Week 4

banknote forgery

This week, we began by looking at a hands-on example of descriptive statistics -as discussed in chapter 5- and looked at some code. You can find a link to this code by clicking the button above

First off, you should know the code written in the Python programming language, which arguably is the most important prgramming langugae used in tech today.

We also spent some time introducing chapter 6: likelihood inference. We started by observing a fatal flaw in the idea of descriptive statistics: indeed, given \(\Delta \subset S\) and some statistical model \(\Theta\Longrightarrow S\):
the construction of \(P_\theta\) in descriptive statistics does not use the model \(\Theta\Longrightarrow S\) in any way, and is dependent solely on the data \(\Delta\)!
In the coming chapters, we will consider other approaches to statistics that do rely on the statistical model
Chapter 6 is a first approach: likelihood inference.
The idea here is to find a way to model the likelihood that the data \(\Delta\) will be obtained given that it is the result of a random sample with some probability measure \(P_\theta\). More precisely, for each \(\theta \in \Theta\) write the data as \(\Delta=(x_1,\ldots , x_n)\) and consider a random sample \(X_1,\ldots, X_n\) with probability \(P_\theta\).
The likelihood that the data \(x_1,\ldots x_n\) is the result of the random sample assuming the random sample has distribution \(P_\theta\) is simple the density function of the joined random variable \[X_1\wedge\ldots \wedge X_n:S\longrightarrow \mathbb{R}^n:s \mapsto X_1(s),\ldots X_n(s)\] evaluated at the data:, i.e \( f_{X_1\wedge X_2\wedge \ldots \wedge X_n}(x_1,\ldots ,x_n)\). Now, using independence of the RV's and the fact that \(f_{X_i}=f_\theta\) we obtain \(f_{X_1\wedge X_2\wedge \ldots \wedge X_n}(x_1,\ldots ,x_n)=\prod_i \, f_\theta(x_i) \) We thus define the likelihood as the function \[ L: \Theta\longrightarrow \mathbb{R}:\theta \longrightarrow L(\Delta \vert \theta)=\prod_i f_\theta (x_is) \textrm{ and } \Theta \] If the likelihod reaches a unique maximum for a value \(\theta\), we say that \(\theta\) is the MLE for the data. As an example
Assume we throw 10 coins which may be biased the probability of heads being \(\theta\). Let \(k\) be amount of heads in the experiment. Then \[L(k,\theta)=\binom{n}{k}\theta^k\]
The curve is drawn below. for \(k=4\). Can you find an MLE? Does it make sense?