Aug 04

Bayes’ Theorem (Quant: Stat)

by David Harper, CFA, FRM, CIPM

FRM |

Learning objective: Explain Bayes’ Theorem and use Bayes' formula to determine the probability of causes for a given event

Bayes Theorem updates a probability that that lacks prior information (i.e., an unconditional), into an “improved” or “more informed” probability based on additional information.

Example of Bayes’ Theorem

The following example is an extremely simplified model. Assume two variables:

  • The economy can either grow (G) or slow (S). Additionally,
  • A stock can either go up (U) or (D).

The combination of these two variables gives rise to four possible outcomes:

  • Economy grows, stock goes up (G then U)
  • Economy slows, stock goes up (S then U)
  • Economy grows, stock goes down (G then D)
  • Economy slows, stock goes down (S then D)

image

Bayes’ Theorem solves for a conditional probability. In this case, the probability that the economy grew given that (conditional on) the stock going up:

bayes_short

The denominator can be expanded. This gives the full version of Bayes’ formula:

bayes_long 

And we can refer to the following probabilities:

  • Unconditional probability the economy will grow: P(G) = 70%
  • Unconditional probability the economy will slow: P(S) = 30%
  • Conditional probability that stock goes up if economy grows: P(U|G) = 80%
  • Conditional probability that stock goes down if economy grows: P(D|G) = 20%
  • Conditional probability that stock goes up if economy slows: P(U|S) = 30%
  • Conditional probability that stock goes down if economy slows: P(D|S) = 70%
  • Joint probability that economy grows and stock goes up = P(GU) = P(U|G)P(G) = 56%
  • Joint probability that economy grows and stock goes down = P(GD) = P(D|G)P(G) = 14%
  • Note something about this two-step binomial tree: the four terminal nodes are mutually exclusive and cumulatively exhaustive. Specifically, P(GU) + P(GD) + P(SU) + P(SD) = 56% + 14% + 9% + 21% = 100%.

Before Bayes’ theorem: the marginal (unconditional) probability the economy grows: P(G) = 70%.

After Bayes’ theorem: we add information. Specifically, we are told “the stock went up.” Now, what is the probability the economy grew given (conditional on) the stock went up?

bayes_example

With Bayes’ theorem, we improved our foresight by going from an unconditional probability that the economy will grow [P(G) = 70%] to a probability the economy will grow conditional on prior information [P(G|U) = 86%].

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