Jan 10

Bayes’ Formula (binomial example)

by David Harper, CFA, FRM, CIPM

FRM |

probApproaches

I just started to read Plight of the Fortune Tellers by Riccardo Rebonato. I was taught three approaches to probability (classical, empirical and subjective; pictured), but he makes a case that, in practice, they are misapplied as we forget the limits of empiricism (he calls this the  'frequentist' view).  Rebonator says historical frequencies don't necessarily tell us about probabilities. Put another way, past isn't prologue. Interestingly, he elevates the subjective approach to the general case and the empirical approach to a special case of the subjective! I personally think repackaging empirical probabilities into (inside) an overall subjectivity container is brilliant and indisputably correct.

And, regardless, he is a Bayesian. Bayesians have a habit of making sense, so is hard to argue with them:

According to the Bayesian view of the world, we always start from some prior belief at the problem at hand. We then acquire new evidence. If we are Bayesians, we neither accept in full this new piece of information, nor do we stick to our prior belief as if nothing had happened. Instead, we modify our initial views to a degree commensurate with the weight and reliability both of the evidence and of our prior belief. - Plight of the Fortune Tellers, Riccardo Rebonato, p.44

A simple example of Bayes' Formula

Consider the binomial tree below, and assume we start at the left. It's a binomial tree, so right-away we know it will be too simple for reality. (it rarely helps to say "but the model is not reality" because models never are, it is not per se a refutation of them)

But say we own a stock. And the stock is sensitive to the economy. There is a 70% chance the economy will grow next year; a 30% chance the economy will slow (one year = one period). If the economy grows, there is an 80% chance the stock will go up; otherwise (1- 80%) it will go down. On the other hand, if the economy slows, there is only a 30% chance the stock will go up; otherwise (1-30%) it will go down:

image

A few terms:

  • An unconditional probability is not conditional on anything. In the above, there is an (unconditional) 70% chance the economy will grow. The unconditional probability is given by P(G) = 70%
  • A conditional probability "depends" on something. In the above, if the economy grows (the condition), then there is an 80% chance the stock will grow. This conditional probability is given by P(U|G) = 80%. As in", the probability that the stock will go up (U) conditional on an growing economy (G)."
  • The joint probability is the probability that both events occur. In this case, what is the probability that both the economy will grow and the stock will go up. This joint probability is given by P(GU) = (70%)(80%) = 56%

Bayes' Formula

Bayes formula adjusts a probability to handle the addition of new (but a priori) information. In this case, we can ask "if we know the stock went up, what are the odds that the economy grew?" Notice this is an improvement on the unconditional "what is the odds the economy grew?" (70%). This is expressed with the conditional probability P(G|U). And the answer, Bayes' Formula, is given by:

image

 

The intuition is easier, in my opinion, than the formula. (personally, I forget the formula. I have to think backwards from the picture). In the numerator, we have P(U|G)P(G) = (80%)(70%) = 56%. In the denominator, we have P(U) which is P(U|G)P(G) + P(U|G')P(G'). G' just means "not G" which is (S) in this case. So this is the same as: P(U|G)P(G) + P(U|S)P(S).

So, we have, P(G|U) = P(U|G)P(G) /[P(U|G)P(G) + P(U|S)P(S)] = 56%/[56%+9%] = 56%/65% = 86.2%. So we go from an unconditional probability of 70% to a conditional ("if we know the stock goes up") probability of 86.2%. That's a simple case of Bayes' formula.

The intuition is even easier: we are just calculating the fraction represented by the outcome in the upper node. There are four node outcomes on the right: (i) economy grows, stock goes up [odds = (70%)(80%) = 56%], (ii) economy grows, stock goes down [odds = (70%)(20%) = 14%], (iii) economy slows, stock goes up [odds = (30%)(30%) = 9%], and (iv) economy slows, stock goes down [odds = (30%)(70%) = 21%]. They are mutually exclusive (and cumulatively exhaustive) outcomes, so they must add to 100%: 56% + 14% + 9% + 21% = 100%.

So if we know the stock goes up, then we can have narrowed the possibilities to two of the four outcomes. The probability of the occurrence of either of these two outcomes (the "or") is the denominator of the Bayes' Formula: 56% + 9% = 65%. Again, that is all possible outcomes where the stock goes up, P(U). The numerator is just the share of those where the economy grows. In other words, we are doing this: P[both economy grows and stock goes up]/P[stock goes up] must equal the odds the economy grew, if the stock goes up.

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