Probability and Statistics explained in the context of deep learning


This article is intended for beginners in deep learning who wish to gain knowledge about probability and statistics and also as a reference for practitioners. In my previous article, I wrote about the concepts of linear algebra for deep learning in a top down approach ( link for the article) (If you do not have enough idea about linear algebra, please read that first).The same top down approach is used here.Providing the description of use cases first and then the concepts. All the example code uses python and numpy.Formulas are provided as images for reuse. Probability is the science of quantifying uncertain things.Most of machine learning and deep learning systems utilize a lot of data to learn about patterns in the data.Whenever data is utilized in a system rather than sole logic, uncertainty grows up and whenever uncertainty grows up, probability becomes relevant. By introducing probability to a deep learning system, we introduce common sense to the system.Otherwise the system would be very brittle and will not be useful.In deep learning, several models like bayesian models, probabilistic graphical models, hidden markov models are used.They depend entirely on probability concepts.

Gaussian-binary Restricted Boltzmann Machines on Modeling Natural Image Statistics Machine Learning

We present a theoretical analysis of Gaussian-binary restricted Boltzmann machines (GRBMs) from the perspective of density models. The key aspect of this analysis is to show that GRBMs can be formulated as a constrained mixture of Gaussians, which gives a much better insight into the model's capabilities and limitations. We show that GRBMs are capable of learning meaningful features both in a two-dimensional blind source separation task and in modeling natural images. Further, we show that reported difficulties in training GRBMs are due to the failure of the training algorithm rather than the model itself. Based on our analysis we are able to propose several training recipes, which allowed successful and fast training in our experiments. Finally, we discuss the relationship of GRBMs to several modifications that have been proposed to improve the model.

will wolf


Bayesian probabilistic models provide a nimble and expressive framework for modeling "small-world" data. In contrast, deep learning offers a more rigid yet much more powerful framework for modeling data of massive size. Edward is a probabilistic programming library that bridges this gap: "black-box" variational inference enables us to fit extremely flexible Bayesian models to large-scale data. Furthermore, these models themselves may take advantage of classic deep-learning architectures of arbitrary complexity. Edward uses TensorFlow for symbolic gradients and data flow graphs.

Unsupervised Variational Bayesian Learning of Nonlinear Models

Neural Information Processing Systems

In this paper we present a framework for using multi-layer perceptron (MLP)networks in nonlinear generative models trained by variational Bayesian learning. The nonlinearity is handled by linearizing it using a Gauss-Hermite quadrature at the hidden neurons. Thisyields an accurate approximation for cases of large posterior variance.The method can be used to derive nonlinear counterparts forlinear algorithms such as factor analysis, independent component/factor analysis and state-space models. This is demonstrated witha nonlinear factor analysis experiment in which even 20 sources can be estimated from a real world speech data set.