We comment on the fact that gradient ascent for logistic regression has a connection with the perceptron learning algorithm. Logistic learning is the "soft" variant of perceptron learning.

We consider the problem of parametric statistical inference when likelihood computations are prohibitively expensive but sampling from the model is possible. Several so-called likelihood-free methods have been developed to perform inference in the absence of a likelihood function. The popular synthetic likelihood approach infers the parameters by modelling summary statistics of the data by a Gaussian probability distribution. In another popular approach called approximate Bayesian computation, the inference is performed by identifying parameter values for which the summary statistics of the simulated data are close to those of the observed data. Synthetic likelihood is easier to use as no measure of "closeness" is required but the Gaussianity assumption is often limiting. Moreover, both approaches require judiciously chosen summary statistics. We here present an alternative inference approach that is as easy to use as synthetic likelihood but not as restricted in its assumptions, and that, in a natural way, enables automatic selection of relevant summary statistic from a large set of candidates. The basic idea is to frame the problem of estimating the posterior as a problem of estimating the ratio between the data generating distribution and the marginal distribution. This problem can be solved by logistic regression, and including regularising penalty terms enables automatic selection of the summary statistics relevant to the inference task. We illustrate the general theory on toy problems and use it to perform inference for stochastic nonlinear dynamical systems.

Research in statistical relational learning has produced a number of methods for learning relational models from large-scale network data. While these methods have been successfully applied in various domains, they have been developed under the unrealistic assumption of full data access. In practice, however, the data are often collected by crawling the network, due to proprietary access, limited resources, and privacy concerns. Recently, we showed that the parameter estimates for relational Bayes classifiers computed from network samples collected by existing network crawlers can be quite inaccurate, and developed a crawl-aware estimation method for such models (Yang, Ribeiro, and Neville, 2017). In this work, we extend the methodology to learning relational logistic regression models via stochastic gradient descent from partial network crawls, and show that the proposed method yields accurate parameter estimates and confidence intervals.

We consider the recovery of regression coefficients, denoted by $\boldsymbol{\beta}_0$, for a single index model (SIM) relating a binary outcome $Y$ to a set of possibly high dimensional covariates $\boldsymbol{X}$, based on a large but 'unlabeled' dataset $\mathcal{U}$, with $Y$ never observed. On $\mathcal{U}$, we fully observe $\boldsymbol{X}$ and additionally, a surrogate $S$ which, while not being strongly predictive of $Y$ throughout the entirety of its support, can forecast it with high accuracy when it assumes extreme values. Such datasets arise naturally in modern studies involving large databases such as electronic medical records (EMR) where $Y$, unlike $(\boldsymbol{X}, S)$, is difficult and/or expensive to obtain. In EMR studies, an example of $Y$ and $S$ would be the true disease phenotype and the count of the associated diagnostic codes respectively. Assuming another SIM for $S$ given $\boldsymbol{X}$, we show that under sparsity assumptions, we can recover $\boldsymbol{\beta}_0$ proportionally by simply fitting a least squares LASSO estimator to the subset of the observed data on $(\boldsymbol{X}, S)$ restricted to the extreme sets of $S$, with $Y$ imputed using the surrogacy of $S$. We obtain sharp finite sample performance bounds for our estimator, including deterministic deviation bounds and probabilistic guarantees. We demonstrate the effectiveness of our approach through multiple simulation studies, as well as by application to real data from an EMR study conducted at the Partners HealthCare Systems.

