# An EM Exercise

The Expectation-Maximization (EM) algorithm is a popular method to obtain the Maximum Likelihood Estimate (MLE) when some of the data may be missing. See [Roche, 2012] for a nice tutorial on EM. The following problem is a nice exercise in working out the algorithm details to reinforce the concepts. Read more...

# Essential PCA

Assume that we are given a matrix $X \in \mathbb{R}^{n \times p}$. Each row of the matrix $X$ is considered to be an observation represented by a data vector that measures $p$ features of some phenomenon. We can think of Principal Component Analysis (PCA) as trying to trying to solve two related problems. Read more...

# Bayes Classifier with Asymmetric Costs

Thanks to Prof. Larry for this problem! Consider the following binary classification problem. Every individual of a population is associated with an independent replicate of the pair $(\mathbf{X}, Y)$, having known joint distribution and where the (observed) covariate $\mathbf{X}$ has a (marginal) distribution $\pi$, and the (unobserved) response $Y \in \{-1, 1\}$. Suppose the costs of misclassifying an individual with $Y = 1$ and $Y = -1$ are $a > 0$ and $b > 0$, respectively. What’s the Bayes decision rule? Read more...