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Bayes' Theorem and Base Rates

Install and import#

bash
npm install fintech-algorithms
ts
import { bayesTheoremAndBaseRates } from "fintech-algorithms/foundations/probability-and-random-variables/bayes-theorem-and-base-rates";

Signature#

bayesTheoremAndBaseRates(input)

Turns a prior, a true-positive rate, and a false-positive rate into the posterior probability of the condition given a positive result, along with the unconditional probability of that result.

Parameters#

NameTypeNotes
input{ pA: number; pB: number; pAB: number; prior: number; sensitivity: number; falsePositiveRate: number }prior is the base rate of the condition, sensitivity the chance of a positive result when the condition holds, and falsePositiveRate the chance of a positive result when it does not. The family-wide pA, pB, and pAB are validated on every call even though this topic does not use their values.
pA: 0 <= pA <= 1 · pB: 0 <= pB <= 1 · pAB: 0 <= pAB <= min(pA, pB) · prior: 0 <= prior <= 1 · sensitivity: 0 <= sensitivity <= 1 · falsePositiveRate: 0 <= falsePositiveRate <= 1

Returns#

{ posterior: number; evidenceProbability: number }

evidenceProbability is sensitivity * prior + falsePositiveRate * (1 - prior), the total chance of a positive result, and posterior is sensitivity * prior divided by it.

Errors#

  • When input is null, an array, or not an object — throws TypeError
  • When pA, pB, or pAB falls outside [0, 1], or pAB exceeds min(pA, pB) — throws RangeError
  • When prior, sensitivity, or falsePositiveRate falls outside [0, 1] — throws RangeError
  • When the evidence probability works out to exactly zero, leaving the posterior undefined — throws RangeError

Complexity: time O(1), space O(1).

Worked example#

verified This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

Input#

input
{
  "pA": 0.6,
  "pB": 0.5,
  "pAB": 0.3,
  "outcomes": ["up", "flat", "down"],
  "event": ["up", "flat"],
  "prior": 0.01,
  "sensitivity": 0.9,
  "falsePositiveRate": 0.05,
  "randomValues": [0, 1, 2],
  "probabilities": [0.2, 0.5, 0.3],
  "randomVariableKind": "discrete",
  "joint": [
    {
      "x": 0,
      "y": 0,
      "p": 0.3
    },
    {
      "x": 0,
      "y": 1,
      "p": 0.2
    },
    {
      "x": 1,
      "y": 0,
      "p": 0.1
    }
  ],
  "conditionY": 1
}

Call#

bayesTheoremAndBaseRates(input)

Returns#

object with 2 fields: posterior, evidenceProbability

{
  "posterior": 0.15384615384615385,
  "evidenceProbability": 0.0585
}

Diagrams#

Bayes' Theorem and Base Rates — article hero
Bayes' Theorem and Base Rates — calculation ledger
Bayes' Theorem and Base Rates — concept anatomy
Bayes' Theorem and Base Rates — failure boundary
Bayes' Theorem and Base Rates — method map
Bayes' Theorem and Base Rates — scenario contrast

How it works#

This page states the contract — how to call it correctly. The article explains the concept: why it works, and where it breaks.

Read the article →

References#

The rest of the Probability and Random Variables family#