The residual plot above shows the prediction error of the test dataset plotted against a selected feature. We built this model just before the wild-card round of the NFL playoffs, and we wanted to test the model against 10 previous games. Of our 10 predictions, seven were correct, and two of the three incorrect predictions were very close to margin (50 percent), as seen in the table below. So, we were comfortable with this model. Next, our model correctly predicted the outcome of three out of four playoff games.

Predictive modeling is increasingly being employed to assist human decision-makers. One purported advantage of replacing human judgment with computer models in high stakes settings-- such as sentencing, hiring, policing, college admissions, and parole decisions-- is the perceived "neutrality" of computers. It is argued that because computer models do not hold personal prejudice, the predictions they produce will be equally free from prejudice. There is growing recognition that employing algorithms does not remove the potential for bias, and can even amplify it, since training data were inevitably generated by a process that is itself biased. In this paper, we provide a probabilistic definition of algorithmic bias. We propose a method to remove bias from predictive models by removing all information regarding protected variables from the permitted training data. Unlike previous work in this area, our framework is general enough to accommodate arbitrary data types, e.g. binary, continuous, etc. Motivated by models currently in use in the criminal justice system that inform decisions on pre-trial release and paroling, we apply our proposed method to a dataset on the criminal histories of individuals at the time of sentencing to produce "race-neutral" predictions of re-arrest. In the process, we demonstrate that the most common approach to creating "race-neutral" models-- omitting race as a covariate-- still results in racially disparate predictions. We then demonstrate that the application of our proposed method to these data removes racial disparities from predictions with minimal impact on predictive accuracy.

Cross-validation (CV) is a technique for evaluating the ability of statistical models/learning systems based on a given data set. Despite its wide applicability, the rather heavy computational cost can prevent its use as the system size grows. To resolve this difficulty in the case of Bayesian linear regression, we develop a formula for evaluating the leave-one-out CV error approximately without actually performing CV. The usefulness of the developed formula is tested by statistical mechanical analysis for a synthetic model. This is confirmed by application to a real-world supernova data set as well.

In this paper we study the problem of recovering a structured but unknown parameter ${\bf{\theta}}^*$ from $n$ nonlinear observations of the form $y_i=f(\langle {\bf{x}}_i,{\bf{\theta}}^*\rangle)$ for $i=1,2,\ldots,n$. We develop a framework for characterizing time-data tradeoffs for a variety of parameter estimation algorithms when the nonlinear function $f$ is unknown. This framework includes many popular heuristics such as projected/proximal gradient descent and stochastic schemes. For example, we show that a projected gradient descent scheme converges at a linear rate to a reliable solution with a near minimal number of samples. We provide a sharp characterization of the convergence rate of such algorithms as a function of sample size, amount of a-prior knowledge available about the parameter and a measure of the nonlinearity of the function $f$. These results provide a precise understanding of the various tradeoffs involved between statistical and computational resources as well as a-prior side information available for such nonlinear parameter estimation problems.

Hidden Decision Trees is a statistical and data mining methodology (just like logistic regression, SVM, neural networks or decision trees) to handle problems with large amounts of data, non-linearities and strongly correlated dependent variables. The technique is easy to implement in any programming language. It is more robust than decision trees or logistic regression, and help detect natural final nodes. Implementations typically rely heavily on large, granular hash tables. No decision tree is actually built (thus the name hidden decision trees), but the final output of an hidden decision tree procedure consists of a few hundred nodes from multiple non-overlapping small decision trees.

In this paper we study the problem of exact recovery of the pure-strategy Nash equilibria (PSNE) set of a graphical game from noisy observations of joint actions of the players alone. We consider sparse linear influence games --- a parametric class of graphical games with linear payoffs, and represented by directed graphs of n nodes (players) and in-degree of at most k. We present an $\ell_1$-regularized logistic regression based algorithm for recovering the PSNE set exactly, that is both computationally efficient --- i.e. runs in polynomial time --- and statistically efficient --- i.e. has logarithmic sample complexity. Specifically, we show that the sufficient number of samples required for exact PSNE recovery scales as $\mathcal{O}(\mathrm{poly}(k) \log n)$. We also validate our theoretical results using synthetic experiments.

Python is one of the fastest-growing platforms for applied machine learning. In this mini-course, you will discover how you can get started, build accurate models and confidently complete predictive modeling machine learning projects using Python in 14 days. This is a big and important post. You might want to bookmark it. Python Machine Learning Mini-Course Photo by Dave Young, some rights reserved.

Let's go through an example of Cancer Tissue Observations: Logistic regression is a popular method to predict a binary response. It is a special case of Generalized Linear models that predicts the probability of the outcome. Logistic regression measures the relationship between the Y "Label" and the X "Features" by estimating probabilities using a logistic function. The model predicts a probability which is used to predict the label class. Our data is from the Wisconsin Diagnostic Breast Cancer (WDBC) Data Set which categorizes breast tumor cases as either benign or malignant based on 9 features to predict the diagnosis.

We study model evaluation and model selection from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. We believe that GA is one way formally to address concerns about the external validity of a model. The GA of a model estimated on a sample can be measured by its empirical out-of-sample errors, called the generalization errors (GE). We derive upper bounds for the GE, which depend on sample sizes, model complexity and the distribution of the loss function. The upper bounds can be used to evaluate the GA of a model, ex ante. We propose using generalization error minimization (GEM) as a framework for model selection. Using GEM, we are able to unify a big class of penalized regression estimators, including lasso, ridge and bridge, under the same set of assumptions. We establish finite-sample and asymptotic properties (including $\mathcal{L}_2$-consistency) of the GEM estimator for both the $n \geqslant p$ and the $n < p$ cases. We also derive the $\mathcal{L}_2$-distance between the penalized and corresponding unpenalized regression estimates. In practice, GEM can be implemented by validation or cross-validation. We show that the GE bounds can be used for selecting the optimal number of folds in $K$-fold cross-validation. We propose a variant of $R^2$, the $GR^2$, as a measure of GA, which considers both both in-sample and out-of-sample goodness of fit. Simulations are used to demonstrate our key results.

If you are modeling binomial data; ie a numerator consisting of the number of 1/0 successes you have for a given pattern of covariates, and a denominator that gives the value of the total number of observations having that covariate pattern (a specific profile of predictor values; eg age 23, married 1, working 0), a logistic regresson is generally appropriate. But when the mean values of the numerators are less than 10% of the mean values of the denominator, it is likely that a Poisson model is preferred. The otherwise logistic numerator is the count response variable (dependent variable) and the natural log of the denominator is the offset. Generally the Poisson model will fit the data better. Logistic models are not indended for rare occurrences.