We consider the linear regression problem under semi-supervised settings wherein the available data typically consists of: (i) a small or moderate sized 'labeled' data, and (ii) a much larger sized 'unlabeled' data. Such data arises naturally from settings where the outcome, unlike the covariates, is expensive to obtain, a frequent scenario in modern studies involving large databases like electronic medical records (EMR). Supervised estimators like the ordinary least squares (OLS) estimator utilize only the labeled data. It is often of interest to investigate if and when the unlabeled data can be exploited to improve estimation of the regression parameter in the adopted linear model. In this paper, we propose a class of 'Efficient and Adaptive Semi-Supervised Estimators' (EASE) to improve estimation efficiency. The EASE are two-step estimators adaptive to model mis-specification, leading to improved (optimal in some cases) efficiency under model mis-specification, and equal (optimal) efficiency under a linear model. This adaptive property, often unaddressed in the existing literature, is crucial for advocating 'safe' use of the unlabeled data. The construction of EASE primarily involves a flexible 'semi-non-parametric' imputation, including a smoothing step that works well even when the number of covariates is not small; and a follow up 'refitting' step along with a cross-validation (CV) strategy both of which have useful practical as well as theoretical implications towards addressing two important issues: under-smoothing and over-fitting. We establish asymptotic results including consistency, asymptotic normality and the adaptive properties of EASE. We also provide influence function expansions and a 'double' CV strategy for inference. The results are further validated through extensive simulations, followed by application to an EMR study on auto-immunity.

The logarithm is an interesting character. Let us only understand its personality applicable for regression models. The fundamental property of a logarithm is its base. The typical base of the logarithm is 2, 10 or e. There is another common base for logarithms.

One of most excruciating pain points during Data Exploration and Preparation stage of an Analytics project are missing values. How do you deal with missing values - ignore or treat them? The answer would depend on the percentage of those missing values in the dataset, the variables affected by missing values, whether those missing values are a part of dependent or the independent variables, etc. Missing Value treatment becomes important since the data insights or the performance of your predictive model could be impacted if the missing values are not appropriately handled.The 2 tables above give different insights. The inference from the table on the left with the missing data indicates lower count for Android Mobile users and iOS Tablet users and higher Average Transaction Value compared to the inference from the right table with no missing data. The inference from the data with missing values could adversely impact business decisions.

The problem of subgroups is ubiquitous in scientific research (ex. disease heterogeneity, spatial distributions in ecology...), and piecewise regression is one way to deal with this phenomenon. Morse-Smale regression offers a way to partition the regression function based on level sets of a defined function and that function's basins of attraction. This topologically-based piecewise regression algorithm has shown promise in its initial applications, but the current implementation in the literature has been limited to elastic net and generalized linear regression. It is possible that nonparametric methods, such as random forest or conditional inference trees, may provide better prediction and insight through modeling interaction terms and other nonlinear relationships between predictors and a given outcome. This study explores the use of several machine learning algorithms within a Morse-Smale piecewise regression framework, including boosted regression with linear baselearners, homotopy-based LASSO, conditional inference trees, random forest, and a wide neural network framework called extreme learning machines. Simulations on Tweedie regression problems with varying Tweedie parameter and dispersion suggest that many machine learning approaches to Morse-Smale piecewise regression improve the original algorithm's performance, particularly for outcomes with lower dispersion and linear or a mix of linear and nonlinear predictor relationships. On a real actuarial problem, several of these new algorithms perform as good as or better than the original Morse-Smale regression algorithm, and most provide information on the nature of predictor relationships within each partition to provide insight into differences between dataset partitions.

About this course: Case Studies: Analyzing Sentiment & Loan Default Prediction In our case study on analyzing sentiment, you will create models that predict a class (positive/negative sentiment) from input features (text of the reviews, user profile information,...). In our second case study for this course, loan default prediction, you will tackle financial data, and predict when a loan is likely to be risky or safe for the bank. These tasks are an examples of classification, one of the most widely used areas of machine learning, with a broad array of applications, including ad targeting, spam detection, medical diagnosis and image classification. In this course, you will create classifiers that provide state-of-the-art performance on a variety of tasks. You will become familiar with the most successful techniques, which are most widely used in practice, including logistic regression, decision trees and boosting.

Shounak, I haven't worked with Fraud/Risk/Credit data before, and I understand the amount of precision with which modeling works in those domains. I've also heard the very small modeling population like in your case 0.2%. There are ways to weigh your sample using weighted response modeling techniques. Basically, you skew your sample to contain 0.5% or 1% response. Usually, this may not be required in your case, since I believe I've heard modeling with 0.2% is quite common.