Epileptic seizure activity shows complicated dynamics in both space and time. To understand the evolution and propagation of seizures spatially extended sets of data need to be analysed. We have previously described an efficient filtering scheme using variational Laplace that can be used in the Dynamic Causal Modelling (DCM) framework [Friston, 2003] to estimate the temporal dynamics of seizures recorded using either invasive or non-invasive electrical recordings (EEG/ECoG). Spatiotemporal dynamics are modelled using a partial differential equation -- in contrast to the ordinary differential equation used in our previous work on temporal estimation of seizure dynamics [Cooray, 2016]. We provide the requisite theoretical background for the method and test the ensuing scheme on simulated seizure activity data and empirical invasive ECoG data. The method provides a framework to assimilate the spatial and temporal dynamics of seizure activity, an aspect of great physiological and clinical importance.

Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets.

Bayesian neural networks (BNNs) with latent variables are probabilistic models which can automatically identify complex stochastic patterns in the data. We describe and study in these models a decomposition of predictive uncertainty into its epistemic and aleatoric components. First, we show how such a decomposition arises naturally in a Bayesian active learning scenario by following an information theoretic approach. Second, we use a similar decomposition to develop a novel risk sensitive objective for safe reinforcement learning (RL). This objective minimizes the effect of model bias in environments whose stochastic dynamics are described by BNNs with latent variables. Our experiments illustrate the usefulness of the resulting decomposition in active learning and safe RL settings.

Depending on the algorithm/model that generates this dataset metrics present in the dataset will vary. Here is a list of metrics based on the model: Linear Regression, CART numeric, Elastic Net Linear: R-Square, R-Square Adjusted, Mean Absolute Error(MAE), Mean Squared Error(MSE), Relative Absolute Error(RAE), Related Squared Error(RSE), Root Mean Squared Error(RMSE) CART(Classification And Regression Trees), Naive Bayes Classification, Neural Network, Support Vector Machine(SVM), Random Forest, Logistic Regression: Now you know what the Related datasets are and how they can be useful for fine tuning your Machine Learning model or for comparing two different models. .

The Bayes error rate (BER) is a central concept in the statistical theory of classification. It represents the error rate of the Bayes classifier, which assigns a label to an object corresponding to the class with the highest posterior probability. By definition, the Bayes error represents the smallest possible average error rate that can be achieved by any decision rule (Wald, 1947). Because of these properties, the BER is of great interest both for benchmarking classification algorithms as well as for the practical design of classification algorithms. For example, an accurate approximation of the BER can be used for classifier parameter selection, data dimensionality reduction, or variable selection. However, accurate BER approximation is difficult, especially in high dimension, and thus much attention has focused on tight and tractable BER bounds. This paper proposes a model-free approach to designing multiclass classifiers using a bias-corrected BER bound estimated directly from the multiclass data. There exists several useful bounds on the BER that are functions of the class-dependent feature distributions. These include information theoretic divergence measures such as the Chernoffα -divergence (Chernoff, 1952), the Bhattacharyya divergence (Kailath, 1967), or the Jensen-Shannon divergence (Lin, 1991).

We review the problem of defining and inferring a "state" for a control system based on complex, high-dimensional, highly uncertain measurement streams such as videos. Such a state, or representation, should contain all and only the information needed for control, and discount nuisance variability in the data. It should also have finite complexity, ideally modulated depending on available resources. This representation is what we want to store in memory in lieu of the data, as it "separates" the control task from the measurement process. For the trivial case with no dynamics, a representation can be inferred by minimizing the Information Bottleneck Lagrangian in a function class realized by deep neural networks. The resulting representation has much higher dimension than the data, already in the millions, but it is smaller in the sense of information content, retaining only what is needed for the task. This process also yields representations that are invariant to nuisance factors and having maximally independent components. We extend these ideas to the dynamic case, where the representation is the posterior density of the task variable given the measurements up to the current time, which is in general much simpler than the prediction density maintained by the classical Bayesian filter. Again this can be finitely-parametrized using a deep neural network, and already some applications are beginning to emerge. No explicit assumption of Markovianity is needed; instead, complexity trades off approximation of an optimal representation, including the degree of Markovianity.

Many countries are suffering from severe air pollution. Understanding how different air pollutants accumulate and propagate is critical to making relevant public policies. In this paper, we use urban big data (air quality data and meteorological data) to identify the \emph{spatiotemporal (ST) causal pathways} for air pollutants. This problem is challenging because: (1) there are numerous noisy and low-pollution periods in the raw air quality data, which may lead to unreliable causality analysis, (2) for large-scale data in the ST space, the computational complexity of constructing a causal structure is very high, and (3) the \emph{ST causal pathways} are complex due to the interactions of multiple pollutants and the influence of environmental factors. Therefore, we present \emph{p-Causality}, a novel pattern-aided causality analysis approach that combines the strengths of \emph{pattern mining} and \emph{Bayesian learning} to efficiently and faithfully identify the \emph{ST causal pathways}. First, \emph{Pattern mining} helps suppress the noise by capturing frequent evolving patterns (FEPs) of each monitoring sensor, and greatly reduce the complexity by selecting the pattern-matched sensors as "causers". Then, \emph{Bayesian learning} carefully encodes the local and ST causal relations with a Gaussian Bayesian network (GBN)-based graphical model, which also integrates environmental influences to minimize biases in the final results. We evaluate our approach with three real-world data sets containing 982 air quality sensors, in three regions of China from 01-Jun-2013 to 19-Dec-2015. Results show that our approach outperforms the traditional causal structure learning methods in time efficiency, inference accuracy and interpretability.

This work brings together two powerful concepts in Gaussian processes: the variational approach to sparse approximation and the spectral representation of Gaussian processes. This gives rise to an approximation that inherits the benefits of the variational approach but with the representational power and computational scalability of spectral representations. The work hinges on a key result that there exist spectral features related to a finite domain of the Gaussian process which exhibit almost-independent covariances. We derive these expressions for Matern kernels in one dimension, and generalize to more dimensions using kernels with specific structures. Under the assumption of additive Gaussian noise, our method requires only a single pass through the dataset, making for very fast and accurate computation. We fit a model to 4 million training points in just a few minutes on a standard laptop. With non-conjugate likelihoods, our MCMC scheme reduces the cost of computation from O(NM2) (for a sparse Gaussian process) to O(NM) per iteration, where N is the number of data and M is the number of features.

Bayes' theorem finds many uses in the probability theory and statistics. There's a micro chance that you have never heard about this theorem in your life. Turns out that this theorem has found its way into the world of machine learning, to form one of the highly decorated algorithms. In this article, we will learn all about the Naive Bayes Algorithm, along with its variations for different purposes in machine learning. As you might have guessed, this requires us to view things from a probabilistic point of view.

We show how an ensemble of $Q^*$-functions can be leveraged for more effective exploration in deep reinforcement learning. We build on well established algorithms from the bandit setting, and adapt them to the $Q$-learning setting. We propose an exploration strategy based on upper-confidence bounds (UCB). Our experiments show significant gains on the Atari benchmark.