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Orthant Probability

TL;DR

Section titled “TL;DR”
  • Measures the likelihood that all components of a multivariate vector simultaneously satisfy specified bounds.
  • Applies when assessing joint outcomes across multiple variables (e.g., all assets exceeding a threshold).
  • Common estimation approaches include simulation (Monte Carlo) and dependence modeling (copula methods).

Definition

Section titled “Definition”

Orthant probability is the probability of a multivariate random vector falling within a particular region in space; that is, the probability that all of the variables in a vector fall within certain ranges or limits.

Explanation

Section titled “Explanation”

Orthant probability quantifies the chance that every component of a multivariate observation lies in a prescribed range (an orthant of the sample space). Calculating this probability typically requires accounting for the individual likelihoods of each variable meeting its bound and the dependence structure between variables. Methods for obtaining orthant probabilities include simulation-based approaches and analytic or semi-analytic approaches that model dependence.

Examples

Section titled “Examples”

Stocks example

Section titled “Stocks example”

Find the probability that the stock prices of three different companies (A, B, and C) will all be above 50attheendoftheyear.Onecanestimatethelikelihoodofeachindividualstockreachingthisthresholdandthenmultiplytheseprobabilitiestogethertoobtaintheoverallprobabilitythatallthreestockswillbeabove50 at the end of the year. One can estimate the likelihood of each individual stock reaching this threshold and then multiply these probabilities together to obtain the overall probability that all three stocks will be above 50.

Patient health example

Section titled “Patient health example”

Find the probability that a patient will simultaneously have normal blood pressure, cholesterol, and body mass index (BMI). For example, estimate the probability that blood pressure is below 120/80 mmHg, cholesterol is below 200 mg/dL, and BMI is below 25 kg/m2; then multiply the individual probabilities to get the overall probability that all three measures are within the desired ranges at the same time.

Use cases

Section titled “Use cases”
  • Finance: evaluate the risk of a portfolio of stocks or bonds by assessing joint outcomes.
  • Healthcare: assess the likelihood of a patient meeting multiple clinical criteria simultaneously.
  • Risk assessment: determine the probability of multiple adverse conditions or events occurring together.
Section titled “Related terms”
  • Multivariate random vector
  • Monte Carlo method
  • Copula method