# Posterior Probability – Explanation – Calculation – Example

## What Is a Posterior Probability?

Posterior probability is the probability an event will happen after all evidence or background information has been taken into account. It is a revised probability that takes into account new available information.

In Bayesian statistics, it is the revised or updated probability of an event.  However, the revision occurs after taking into consideration the existing as well as the new information. Specifically, it is the conditional probability of a given event calculated after observing a second event whose conditional and unconditional probabilities were known in advance. It is computed by revising the prior probability, that is, the probability assigned to the first event before observing the second event using Bayes’ theorem. In statistical terms, it becomes the probability of event A occurring given that event B has occurred.

### Prior Probability vs Posterior Probability

Prior probability is the probability an event will happen before you take any new evidence into account. You can think of posterior probability as an adjustment on prior probability.  For example, historical data suggests that around 50% of students who start college will graduate within 4 years. This is the prior probability. However, you think that figure is actually much lower, so set out to collect new data. The evidence you collect suggests that the true figure is actually closer to 40%.  This is the posterior probability.

For example, let’s say you bumped into another person on the street anywhere in the world. What is the probability that the person you ran into was of Asian Indian ethnic descent? With no other information to go by, you might say that the probability was around 18%.  This is simply because you estimate there are approximately 1.4 billion Asian Indians out of a world population of 7.6 billion.  From this statistic, you made your best estimate at a prior probability with no other context.

However, let’s add the information that this event took place somewhere in Shanghai, China. With this additional knowledge, you could calculate a new probability.  This time, you would include the population distribution of ethnically Indian people living and traveling in Shanghai. In other words, you would need to update your prior probability which was unconditional given a new condition of occurring in the China.  You would then be able to calculate a posterior probability, or conditional probability based on new information or evidence.

## Bayes’ Theorem Formula to Calculate Posterior Probability

The formula to calculate a posterior probability of A occurring given that B occurred:

## Example

Suppose an individual is chosen from a high school population at random.  The probability of choosing a female individual is 50%.  The probability of choosing an individual with brown hair is 40%.  The conditional probability that the individual is female and with brown hair is 20%.  If the individual extracted at random from the high school population is female, what is the conditional probability that she also has brown hair?  This is an example of posterior probability and it can be calculated using Bayes’ Formula.

• P(female) = 0.5
• P(brown hair) = 0.4
• P(female/brown hair) = 0.2

Probability(female/brown hair) = {P(brown hair/female) x P(female)} / P(brown hair)

P(female/brown hair) = (0.2 x 0.5) / (0.4)

The posterior probability of choosing a female with brown hair = 0.25 or a 1 in 4 chance.

## What Posterior Probability Tells You

Bayes’ theorem is used in many applications including medicine, finance, and economics. In finance, Bayes’ theorem can be used to update a previous belief once new information is obtained. Prior probability represents what is originally believed before new evidence is introduced.  Posterior probability takes this new information into account to generate a new probability.  Posterior probability distributions should be a better reflection of the underlying truth of a data generating process than the prior probability.  Obviously, this is due to the posterior including more information. A posterior probability can subsequently become a prior for a new updated posterior probability.  This occurs as new information becomes available and is incorporated into the analysis.

## Up Next: What Is DVP – Delivery Versus Payment?

DVP – Delivery versus payment is a method of settlement for securities. It guarantees the transfer of securities only after payment is made.  It requires that the buyer fulfills their payment obligations Payment must occur before or immediately at the time of the delivery of the purchased securities.

DVP stipulates that the buyer’s cash payment for securities must be made prior to or at the same time as the delivery of the security. Delivery versus payment is the settlement process from the buyer’s perspective.  Receive versus payment (RVP) is from the seller’s perspective.  DVP/RVP requirements emerged in the aftermath of 1987.  Thereafter, institutions were not required to pay money for securities before the securities were held in negotiable form. There are other terms that have the same essential meaning as DVP.  Terms include delivery against payment (DAP), delivery against cash (DAC), and cash on delivery COD).