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4 - Multiple Discrete Variables
Carlos Fernandez-Granda, New York University
Book:

Probability and Statistics for Data Science

Published online:

19 June 2025

Print publication:

03 July 2025, pp 109-160
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Summary

This chapter describes how to model multiple discrete quantities as discrete random variables within the same probability space and manipulate them using their joint pmf. We explain how to estimate the joint pmf from data, and use it to model precipitation in Oregon. Then, we introduce marginal distributions, which describe the individual behavior of each variable in a model, and conditional distributions, which describe the behavior of a variable when other variables are fixed. Next, we generalize the concepts of independence and conditional independence to random variables. In addition, we discuss the problem of causal inference, which seeks to identify causal relationships between variables. We then turn our attention to a fundamental challenge: It is impossible to completely characterize the dependence between all variables in a model, unless they are very few. This phenomenon, known as the curse of dimensionality, is the reason why independence assumptions are needed to make probabilistic models tractable. We conclude the chapter by describing two popular models based on such assumptions: Naive Bayes and Markov chains.

Can Factor Investing Become Scientific?
Marcos M. López de Prado
Published online:

12 October 2023

Print publication:

09 November 2023
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Virtually all journal articles in the factor investing literature make associational claims, in denial of the causal content of factor models. Authors do not identify the causal graph consistent with the observed phenomenon, they justify their chosen model specification in terms of correlations, and they do not propose experiments for falsifying causal mechanisms. Absent a causal theory, their findings are likely false, due to rampant backtest overfitting and incorrect specification choices. This Element differentiates between type-A and type-B spurious claims, and explains how both types prevent factor investing from advancing beyond its current phenomenological stage. It analyzes the current state of causal confusion in the factor investing literature, and proposes solutions with the potential to transform factor investing into a truly scientific discipline. This title is also available as Open Access on Cambridge Core.

1 - Clinical research design: analytical studies
from Part I - Quantitative methods in clinical neurology
- By Richard Mayeux
Albert Hofman, Erasmus Universiteit Rotterdam, Richard Mayeux, Columbia University, New York
Book:

Investigating Neurological Disease

Published online:

29 September 2009

Print publication:

30 August 2001, pp 3-10
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Summary

Analytic studies are increasingly important for understanding relationships between diseases and their causes. They usually take the form of observational investigations, but can also include randomized clinical trials. Risk factors are antecedents that are considered to be components of the disease pathway. Appropriate study design and consideration to systematic bias and confounding helps to establish association between the exposure and disease. The purpose for maintaining the principles of causal inference and eliminating chance, bias, and confounding is to establish validity. The most serious concern in analytic studies is maintaining validity. Confounders are extraneous factors that are related to the disease and to a risk factor or exposure related to the disease. The confounder usually predicts disease in the absence of any risk factor. Investigators have to consider cost and efficiency in their design as well as the potential public health impact of any observed association.