Bayes' formula: updating a probability with new information

What are the parts of Bayes’ formula?

News arrives and you want to revise a probability. Start with the prior probability , your view of the event before the news. 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.

Updating a probabilitynews arrives, so revise the view

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

Bayes’ formula connects them. The denominator, P(I), is the unconditional probability of the information.

$$P(E \mid I) = \dfrac{P(I \mid E)}{P(I)}\,P(E)$$

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

$$P(I) = P(I \mid E)\,P(E) + P(I \mid E^c)\,P(E^c)$$

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

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.

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	0.30	0.10	0.40
Does not beat	0.09	0.51	0.60
Column total	0.39	0.61	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, 0.30 ÷ 0.39 = 0.769. Had the company not raised, it would be 0.10 ÷ 0.61 = 0.164.

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

$$P(S_i \mid I) = \dfrac{P(I \mid S_i)\,P(S_i)}{\sum_{j} P(I \mid S_j)\,P(S_j)}$$

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 0.6 and 0.2, equal priors of 0.5 give posteriors of 0.6 ÷ 0.8 = 0.75 and 0.2 ÷ 0.8 = 0.25.

Exam tip When the question gives you P(I | E) but you need P(E | I), 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 90% of firms that will default and also flags 9% of firms that will not. A flagged firm looks alarming, yet P(flag) = 0.9 × 0.01 + 0.09 × 0.99 = 0.0981, so the posterior is 0.009 ÷ 0.0981 = 9.2%. The low prior limits the update. Ignoring the prior and treating the 90% 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 0.769. If a second signal, an analyst upgrade, then arrives, appearing for 60% of beaters and 20% of non-beaters, and the two signals are independent given the outcome, then P(upgrade) = 0.6 × 0.769 + 0.2 × 0.231 = 0.508 and the new posterior is 0.462 ÷ 0.508 = 0.909.