We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance kernels, the core of our proposal is to model power spectral densities through a rectangular function, which results in a kernel based on the sinc function with straightforward extensions to non-centred (around zero frequency) and frequency-varying cases. In addition to its use in regression, the relationship between the sinc kernel and the classic theory is illuminated, in particular, the Shannon-Nyquist theorem is interpreted as posterior reconstruction under the proposed kernel. Additionally, we show that the sinc kernel is instrumental in two fundamental signal processing applications: first, in stereo amplitude modulation, where the non-centred sinc kernel arises naturally. Second, for band-pass filtering, where the proposed kernel allows for a Bayesian treatment that is robust to observation noise and missing data. The developed theory is complemented with illustrative graphic examples and validated experimentally using real-world data.

We address the problem of learning to benchmark the best achievable classifier performance. In this problem the objective is to establish statistically consistent estimates of the Bayes misclassification error rate without having to learn a Bayes-optimal classifier. Our learning to benchmark framework improves on previous work on learning bounds on Bayes misclassification rate since it learns the {\it exact} Bayes error rate instead of a bound on error rate. We propose a benchmark learner based on an ensemble of $\epsilon$-ball estimators and Chebyshev approximation. Under a smoothness assumption on the class densities we show that our estimator achieves an optimal (parametric) mean squared error (MSE) rate of $O(N^{-1})$, where $N$ is the number of samples. Experiments on both simulated and real datasets establish that our proposed benchmark learning algorithm produces estimates of the Bayes error that are more accurate than previous approaches for learning bounds on Bayes error probability.

Given two distinct datasets, an important question is if they have arisen from the the same data generating function or alternatively how their data generating functions diverge from one another. In this paper, we introduce an approach for measuring the distance between two datasets with high dimensionality using variational autoencoders. This approach is augmented by a permutation hypothesis test in order to check the hypothesis that the data generating distributions are the same within a significance level. We evaluate both the distance measurement and hypothesis testing approaches on generated and on public datasets. According to the results the proposed approach can be used for data exploration (e.g. by quantifying the discrepancy/separability between categories of images), which can be particularly useful in the early phases of the pipeline of most machine learning projects.

Uncertainty involves making decisions with incomplete information, and this is the way we generally operate in the world. Handling uncertainty is typically described using everyday words like chance, luck, and risk. Probability is a field of mathematics that gives us the language and tools to quantify the uncertainty of events and reason in a principled manner. In this post, you will discover a gentle introduction to probability. Photo by Emma Jane Hogbin Westby, some rights reserved.

In many scientific and engineering applications, we are tasked with the maximisation of an expensive to evaluate black box function f. Traditional settings for this problem assume just the availability of this single function. However, in many cases, cheap approximations to f may be obtainable. For example, the expensive real world behaviour of a robot can be approximated by a cheap computer simulation. We can use these approximations to eliminate low function value regions cheaply and use the expensive evaluations of f in a small but promising region and speedily identify the optimum. We formalise this task as a multi-fidelity bandit problem where the target function and its approximations are sampled from a Gaussian process. We develop MF-GP-UCB, a novel method based on upper confidence bound techniques. In our theoretical analysis we demonstrate that it exhibits precisely the above behaviour and achieves better bounds on the regret than strategies which ignore multi-fidelity information. Empirically, MF-GP-UCB outperforms such naive strategies and other multi-fidelity methods on several synthetic and real experiments.

In many real-world scenarios where data is high dimensional, test time acquisition of features is a non-trivial task due to costs associated with feature acquisition and evaluating feature value. The need for highly confident models with an extremely frugal acquisition of features can be addressed by allowing a feature selection method to become target aware. We introduce an approach to feature selection that is based on Bayesian learning, allowing us to report target-specific levels of uncertainty, false positive, and false negative rates. In addition, measuring uncertainty lifts the restriction on feature selection being target agnostic, allowing for feature acquisition based on a single target of focus out of many. We show that acquiring features for a specific target is at least as good as common linear feature selection approaches for small non-sparse datasets, and surpasses these when faced with real-world healthcare data that is larger in scale and in sparseness.

