Null hypothesis

In inferential statistics the null hypothesis usually refers to a general statement or default position that there is no relationship between measured phenomena, or no difference among groups.

The concept of a null hypothesis is used differently in two approaches to Statistical inference. In the significance testing approach, a null hypothesis is rejected on the basis of data that is significantly unlikely if the null is true, but the null hypothesis is never accepted or proved. This is analogous to a criminal trial, in which the defendant is assumed to be innocaent (null is not rejected) until proven guilty (null is rejected) beyond a reasonable double.

• Covers topics in the area of data-intensive science, programming to computational methods
• Computational science rather than computer science

One of the important point of the report Frontiers in the Analysis of Massive Data is that The need to look at the end-to-end data life cycle. It is going to be an important scence in all material.

This post only shows the intuition of multivariate gaussian. It does not contain any detailed explaination.

Stochastic matrix

In math, a Stochastic matrix (also termed probability matrix, transition matrix, substitution matrix or Markov matrix) is a matrix used to describe the transitions of a Markov Chain.

Each of its entries is a nonnegative real number representing a probability.

A right Stochastic matrix is a real square matrix, with row summing to 1.

A Left Stochastic matrix is a real square matrix, with each column summing to 1.

A stationary probability vector $\pi$ is defined as a distribution, that does not change under application of the transition matrix. It is defined as a probability distribution on the set ${1,\dots,n}$ which is also a row eigenvector of the probability matrix, associated with eigenvalue 1.

$\pi P = \pi$