Being able to reliably assess not only the accuracy but also the uncertainty of models' predictions is an important endeavour in modern machine learning. Even if the model generating the data and labels is known, computing the intrinsic uncertainty after learning the model from a limited number of samples amounts to sampling the corresponding posterior probability measure. Such sampling is computationally challenging in high-dimensional problems and theoretical results on heuristic uncertainty estimators in high-dimensions are thus scarce. In this manuscript, we characterise uncertainty for learning from limited number of samples of high-dimensional Gaussian input data and labels generated by the probit model. We prove that the Bayesian uncertainty (i.e. the posterior marginals) can be asymptotically obtained by the approximate message passing algorithm, bypassing the canonical but costly Monte Carlo sampling of the posterior. We then provide a closed-form formula for the joint statistics between the logistic classifier, the uncertainty of the statistically optimal Bayesian classifier and the ground-truth probit uncertainty. The formula allows us to investigate calibration of the logistic classifier learning from limited amount of samples. We discuss how over-confidence can be mitigated by appropriately regularising, and show that cross-validating with respect to the loss leads to better calibration than with the 0/1 error.

Variational inference often minimizes the "reverse" Kullbeck-Leibler (KL) KL(q||p) from the approximate distribution q to the posterior p. Recent work studies the "forward" KL KL(p||q), which unlike reverse KL does not lead to variational approximations that underestimate uncertainty. This paper introduces Transport Score Climbing (TSC), a method that optimizes KL(p||q) by using Hamiltonian Monte Carlo (HMC) and a novel adaptive transport map. The transport map improves the trajectory of HMC by acting as a change of variable between the latent variable space and a warped space. TSC uses HMC samples to dynamically train the transport map while optimizing KL(p||q). TSC leverages synergies, where better transport maps lead to better HMC sampling, which then leads to better transport maps. We demonstrate TSC on synthetic and real data. We find that TSC achieves competitive performance when training variational autoencoders on large-scale data.

The aim of this paper is to discuss a recent result which shows that probabilistic inference in the presence of (unknown) causal mechanisms can be tractable for models that have traditionally been viewed as intractable. This result was reported recently in [15] to facilitate model-based supervised learning but it can be interpreted in a causality context as follows. One can compile a non-parametric causal graph into an arithmetic circuit that supports inference in time linear in the circuit size. The circuit is also non-parametric so it can be used to estimate parameters from data and to further reason (in linear time) about the causal graph parametrized by these estimates. Moreover, the circuit size can sometimes be bounded even when the treewidth of the causal graph is not, leading to tractable inference on models that have been deemed intractable previously. This has been enabled by a new technique that can exploit causal mechanisms computationally but without needing to know their identities (the classical setup in causal inference). Our goal is to provide a causality-oriented exposure to these new results and to speculate on how they may potentially contribute to more scalable and versatile causal inference.

The empirical utility of some fields of machine learning has rapidly outpaced our understanding of the underlying theory: the models are unreasonably effective, but we're not entirely sure why. Conversely, other areas of research that are relatively well understood are difficult to implement or have limited applicability in practice. This article attempts to map different fields of machine learning with respect to their utility and understanding, and explores how scientific and technological progress manifests within this framework. Constructing this matrix is a highly subjective exercise, that reduces multi-faceted fields to undefined, single values on one-dimensional scales, that themselves are comprised of multiple factors. This matrix represents my personal view - one in which fields are crudely assessed only by their general characteristics.

For the last two decades, oversampling has been employed to overcome the challenge of learning from imbalanced datasets. Many approaches to solving this challenge have been offered in the literature. Oversampling, on the other hand, is a concern. That is, models trained on fictitious data may fail spectacularly when put to real-world problems. The fundamental difficulty with oversampling approaches is that, given a real-life population, the synthesized samples may not truly belong to the minority class. As a result, training a classifier on these samples while pretending they represent minority may result in incorrect predictions when the model is used in the real world. We analyzed a large number of oversampling methods in this paper and devised a new oversampling evaluation system based on hiding a number of majority examples and comparing them to those generated by the oversampling process. Based on our evaluation system, we ranked all these methods based on their incorrectly generated examples for comparison. Our experiments using more than 70 oversampling methods and three imbalanced real-world datasets reveal that all oversampling methods studied generate minority samples that are most likely to be majority. Given data and methods in hand, we argue that oversampling in its current forms and methodologies is unreliable for learning from class imbalanced data and should be avoided in real-world applications.

