The motive behind Creating this repo is to feel the fear of mathematics and do what ever you want to do in Machine Learning, Deep Learning and other fields of AI . So, try this Code in your python notebook which is provided in edx Course. In this Repo you will also learn the Libraries which are essential like numpy, pandas, matplotlib... I am going to upload new material when i find those material useful, you can also help me in keeping this repo fresh. Selecting the right algorithm which includes giving considerations to accuracy, training time, model complexity, number of parameters and number of features.

In a number of data-driven applications such as detection of arrhythmia, interferometry or audio compression, observations are acquired indistinctly in the time or frequency domains: temporal observations allow us to study the spectral content of signals (e.g., audio), while frequency-domain observations are used to reconstruct temporal/spatial data (e.g., MRI). Classical approaches for spectral analysis rely either on i) a discretisation of the time and frequency domains, where the fast Fourier transform stands out as the \textit{de facto} off-the-shelf resource, or ii) stringent parametric models with closed-form spectra. However, the general literature fails to cater for missing observations and noise-corrupted data. Our aim is to address the lack of a principled treatment of data acquired indistinctly in the temporal and frequency domains in a way that is robust to missing or noisy observations, and that at the same time models uncertainty effectively. To achieve this aim, we first define a joint probabilistic model for the temporal and spectral representations of signals, to then perform a Bayesian model update in the light of observations, thus jointly reconstructing the complete (latent) time and frequency representations. The proposed model is analysed from a classical spectral analysis perspective, and its implementation is illustrated through intuitive examples. Lastly, we show that the proposed model is able to perform joint time and frequency reconstruction of real-world audio, healthcare and astronomy signals, while successfully dealing with missing data and handling uncertainty (noise) naturally against both classical and modern approaches for spectral estimation.

Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the unconstrained LME models to allow for sign constraints on its overall coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN) distribution on the random effects instead of the unconstrained Normal distribution which is often found in classical literature. With the aforementioned change, difficulty has dramatically increased as the exact distribution of the dependent variable becomes analytically intractable. We then develop likelihood-based approaches to estimate the unknown model parameters utilizing the approximation of its exact distribution. Hypothesis testing under the new model specification is also discussed and studied empirically. Simulation studies have shown that the proposed constrained model not only improves real-world interpretations of results, but also achieves satisfactory performance on model fits as compared to the existing model.

This article deals with the problem of distributed machine learning, in which agents update their models based on their local datasets, and aggregate the updated models collaboratively and in a fully decentralized manner. In this paper, we tackle the problem of information heterogeneity arising in multi-agent networks where the placement of informative agents plays a crucial role in the learning dynamics. Specifically, we propose BayGo, a novel fully decentralized joint Bayesian learning and graph optimization framework with proven fast convergence over a sparse graph. Under our framework, agents are able to learn and communicate with the most informative agent to their own learning. Unlike prior works, our framework assumes no prior knowledge of the data distribution across agents nor does it assume any knowledge of the true parameter of the system. The proposed alternating minimization based framework ensures global connectivity in a fully decentralized way while minimizing the number of communication links. We theoretically show that by optimizing the proposed objective function, the estimation error of the posterior probability distribution decreases exponentially at each iteration. Via extensive simulations, we show that our framework achieves faster convergence and higher accuracy compared to fully-connected and star topology graphs.

The framework of differential privacy (DP) upper bounds the information disclosure risk involved in using sensitive datasets for statistical analysis. A DP mechanism typically operates by adding carefully calibrated noise to the data release procedure. Generalized linear models (GLMs) are among the most widely used arms in data analyst's repertoire. In this work, with logistic and Poisson regression as running examples, we propose a generic noise-aware Bayesian framework to quantify the parameter uncertainty for a GLM at hand, given noisy sufficient statistics. We perform a tight privacy analysis and experimentally demonstrate that the posteriors obtained from our model, while adhering to strong privacy guarantees, are similar to the non-private posteriors.

We consider a commonly studied supervised classification of a synthetic dataset whose labels are generated by feeding a one-layer neural network with random iid inputs. We study the generalization performances of standard classifiers in the high-dimensional regime where $\alpha=n/d$ is kept finite in the limit of a high dimension $d$ and number of samples $n$. Our contribution is three-fold: First, we prove a formula for the generalization error achieved by $\ell_2$ regularized classifiers that minimize a convex loss. This formula was first obtained by the heuristic replica method of statistical physics. Secondly, focussing on commonly used loss functions and optimizing the $\ell_2$ regularization strength, we observe that while ridge regression performance is poor, logistic and hinge regression are surprisingly able to approach the Bayes-optimal generalization error extremely closely. As $\alpha \to \infty$ they lead to Bayes-optimal rates, a fact that does not follow from predictions of margin-based generalization error bounds. Third, we design an optimal loss and regularizer that provably leads to Bayes-optimal generalization error.

