show that, (i) the GL+ function is superior to the GL function, (ii) the active set of nuisance variables exerts a large control on the performance of the GL+, SL and UL functions, (iii) the treatment of autocorrelation deteriorates the scoring rules and performance metrics of the forecast distribution, (iv) a leptokurtic distribution is favored for discharge residuals, (v) scoring rules are indispensable in our search for the true forecast distribution, and (vi) the use of multiple strictly proper scoring rules turns the selection of an adequate likelihood function into a multi-criteria problem.

The Most Probable Curve method - A robust approach to estimate kinetic models from low plate count data resulting in reduced uncertainty

A novel method is proposed for fitting microbial inactivation models to data on liquid media: the Most Probable Curve (MPC) method. It is a multilevel model that makes a separation between the “true” microbial concentration according to the model, the “actual” concentration in the media considering chance, and the actual counts on the plate. It is based on the assumptions that stress resistance is homogeneous within a microbial population, and that there is no aggregation of microbial cells. Under these assumptions, the number of colonies in/on a plate follows a Poisson distribution with expected value depending on the proposed kinetic model, the number of dilutions and the plated volume.The novel method is compared against (non)linear regression based on a normal likelihood distribution (traditional method), Poisson regression and gamma-Poisson regression using data on the inactivation of Listeria monocytogenes. The conclusion is that the traditional method has limitations when the data includes plates with low (or zero) cell counts, which can be mitigated using more complex (discrete) likelihoods. However, Poisson regression uses an unrealistic likelihood function, making it unsuitable for survivor curves with several log-reductions. Gamma-Poisson regression uses a more realistic likelihood function, even though it is based mostly on empirical hypotheses. We conclude that the MPC method can be used reliably, especially when the data includes plates with low or zero counts. Furthermore, it generates a more realistic description of uncertainty, integrating the contribution of the plating error and reducing the uncertainty of the primary model parameters. Consequently, although it increases modelling complexity, the MPC method can be of great interest in predictive microbiology, especially in studies focused on variability analysis.

Fused-Lasso Regularized Cholesky Factors of Large Nonstationary Covariance Matrices of Replicated Time Series

The smoothness of subdiagonals of the Cholesky factor of large covariance matrices is closely related to the degree of nonstationarity of autoregressive models for time series data. Heuristically, one expects for nearly stationary covariance matrix entries in each subdiagonal of the Cholesky factor of its inverse to be approximately the same in the sense that the sum of the absolute values of successive differences is small. Statistically, such smoothness is achieved by regularizing each subdiagonal using fused-type lasso penalties. We rely on the standard Cholesky factor as the new parameter within a regularized normal likelihood setup which guarantees: (a) joint convexity of the likelihood function, (b) strict convexity of the likelihood function restricted to each subdiagonal even when n < p, and (c) positive-definiteness of the estimated covariance matrix. A block coordinate descent algorithm, where each block is a subdiagonal, is proposed, and its convergence is established under mild conditions. Lack of decoupling of the penalized likelihood function into a sum of functions involving individual subdiagonals gives rise to some computational challenges and advantages relative to two recent algorithms for sparse estimation of the Cholesky factor, which decouple row-wise. Simulation results and real data analysis show the scope and good performance of the proposed methodology. Software for our method is freely available in R language. Supplementary materials for this article are available online.

A Quantile-Conserving Ensemble Filter Framework. Part I: Updating an Observed Variable

Abstract A general framework for deterministic univariate ensemble filtering is presented. The framework fits a continuous prior probability density function (PDF) to the prior ensemble. A functional representation for the observation likelihood is combined with the prior PDF to get a continuous analysis (posterior) PDF. Cumulative distribution functions for the prior and analysis are also required. The key innovation is that an analysis ensemble is computed so that the quantile of each ensemble member is the same as its prior quantile. Many choices for the prior PDF family and the likelihood function are described. A choice of normal prior with normal likelihood is equivalent to the ensemble adjustment Kalman filter. Some other choices for the prior include gamma, inverse gamma, beta, beta prime, lognormal, and exponential distributions. Both prior distributions and likelihoods can be defined over a set of intervals giving additional flexibility that can be used to implement methods like a Huber likelihood for observations with occasional outliers. Priors and likelihoods can also be defined as sums of distributions allowing choices like bivariate normals or kernel filters. Empirical distributions, for instance piecewise linear approximations to arbitrary PDFs and functions can be used. Another empirical choice leads to the rank histogram filter. Results here are univariate and can be used to compute increments for observed variables or marginal distributions for any variable for a reanalysis. Linear regression of increments can be used to update state variables in a serial filter to build a comprehensive data assimilation system. Part 2 will discuss other methods for extending the framework to multivariate data assimilation. Significance Statement Data assimilation is used to combine information from model forecasts with subsequent observations to obtain better estimates of the current state of the atmosphere or other parts of the Earth system. Ensemble data assimilation uses a number of forecasts to get more information about uncertainty. A new method allows much more flexibility in the assumptions that must be made when doing ensemble data assimilation. As an example, the method can be better for quantities that are bounded like the amount of an atmospheric trace pollutant.

