Author: marmara

The standard scenario of hypothesis testing is this: We have a pdf at hand representing the “normal” operation of a system, and we know all…

Expectation Maximization Assume we have some observed data $x=(x_1,\ldots,x_n)$. Assume that this data is generated by a probability distribution $p_{\theta}(x,z)$ where the model parameter $\theta$…

k-Means

Parametric Inference

Sufficient Statistics

Statistical Inference Statistical inference, or “learning” as it is called in CS, is the process of using data to infer the distribution that generated the…

Limit Theorems Weak law of large numbers: casinos work for small mean but large variance–a lot of nerve is required.. Central Limit Theorem In this…

Stochastic Convergence In order to understand what ”stochastic converge“ is, we have to remember first what ”deterministic convergence“ means. Deterministic Convergence Definition: A deterministic sequence…

Probabilistic Inequalities Markov’s Inequality Theorem: If $X$ is a random variable, $X>0$ and $a>0$ is a positive real number, then \begin{eqnarray} \mathrm{P}(X>a) \leqslant \frac{\mathrm{E}(X)}{a} \end{eqnarray}…