Linear Regression

Today we'll talk about linear regression. We use linear regression when we wan to predict future outcomes. Linear regression basically assumes that we have data($latex x_{ i }, y_{ i } $) for i=1,...,n. Furthermore $latex x_{ i } $ is not random and is called the predictor variable. $latex y_{ i } $ on …

Frequentist Inference III – Confidence Intervals

Confidence Intervals are often used and are a great way to not just give a point estimate but to tell where else the point might be. We will focus on confidence intervals based on normal data today. z-confidence intervals for the mean Suppose data $latex x_{ 1 },...,x_{ n } \sim N(\mu,\sigma^{ 2 })$ with …

Frequentist Inference II – NHST II

This here shall be a summary of the most common significance tests. Designing NHST specify $latex H_{ 0 } $ and $latex H_{ 1 } $ choose test statistic of which we know the null distribution and alternative distribution specify rejection region, significance level and decide if the rejection region is one or two-sided compute …

Frequentist Inference I – Null Hypothesis Significance Testing

Beside Bayesian Inference there is Frequentist Inference, a well established way of making assumptions about an experiment. Frequentist Inference is especially used in medical research and similar fields, while Bayesian inference is mostly used in modern technology and had a comeback when computers became more powerful. Null Hypothesis Significance Test (NHST) Suppose a friend tosses …

Bayesian Inference VII – Bayesian Updating Continuous Priors and Data

We finally got to the part of Bayesian Updating when both data and prior are continuous. It is, like always, not very different from Bayesian Updating with discrete data and priors except of that we use the PDFs and not the PMFs. Continuous Data, Continuous Prior When we use probability density functions we have to …

Bayesian Inference VI – Beta Distribution

We’ve seen last time how Bayesian Updating with continuous priors works. As mentioned back then, it isn’t always easy to calculate the total probability of X (p(x)). Today I will introduce the Beta Distribution which makes these calculations easier. Beta Distribution The beta(a,b) distribution is a two parameter distribution on range [0,1] and is therefore …

Bayesian Inference V – Bayesian Updating Continuous Priors

We already know how to do Bayesian Updating with discrete priors. Today we will learn how to do Bayesian Updating with continuous priors. Continous Priors To do Bayesian Updating with continuous priors but with discrete data - we will look at the case that both is discrete next time - we just change sums to …