In recent times, neural networks have become a powerful tool for the analysis of complex and abstract data models. However, their introduction intrinsically increases our uncertainty about which features of the analysis are model-related and which are due to the neural network. This means that predictions by neural networks have biases which cannot be trivially distinguished from being due to the true nature of the creation and observation of data or not. In order to attempt to address such issues we discuss Bayesian neural networks: neural networks where the uncertainty due to the network can be characterised. In particular, we present the Bayesian statistical framework which allows us to categorise uncertainty in terms of the ingrained randomness of observing certain data and the uncertainty from our lack of knowledge about how data can be created and observed. In presenting such techniques we show how errors in prediction by neural networks can be obtained in principle, and provide the two favoured methods for characterising these errors. We will also describe how both of these methods have substantial pitfalls when put into practice, highlighting the need for other statistical techniques to truly be able to do inference when using neural networks.

Learning models of the environment from data is often viewed as an essential component to building intelligent reinforcement learning (RL) agents. The common practice is to separate the learning of the model from its use, by constructing a model of the environment's dynamics that correctly predicts the observed state transitions. In this paper we argue that the limited representational resources of model-based RL agents are better used to build models that are directly useful for value-based planning. As our main contribution, we introduce the principle of value equivalence: two models are value equivalent with respect to a set of functions and policies if they yield the same Bellman updates. We propose a formulation of the model learning problem based on the value equivalence principle and analyze how the set of feasible solutions is impacted by the choice of policies and functions. Specifically, we show that, as we augment the set of policies and functions considered, the class of value equivalent models shrinks, until eventually collapsing to a single point corresponding to a model that perfectly describes the environment. In many problems, directly modelling state-to-state transitions may be both difficult and unnecessary. By leveraging the value-equivalence principle one may find simpler models without compromising performance, saving computation and memory. We illustrate the benefits of value-equivalent model learning with experiments comparing it against more traditional counterparts like maximum likelihood estimation. More generally, we argue that the principle of value equivalence underlies a number of recent empirical successes in RL, such as Value Iteration Networks, the Predictron, Value Prediction Networks, TreeQN, and MuZero, and provides a first theoretical underpinning of those results.

Dynamic Uncertain Causality Graph(DUCG) is a recently proposed model for diagnoses of complex systems. It performs well for industry system such as nuclear power plants, chemical system and spacecrafts. However, the variable state combination explosion in some cases is still a problem that may result in inefficiency or even disability in DUCG inference. In the situation of clinical diagnoses, when a lot of intermediate causes are unknown while the downstream results are known in a DUCG graph, the combination explosion may appear during the inference computation. Monte Carlo sampling is a typical algorithm to solve this problem. However, we are facing the case that the occurrence rate of the case is very small, e.g. $10^{-20}$, which means a huge number of samplings are needed. This paper proposes a new scheme based on conditional stochastic simulation which obtains the final result from the expectation of the conditional probability in sampling loops instead of counting the sampling frequency, and thus overcomes the problem. As a result, the proposed algorithm requires much less time than the DUCG recursive inference algorithm presented earlier. Moreover, a simple analysis of convergence rate based on a designed example is given to show the advantage of the proposed method. % In addition, supports for logic gate, logic cycles, and parallelization, which exist in DUCG, are also addressed in this paper. The new algorithm reduces the time consumption a lot and performs 3 times faster than old one with 2.7% error ratio in a practical graph for Viral Hepatitis B.

A computational method based on the non-linear Gaussian process (GP), known as deep Gaussian processes (deep GPs) for uncertainty quantification & propagation in modelling of flow through heterogeneous porous media is presented. The method is also used for reducing dimensionality of model output and consequently emulating highly complex relationship between hydrogeological properties and reduced order fluid velocity field in a tractable manner. Deep GPs are multi-layer hierarchical generalisations of GPs with multiple, infinitely wide hidden layers that are very efficient models for deep learning and modelling of high-dimensional complex systems by tackling the complexity through several hidden layers connected with non-linear mappings. According to this approach, the hydrogeological data is modelled as the output of a multivariate GP whose inputs are governed by another GP such that each single layer is either a standard GP or the Gaussian process latent variable model. A variational approximation framework is used so that the posterior distribution of the model outputs associated to given inputs can be analytically approximated. In contrast to the other dimensionality reduction, methods that do not provide any information about the dimensionality of each hidden layer, the proposed method automatically selects the dimensionality of each hidden layer and it can be used to propagate uncertainty obtained in each layer across the hierarchy. Using this, dimensionality of the full input space consists of both geometrical parameters of modelling domain and stochastic hydrogeological parameters can be simultaneously reduced without the need for any simplifications generally being assumed for stochastic modelling of subsurface flow problems. It allows estimation of the flow statistics with greatly reduced computational efforts compared to other stochastic approaches such as Monte Carlo method.