Stock Market can be easily seen as one of the most attractive places for investors, but it is also very complex in terms of making trading decisions. Predicting the market is a risky venture because of the uncertainties and nonlinear nature of the market. Deciding on the right time to trade is key to every successful trader as it can lead to either a huge gain of money or totally a loss in investment that will be recorded as a careless trade. The aim of this research is to develop a prediction system for stock market using Fuzzy Logic Type2 which will handle these uncertainties and complexities of human behaviour in general when it comes to buy, hold or sell decision making in stock trading. The proposed system was developed using VB.NET programming language as frontend and Microsoft SQL Server as backend. A total of four different technical indicators were selected for this research. The selected indicators are the Relative Strength Index, William Average, Moving Average Convergence and Divergence, and Stochastic Oscillator. These indicators serve as input variable to the Fuzzy System. The MACD and SO are deployed as primary indicators, while the RSI and WA are used as secondary indicators. Fibonacci retracement ratio was adopted for the secondary indicators to determine their support and resistance level in terms of making trading decisions. The input variables to the Fuzzy System is fuzzified to Low, Medium, and High using the Triangular and Gaussian Membership Function. The Mamdani Type Fuzzy Inference rules were used for combining the trading rules for each input variable to the fuzzy system. The developed system was tested using sample data collected from ten different companies listed on the Nigerian Stock Exchange for a total of fifty two periods. The dataset collected are Opening, High, Low, and Closing prices of each security.

We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected posterior features, based on regression from kernel or neural net features of the observations. All quantities involved in the Bayesian update are learned from observed data, making the method entirely model-free. The resulting algorithm is a novel instance of a kernel Bayes' rule (KBR). Our approach is based on importance weighting, which results in superior numerical stability to the existing approach to KBR, which requires operator inversion. We show the convergence of the estimator using a novel consistency analysis on the importance weighting estimator in the infinity norm. We evaluate our KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. The proposed method yields uniformly better empirical performance than the existing KBR, and competitive performance with other competing methods.

Particle smoothers are SMC (Sequential Monte Carlo) algorithms designed to approximate the joint distribution of the states given observations from a state-space model. We propose dSMC (de-Sequentialized Monte Carlo), a new particle smoother that is able to process $T$ observations in $\mathcal{O}(\log T)$ time on parallel architecture. This compares favourably with standard particle smoothers, the complexity of which is linear in $T$. We derive $\mathcal{L}_p$ convergence results for dSMC, with an explicit upper bound, polynomial in $T$. We then discuss how to reduce the variance of the smoothing estimates computed by dSMC by (i) designing good proposal distributions for sampling the particles at the initialization of the algorithm, as well as by (ii) using lazy resampling to increase the number of particles used in dSMC. Finally, we design a particle Gibbs sampler based on dSMC, which is able to perform parameter inference in a state-space model at a $\mathcal{O}(\log(T))$ cost on parallel hardware.

Causal inference is essential for data-driven decision making across domains such as business engagement, medical treatment or policy making. However, research on causal discovery and inference has evolved separately, and the combination of the two domains is not trivial. In this work, we develop Deep End-to-end Causal Inference (DECI), a single flow-based method that takes in observational data and can perform both causal discovery and inference, including conditional average treatment effect (CATE) estimation. We provide a theoretical guarantee that DECI can recover the ground truth causal graph under mild assumptions. In addition, our method can handle heterogeneous, real-world, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Moreover, the design principle of our method can generalize beyond DECI, providing a general End-to-end Causal Inference (ECI) recipe, which enables different ECI frameworks to be built using existing methods. Our results show the superior performance of DECI when compared to relevant baselines for both causal discovery and (C)ATE estimation in over a thousand experiments on both synthetic datasets and other causal machine learning benchmark datasets.

We benchmark the robustness of maximum likelihood based uncertainty estimation methods to outliers in training data for regression tasks. Outliers or noisy labels in training data results in degraded performances as well as incorrect estimation of uncertainty. We propose the use of a heavy-tailed distribution (Laplace distribution) to improve the robustness to outliers. This property is evaluated using standard regression benchmarks and on a high-dimensional regression task of monocular depth estimation, both containing outliers. In particular, heavy-tailed distribution based maximum likelihood provides better uncertainty estimates, better separation in uncertainty for out-of-distribution data, as well as better detection of adversarial attacks in the presence of outliers.