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Black-box variational inference (BBVI) is a technique to approximate the posterior of Bayesian models by optimization. Similar to MCMC, the user only needs to specify the model; then, the inference procedure is done automatically. In contrast to MCMC, BBVI scales to many observations, is faster for some applications, and can take advantage of highly optimized deep learning frameworks since it can be formulated as a minimization task. In the case of complex posteriors, however, other state-of-the-art BBVI approaches often yield unsatisfactory posterior approximations. This paper presents Bernstein flow variational inference (BF-VI), a robust and easy-to-use method flexible enough to approximate complex multivariate posteriors. BF-VI combines ideas from normalizing flows and Bernstein polynomial-based transformation models. In benchmark experiments, we compare BF-VI solutions with exact posteriors, MCMC solutions, and state-of-the-art BBVI methods, including normalizing flow-based BBVI. We show for low-dimensional models that BF-VI accurately approximates the true posterior; in higher-dimensional models, BF-VI compares favorably against other BBVI methods. Further, using BF-VI, we develop a Bayesian model for the semi-structured melanoma challenge data, combining a CNN model part for image data with an interpretable model part for tabular data, and demonstrate, for the first time, the use of BBVI in semi-structured models.
Outcomes with a natural order commonly occur in prediction problems and often the available input data are a mixture of complex data like images and tabular predictors. Deep Learning (DL) models are state-of-the-art for image classification tasks but frequently treat ordinal outcomes as unordered and lack interpretability. In contrast, classical ordinal regression models consider the outcome’s order and yield interpretable predictor effects but are limited to tabular data. We present ordinal neural network transformation models (ontrams), which unite DL with classical ordinal regression approaches. ontrams are a special case of transformation models and trade off flexibility and interpretability by additively decomposing the transformation function into terms for image and tabular data using jointly trained neural networks. The performance of the most flexible ontram is by definition equivalent to a standard multi-class DL model trained with cross-entropy while being faster in training when facing ordinal outcomes. Lastly, we discuss how to interpret model components for both tabular and image data on two publicly available datasets.