## Introduction.

In this post, we derive the nonnegative matrix factorization (NMF). We derive the multiplicative updates from a gradient descent point of view by using the treatment of Lee and Seung, Algorithms for Nonnegative Matrix Factorization. The code for this blogpost can be accessed from here.

## Introduction.

In this post, we give the definitions of sample space, probability measure, random variable. We give these definitions on a very simple example of the space of two coin tosses. Note that definitions in this note are for finite probability spaces and the example simplifies everything significantly. This note is mostly based on the Shreve's Stochastic Calculus for Finance, vol. I, Chapter 2 and vol. II, Chapter 2.