# Math

## Understanding the Metropolis Hasting algorithm - A tutorial

The Problem Say you want to evaluate the expectation of a function over a random variable $E[f(x)] = \int p(x)f(x)dx$, or perhaps find the max of a probability distribution, typically the posterior, $\arg \max p(x|y)$, where calculating the derivate is intractable since it depends on some feature that comes from some algorithm or some unknown […]

## A short tutorial on Kernel Density Estimation (KDE)

The aim of Kernel Density Estimation(KDE) is: Given a set of $N$ samples from a random variable, $\mathbf{X}$, possibly multivariate and continuous, estimate the random variables probability density function(pdf) The univariate case To get a rough idea how one can think about the problem, we start out with a set of samples, $X=[x_1,x_2,...,x_N]$, of a […]

## Partially fixing numerical underflow for mixture models

In a mixture model the probability of an event $x$ is written $P(x=X) =\sum_{i}\pi_{i}P_{i}(x=X)$, where $\pi_{i}$ is the probability of the point belonging to mixture $i$ and $P_{i}(x=X)$ is the probability of the event $x=X$ in the $i$-th mixture. The problem is usually that the $P_i$ are small which makes underflow happen when […]

## The Trace Trick for Gaussian Log Likelihood

Maybe you have seen something like this when observing the log likelihood derivations for multivariate Gaussians $\ln p(X|\mu, \Sigma) = \frac{1}{2}\ln|\Sigma|- \frac{1}{2}X^{T}\Sigma^{-1}X + const = \frac{1}{2}\ln|\Sigma| - \frac{1}{2}Tr(\Sigma^{-1}XX^{T}) + const$ and you wondered where that $Tr$ came from. Here you can find a great explanation but I thought I would write it down […]

## Calculating the indices for n choose k

So if you want to calculate n choose k, or n over k or all k different combinations of n unique items or binomial coefficients in Matlab or in math language

you can write nchoosek(n,k)nchoosek(n,k) But lets say you want to generate a list of these possible combinations? Well then you just write […]