We already derive the posterior update formula for Bayesian regression here, telling us that it is distribution of our parameter regression given data set . We are not directly interested in the value of , but, we are interested in the value of itself given new value of new . This is exactly same with regression problem, given new value , we want to predict output value of , which is in continuous value mode. And we already did linear regression problem using LSE (Least Square Error) here. During this post, we will do regression from Bayesian point of view. Using Bayesian in regression, we will have additional benefit. We will see later in the end of this post.
From #Part1 here, we already get . To do regression in Bayesian point of view, we have to derive predictive distribution, so that we will have probability of , . We can achieve that by doing marginalization. Here we go.
where is likelihood and is posterior we derive here Continue reading “Bayesian Linear / Polynomial Regression #Part2: Deriving Predictive Distribution”
Gaussian is very important distribution. During this post, we will discuss the detail of Gaussian distribution by deriving it, calculate the integral value and do MLE (Maximum Likelihood Estimation). To derive Gaussian distribution, it is more difficult if we do it in cartesian coordinate. Thus, we will use polar coordinate. Before we derive the Gaussian using polar coordinate, let’s talk about how to change the coordinate system from cartesian to polar coordinate system first.
(1) Changing coordinate system from cartesian to polar coordinate
Changing coordinate system from catersian to polar coordinate is useful, such as when we calculate integral of certain function, in certain case, we prefer to use polar coordinate system because it will be away easier to calculate. To do that, we can use Jacobian matrix. Jacobian matrix actually defines partial derivative of a vector with respect to another vector. In our case changing cartesian coordinate to polar coordinate, the Jacobian matrix of in cartesian coordinate with respect to in polar coordinate is:
Continue reading “Deriving Gaussian Distribution”