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Joint, Marginal, and Conditional Distributions

Install and import#

bash
npm install fintech-algorithms
ts
import { jointMarginalAndConditionalDistributions } from "fintech-algorithms/foundations/probability-and-random-variables/joint-marginal-and-conditional-distributions";

Signature#

jointMarginalAndConditionalDistributions(input)

Collapses a joint probability table over each axis in turn to give the two marginal distributions, then renormalises one slice of it to give the conditional distribution of X at a chosen value of Y.

Parameters#

NameTypeNotes
input{ pA: number; pB: number; pAB: number; randomValues: number[]; probabilities: number[]; joint: { x: number; y: number; p: number }[]; conditionY: number | string }joint is the probability table as a flat list of cells, each carrying an x level, a y level, and its probability p. conditionY selects the Y level to condition on and is matched by string form, so 1 and "1" select the same slice. The family-wide pA, pB, pAB, randomValues, and probabilities are validated on every call.
pA: 0 <= pA <= 1 · pB: 0 <= pB <= 1 · pAB: 0 <= pAB <= min(pA, pB) · joint: non-empty, every `p` nonnegative, all `p` summing to 1 within 1e-12 · conditionY: must name a Y level whose marginal probability is not zero

Returns#

{ marginalX: Record<string, number>; marginalY: Record<string, number>; conditionalXGivenY: Record<string, number> }

marginalX and marginalY sum the cell probabilities by X level and by Y level, keyed by the level rendered as a string. conditionalXGivenY divides the cells at the selected Y level by that level's marginal, so its entries sum to 1.

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 randomValues or probabilities is missing, empty, or holds a non-finite entry — throws RangeError
  • When randomValues and probabilities differ in length, a probability is negative, or the probabilities do not sum to one within 1e-12 — throws RangeError
  • When joint is empty, a cell probability is negative, or the cell probabilities do not sum to one within 1e-12 — throws RangeError
  • When conditionY names a Y level with zero or missing marginal probability — throws RangeError

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

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#

jointMarginalAndConditionalDistributions(input)

Returns#

object with 2 fields: marginalX, marginalY

{
  "marginalX": {
    "0": 0.5,
    "1": 0.5
  },
  "marginalY": {
    "0": 0.4,
    "1": 0.6000000000000001
  }
}

Diagrams#

Joint, Marginal, and Conditional Distributions — article hero
Joint, Marginal, and Conditional Distributions — calculation ledger
Joint, Marginal, and Conditional Distributions — concept anatomy
Joint, Marginal, and Conditional Distributions — failure boundary
Joint, Marginal, and Conditional Distributions — method map
Joint, Marginal, and Conditional Distributions — 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#