The study presents a general framework for discovering underlying Partial Differential Equations (PDEs) using measured spatiotemporal data. The method, called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides which physical terms are necessary and which can be removed (because they are physically negligible in the sense that they do not affect the dynamics too much) from a pool of candidate functions. The method is built on the recent development of Sparse Bayesian Learning; which enforces the sparsity in the to-be-identified PDEs, and therefore can balance the model complexity and fitting error with theoretical guarantees. Without leveraging prior knowledge or assumptions in the discovery process, we use an automated approach to discover ten types of PDEs, including the famous Navier-Stokes and sine-Gordon equations, from simulation data alone. Moreover, we demonstrate our data-driven discovery process with the Complex Ginzburg-Landau Equation (CGLE) using data measured from a traveling-wave convection experiment. Our machine discovery approach presents solutions that has the potential to inspire, support and assist physicists for the establishment of physical laws from measured spatiotemporal data, especially in notorious fields that are often too complex to allow a straightforward establishment of physical law, such as biophysics, fluid dynamics, neuroscience or nonlinear optics.

We present a new explainable AI (XAI) framework aimed at increasing justified human trust and reliance in the AI machine through explanations. We pose explanation as an iterative communication process, i.e. dialog, between the machine and human user. More concretely, the machine generates sequence of explanations in a dialog which takes into account three important aspects at each dialog turn: (a) human's intention (or curiosity); (b) human's understanding of the machine; and (c) machine's understanding of the human user. To do this, we use Theory of Mind (ToM) which helps us in explicitly modeling human's intention, machine's mind as inferred by the human as well as human's mind as inferred by the machine. In other words, these explicit mental representations in ToM are incorporated to learn an optimal explanation policy that takes into account human's perception and beliefs. Furthermore, we also show that ToM facilitates in quantitatively measuring justified human trust in the machine by comparing all the three mental representations. We applied our framework to three visual recognition tasks, namely, image classification, action recognition, and human body pose estimation. We argue that our ToM based explanations are practical and more natural for both expert and non-expert users to understand the internal workings of complex machine learning models. To the best of our knowledge, this is the first work to derive explanations using ToM. Extensive human study experiments verify our hypotheses, showing that the proposed explanations significantly outperform the state-of-the-art XAI methods in terms of all the standard quantitative and qualitative XAI evaluation metrics including human trust, reliance, and explanation satisfaction.

To give you an example of the impact of machine learning, Man group's AHL Dimension programme is a $5.1 billion dollar hedge fund which is partially managed by AI. After it started off, by the year 2015, its machine learning algorithms were contributing more than half of the profits of the fund even though the assets under its management were far less. After reading this blog, you would be able to understand the basic logic behind some popular and incredibly resourceful machine learning algorithms which have been used by the trading community as well as serve as the foundation stone on which you step on to create the best machine learning algorithm. Initially developed in statistics to study the relationship between input and output numerical variables, it was adopted by the machine learning community to make predictions based on the linear regression equation. The mathematical representation of linear regression is a linear equation that combines a specific set of input data (x) to predict the output value (y) for that set of input values.

To give you an example of the impact of machine learning, Man group's AHL Dimension programme is a $5.1 billion dollar hedge fund which is partially managed by AI. After it started off, by the year 2015, its machine learning algorithms were contributing more than half of the profits of the fund even though the assets under its management were far less. After reading this blog, you would be able to understand the basic logic behind some popular and incredibly resourceful machine learning algorithms which have been used by the trading community as well as serve as the foundation stone on which you step on to create the best machine learning algorithm. Initially developed in statistics to study the relationship between input and output numerical variables, it was adopted by the machine learning community to make predictions based on the linear regression equation. The mathematical representation of linear regression is a linear equation that combines a specific set of input data (x) to predict the output value (y) for that set of input values.