# Probabilistic interpretation of AUC

Unfortunately this was not taught in any of my statistics or data analysis classes at university (wtf it so needs to be ). So it took me some time until I learned that the AUC has a nice probabilistic meaning.

PhD student with focus on statistics, machine learning, and programming, among other things

- a tensor of order one is a vector, which simply is a column of numbers,
- a tensor of order two is a matrix, which is basically numbers arranged in a rectangle,
- a tensor of order three looks like numbers arranged in rectangular box (or a cube, if all modes have the same dimension),
- an
*n*th order (or*n*-way) tensor looks like numbers arranged in an*n*-hyperrectangle… you get the idea…

Unfortunately this was not taught in any of my statistics or data analysis classes at university (wtf it so needs to be ). So it took me some time until I learned that the AUC has a nice probabilistic meaning.

Recently I came across the classical 1983 paper *A note on screening regression equations* by David Freedman. Freedman shows in an impressive way the dangers of data reuse in statistical analyses. The potentially dangerous scenarios include those where the results of one statistical procedure performed on the data are fed into another procedure performed on the same data. As a concrete example Freedman considers the practice of performing variable selection first, and then fitting another model using only the identified variables on the same data that was used to identify them in the first place. Because of the unexpectedly high severity of the problem this phenomenon became known as “Freedman’s paradox”. Moreover, in his paper Freedman derives asymptotic estimates for the resulting errors.

In many applications, data naturally form an *n*-way tensor with *n > 2*, rather than a “tidy” table.
As mentioned in the beginning of my last blog post, a tensor is essentially a multi-dimensional array:

A tensor is essentially a multi-dimensional array:

Many statistical modeling problems reduce to a minimization problem of the general form:

I have always found the common definition of the generalized inverse of a matrix quite unsatisfactory, because it is usually defined by a mere property, $A A^{-} A = A$, which does not really give intuition on when such a matrix exists or on how it can be constructed, etc… But recently, I came across a much more satisfactory definition for the case of symmetric (or more general, normal) matrices.

I spent much of the last two months reading Lehmann & Romano “Testing Statistical Hypotheses” (3rd ed.) and Lehmann & Casella “Theory of Point Estimation” (2nd ed.), abbr. TSH and TPE. The following is a collection of ~~random facts~~ observations I made while reading TSH and TPE. The choice of topics is biased towards application in regression models.

Recently I got surprised by the behaviour of `#permute_columns`

in the *Ruby* gem *NMatrix*.