# Posterior probability distribution

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## What is Posterior Probability?

Posterior probability is the conditional probability an event will happen after all evidence or background information has been taken into account. To put that another way: if we know the conditional and unconditional probabilities of one event in advance, we can calculate the conditional probabilities for a second event.

It is calculated by updating the prior probability with Bayes’ rule.

The formula can be broken down into parts:

• P(A|B) = posterior probability of event A.
• P(B|A) = likelihood of event B, given that A has already happened.
• P(A) = prior probability of event A.
• P(B) = prior probability of event B.

For example, let’s say you wanted to find a patient’s probability of having fatty liver disease, given that they are obese. “Being obese” is the test (in the sense of a litmus test) for fatty liver disease. We know some facts about the following events:

• Event A is the event “Patient has fatty liver disease.” Past data tells you that 10% of patients have fatty liver disease. P(A) = 0.10.
• Event B is the test that “Patient is obese.” Five percent of the clinic’s patients are obese. P(B) = 0.05.
• You also know that among those patients diagnosed with fatty liver disease, 7% are obese. This is B|A (B given A): the probability a patient is obese, given that they have liver disease, is 7%.

Plugging these into the Bayes’ theorem formula:

P(A|B) = (0.07 * 0.1)/0.05 = 0.14

If the patient is obese, their probability of having fatty liver disease is 0.14 (14%). This is a significant increase from the 10% suggested by past data.

## What is a Posterior Distribution?

The posterior distribution is a way to summarize what we know about uncertain quantities in Bayesian statistics. It is a combination of the prior distribution and the likelihood function, which tells you what information is contained in your observed data (the “new evidence”). Thus, the posterior probability distribution is a compromise between the prior distribution and likelihood function. In other words, the posterior distribution summarizes what you know after the data has been observed, where the summary of the evidence from the new observations is the likelihood function.

Posterior Distribution = Prior Distribution + Likelihood Function (“new evidence”)

We can rewrite Bayes’ rule to express it in terms of probability distributions :

Where

• f(θ|data) = posterior distribution for parameter θ,
• f(data|θ) = sampling density for the data. This is proportional to the likelihood function and differs only by a constant which would make it a “proper” density function,
• (θ) = prior distribution for the parameter,
• f(data) = the data’s marginal probability — the probability of observing the data regardless of the parameter’s value.

Posterior distributions are integral to Bayesian analysis. They are, in many ways, the goal of the analysis and can give you:

## Prior vs. Posterior probability

Posterior probability is closely related to prior probability, which is the probability an event will happen before you taken any new evidence into account. You can think of posterior probability as an adjustment on prior probability:

Posterior probability = prior probability + new evidence (called ‘likelihood’)

For example, historical data suggests that around 60% of students who start college will graduate within 6 years. This is the prior probability. However, you think that figure is much lower, so you set out to collect new data. The evidence you collect suggests that the true figure is closer to 50%; This is the posterior probability.

Thus, posterior probability distributions better reflect the true probability than the prior probability because it incorporates more information.

## Origin of the Terms

The words posterior and prior come from the Latin posterior (“later”) and prior (“earlier”). The definition of “a priori” is similar, also stems from Latin. However, it means “from something that came before”.

Merriam Webster defines a priori as:

“…relating to what can be known through an understanding of how certain things work [i.e., a hypothesis] rather than by observation.”

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