Bayes' theorem: updating a probability with new information

What are the parts of Bayes’ theorem?

News arrives and you want to revise a probability. Call the event A and the new information B. Start with the prior probability , your view of the event before the news; it is also called the a priori or ex ante probability. The likelihood is how probable the news is if the event is true. After the update you hold the posterior probability , the probability of the event given the news, also called the a posteriori or ex post probability.

Updating a probabilitynews arrives, so revise the view

• Prior P(A) Memorize your probability before the news
• Likelihood P(B | A) Memorize how likely the news is if the event is true
• Marginal probability P(B) Memorize total chance of the news: add every branch that produces it
  • Event branch P(B | A) × P(A)
  • No-event branch P(B | not A) × P(not A)
• Posterior P(A | B) Memorize event branch ÷ P(B)

Bayes’ theorem (also called Bayes’ rule or Bayes’ formula) connects them. The denominator, P(B), is the marginal probability of the information: its overall chance of showing up.

$$P(A \mid B) = \dfrac{P(B \mid A)}{P(B)}\,P(A)$$

P(B) comes from the total probability rule : add up every route that can produce the information.

$$P(B) = P(B \mid A)\,P(A) + P(B \mid A^c)\,P(A^c)$$

The top of Bayes’ theorem, P(B | A) × P(A), is the joint probability of the event and the information. So the posterior is the joint probability divided by the marginal probability of the information: your share of all the ways the news could have happened.

Go deeper Where the formula comes from

The joint probability can be built in either order: P(A | B) × P(B) = P(A and B) = P(B and A) = P(B | A) × P(A). Divide both ends by P(B) and you have Bayes’ theorem. In effect it runs the total probability rule in reverse: that rule builds P(B) from its pieces, and Bayes’ theorem asks which piece produced it.

Slide the prior down and the two likelihoods together. The posterior moves much less when the prior is low, and much more when the two likelihoods are far apart. The Counts tab shows the same update from a table of whole numbers.

A table method that never fails

Lay out every joint probability in a grid and read the posterior from a column. For the example above:

	Raises guidance	Does not raise	Row total
Beats consensus	0.2925	0.1575	0.45
Does not beat	0.1375	0.4125	0.55
Column total	0.43	0.57	1.00

Each cell is a prior times a likelihood. The posterior is the cell divided by its column total once you know which news you saw: given a raise, 0.2925 ÷ 0.43 = 0.680. Had the company not raised, it would be 0.1575 ÷ 0.57 = 0.276.

The same method works with more than two scenarios. Add a row for each.

$$P(A_i \mid B) = \dfrac{P(B \mid A_i)\,P(A_i)}{\sum_{j} P(B \mid A_j)\,P(A_j)}$$

How do you update from a table of counts?

Counts make a posterior a cell divided by a column total. Here are 400 funds screened at the start of a year. 128 were flagged as likely outperformers. 140 funds outperformed, 91 of them flagged, and 37 of the 260 that did not outperform were flagged too.

	Flagged	Not flagged	Total
Outperformed	91	49	140
Did not outperform	37	223	260
Total	128	272	400

What if P(B) is given and a likelihood is missing?

Use the total probability rule backwards. When the question gives the marginal probability P(B), one likelihood and the prior, solve for the likelihood you do not have, then run Bayes’ theorem.

$$P(B \mid A^c) = \dfrac{P(B) - P(B \mid A)\,P(A)}{P(A^c)}$$

What is a diffuse prior?

Diffuse prior is the name for equal priors: you have no basis to favor any scenario, so each gets the same probability. Then the priors cancel from the top and bottom of the formula, and the posterior follows the likelihoods alone. Each posterior is then its likelihood divided by the sum of the likelihoods. With two scenarios and likelihoods of 0.65 and 0.25, equal priors of 0.5 give posteriors of 0.65 ÷ 0.90 = 0.722 and 0.25 ÷ 0.90 = 0.278.

Exam tip When the question gives you P(B | A) but you need P(A | B), write the table first. If the column total is missing, the question wants the total probability rule.

Go deeper Base-rate neglect: why a rare event stays unlikely

Suppose 1% of firms will default in a year, a screen flags 95% of firms that will default and also flags 7% of firms that will not. A flagged firm looks alarming, yet P(flag) = 0.95 × 0.01 + 0.07 × 0.99 = 0.0788, so the posterior is 0.0095 ÷ 0.0788 = 12.1%. The low prior limits the update. Ignoring the prior and treating the 95% likelihood as the answer is a classic behavioral error.

Go deeper Sequential updating

Yesterday’s posterior becomes today’s prior. After the guidance raise the probability of a beat is 0.680. If a second signal, an analyst upgrade, then arrives, appearing for 85% of beaters and 35% of non-beaters, and the two signals are independent given the outcome, then P(upgrade) = 0.85 × 0.680 + 0.35 × 0.320 = 0.690 and the new posterior is 0.578 ÷ 0.690 = 0.838.