Partition-Based Nonstationary Covariance Estimation Using the Stochastic Score Approximation

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in operations and O(n) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. Supplementary materials for this article are available online.

Robust sparse Bayesian infinite factor models

Most of previous works and applications of Bayesian factor model have assumed the normal likelihood regardless of its validity. We propose a Bayesian factor model for heavy-tailed high-dimensional data based on multivariate Student-t likelihood to obtain better covariance estimation. We use multiplicative gamma process shrinkage prior and factor number adaptation scheme proposed in Bhattacharya and Dunson [Biometrika 98(2):291–306, 2011]. Since a naive Gibbs sampler for the proposed model suffers from slow mixing, we propose a Markov Chain Monte Carlo algorithm where fast mixing of Hamiltonian Monte Carlo is exploited for some parameters in the proposed model. Simulation results illustrate the gain in performance of covariance estimation for heavy-tailed high-dimensional data. We also provide a theoretical result that the posterior of the proposed model is weakly consistent under reasonable conditions. We conclude the paper with the application of the proposed factor model on breast cancer metastasis prediction given DNA signature data of cancer cells.

Network meta-analysis (NMA) of second-line (2L) treatment options in metastatic renal cell carcinoma.

337 Background: Immune checkpoint inhibitors, in combination either with a cytotoxic T-lymphocyte-associated protein 4 or tyrosine kinase inhibitor are approved for first-line (1L) therapy of metastatic renal cell carcinoma (mRCC). Of those, several have been shown to improve median progression-free survival (mPFS) in the 2L. A recent NMA by Irbaz et al. (PMID 33824031) identified the most effective 1L treatments for mRCC. In the absence of head-to-head randomized clinical trials (RCT), we performed an indirect comparison of 2L treatments for mRCC. Methods: We conducted literature searches in Medline, Embase, Cochrane Library, and Google Scholar to identify RCTs evaluating 2L treatments in mRCC. We extracted the mPFS hazard ratios (HR) and 95% confidence intervals (CI). We used a normal likelihood model that incorporates log HRs of the treatment differences for conducting the NMA. P-scores, measuring the certainty that one treatment is better than another averaged over all competing treatments, were used to rank treatments. Higher p scores indicate better outcomes. Data were analyzed with R netmeta package v.1.5-0. Results: All 8 therapies included in this analysis showed benefits in mPFS over placebo (Table). The combination of lenvatinib+everolimus had the lowest hazard of disease progression or death and pazopanib was the highest compared to placebo. Lenvatinib+everolimus ranked highest with a p-score of 0.96 followed by cabozantinib (0.88), and lenvatinib (0.79); with others having p-scores of 0.60 or less (placebo p-score 0.00). Conclusions: This NMA revealed that lenvatinib+everolimus and cabozantinib are most likely to produce the highest PFS benefit in 2L in mRCC. Future studies are needed to evaluate 1L and 2L sequencing options to further inform clinical practice. [Table: see text]

Existence and uniqueness of the Kronecker covariance MLE

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the Kronecker covariance structure allows for standard likelihood inference even when only a very small number of data matrices is available. For instance, in some cases a likelihood ratio test of dependence may be performed with a sample size of one. However, more generally the sample size required to ensure boundedness of the matrix normal likelihood or the existence of a unique maximizer depends in a complicated way on the matrix dimensions. This motivates the study of how large a sample size is needed to ensure that maximum likelihood estimators exist, and exist uniquely with probability one. Our main result gives precise sample size thresholds in the paradigm where the number of rows and the number of columns of the data matrices differ by at most a factor of two. Our proof uses invariance properties that allow us to consider data matrices in canonical form, as obtained from the Kronecker canonical form for matrix pencils.

A unified framework for closed-form nonparametric regression, classification, preference and mixed problems with Skew Gaussian Processes

Skew-Gaussian Processes (SkewGPs) extend the multivariate Unified Skew-Normal distributions over finite dimensional vectors to distribution over functions. SkewGPs are more general and flexible than Gaussian processes, as SkewGPs may also represent asymmetric distributions. In a recent contribution, we showed that SkewGP and probit likelihood are conjugate, which allows us to compute the exact posterior for non-parametric binary classification and preference learning. In this paper, we generalize previous results and we prove that SkewGP is conjugate with both the normal and affine probit likelihood, and more in general, with their product. This allows us to (i) handle classification, preference, numeric and ordinal regression, and mixed problems in a unified framework; (ii) derive closed-form expression for the corresponding posterior distributions. We show empirically that the proposed framework based on SkewGP provides better performance than Gaussian processes in active learning and Bayesian (constrained) optimization. These two tasks are fundamental for design of experiments and in Data Science.