The AUC score is a popular summary statistic that is often used to communicate the performance of a classifier. However, we illustrate here that this score depends not only on the quality of the model in question, but also on the difficulty of the test set considered: If samples are added to a test set that are easily classified, the AUC will go up — even if the model studied has not improved. In general, this behavior implies that isolated, single AUC scores cannot be used to meaningfully qualify a model’s performance. Instead, the AUC should be considered a score that is primarily useful for comparing and ranking multiple models — each at a common test set difficulty.

## Deep reinforcement learning, battleship

Here, we provide a brief introduction to reinforcement learning (RL) — a general technique for training programs to play games efficiently. Our aim is to explain its practical implementation: We cover some basic theory and then walk through a minimal python program that trains a neural network to play the game battleship.

## GPU-accelerated Theano & Keras with Windows 10

There are many tutorials with directions for how to use your Nvidia graphics card for GPU-accelerated Theano and Keras for Linux, but there is only limited information out there for you if you want to set everything up with Windows and the current CUDA toolkit. This is a shame however because there are a large number of computers out there with very nice video cards that are only running windows, and it is not always practical to use a Virtual Machine, or Dual-Boot. So for today’s post we will go over how to get everything running in Windows 10 by saving you all the trial and error I went through. (All of these steps should also work in earlier versions of Windows).

## Hyperparameter sample-size dependence

Here, we briefly review a subtlety associated with machine-learning model selection: the fact that the optimal hyperparameters for a model can vary with training set size, $N.$ To illustrate this point, we derive expressions for the optimal strength for both $L_1$ and $L_2$ regularization in single-variable models. We find that the optimal $L_2$ approaches a finite constant as $N$ increases, but that the optimal $L_1$ decays exponentially fast with $N.$ Sensitive dependence on $N$ such as this should be carefully extrapolated out when optimizing mission-critical models.

## Interpreting the results of linear regression

Our last post showed how to obtain the least-squares solution for linear regression and discussed the idea of sampling variability in the best estimates for the coefficients. In this post, we continue the discussion about uncertainty in linear regression — both in the estimates of individual linear regression coefficients and the quality of the overall fit.

Specifically, we’ll discuss how to calculate the 95% confidence intervals and p-values from hypothesis tests that are output by many statistical packages like python’s statsmodels or R. An example with code is provided at the end.

## Linear Regression

We review classical linear regression using vector-matrix notation. In particular, we derive a) the least-squares solution, b) the fit’s coefficient covariance matrix — showing that the coefficient estimates are most precise along directions that have been sampled over a large range of values (the high variance directions, a la PCA), and c) an unbiased estimate for the underlying sample variance (assuming normal sample variance in this last case). We then review how these last two results can be used to provide confidence intervals / hypothesis tests for the coefficient estimates. Finally, we show that similar results follow from a Bayesian approach.

Last edited July 23, 2016.

## Average queue wait times with random arrivals

Queries ping a certain computer server at random times, on average $\lambda$ arriving per second. The server can respond to one per second and those that can’t be serviced immediately are queued up. What is the average wait time per query? Clearly if $\lambda \ll 1$, the average wait time is zero. But if $\lambda > 1$, the queue grows indefinitely and the answer is infinity! Here, we give a simple derivation of the general result — (9) below.

## Improved Bonferroni correction factors for multiple pairwise comparisons

A common task in applied statistics is the pairwise comparison of the responses of $N$ treatment groups in some statistical test — the goal being to decide which pairs exhibit differences that are statistically significant. Now, because there is one comparison being made for each pairing, a naive application of the Bonferroni correction analysis suggests that one should set the individual pairwise test sizes to $\alpha_i \to \alpha_f/{N \choose 2}$ in order to obtain a desired family-wise type 1 error rate of $\alpha_f$. Indeed, this solution is suggested by many texts. However, implicit in the Bonferroni analysis is the assumption that the comparisons being made are each mutually independent. This is not the case here, and we show that as a consequence the naive approach often returns type 1 error rates far from those desired. We provide adjusted formulas that allow for error-free Bonferroni-like corrections to be made.

[edit (7/4/2016): After posting this article, I’ve since found that the method we suggest here is related to / is a generalization of Tukey’s range test — see here.]

[edit (6/11/2018): I’ve added the notebook used below to our Github, here]