According to Bayesian statistics, Posterior Probability is the modernised or refurbished probability of a particular event occurring after taking new information. “Posterior”, over here, refers to considering the relevant proofs that relate to a particular event being examined. Sometimes, we may be curious to identify the probability of a random event “A” occurring after we take as given that some other random event “B” has just taken place. It is pertinent to clarify here that the Posterior Probability is calculated simply by using Bayes’ theorem to update prior probability. Simply put, the posterior probability refers to the probability of event A occurring, assuming that event B has already occurred.

In this article let us look at:

  1. Posterior Probability Example
  2. Posterior Probability Formula
  3. Prior Probability
  4. Prior and Posterior Probability
  5. Posterior Probability in Machine Learning

1. Posterior Probability Example

Let’s suppose that you are walking through the aisle of a grocery store. The person in front of you has dropped a 100 Rupee note, and you are right behind them. You see the note and pick it up. Of course, you want to return it, but you are confused about whether to address the person as “sir” or “ma’am”.All else being constant and considering the fact that the sex ratio in the world is roughly 1:1, you take a shot and shout “ma’am”.

The conditional probability of a woman turning around after you cry “ma’am” is 0.5. Now, let’s make a note of some new information that is relevant here. Say the person in question is wearing a dress and has long hair. Now we would have to refurbish our conditional probability to the result that the person is indeed a woman. The new information inculcated into the conditional probability gives us the posterior probability.

2. Posterior Probability Formula

Let us now look into the formula for posterior probability. First off, let us look at conditional probability.

The conditional probability function is represented in the following manner;

Posterior = P(H|D)

Here, D = data and H = hypothesis

However, as mentioned above, to calculate posterior probability, we make use of Bayes’ theorem.

The formula for Bayes’ theorem is:

And D = data, H = hypothesis

3. Prior probability

Prior probability, often called the prior, is the probability of a random event occurring before new information comes to light. This is the best judicious evaluation of the probability of an outcome on the basis of the current level of knowledge before an experiment is performed. It can also be drawn with existing information from previously conducted experiments. To put this in context of the above example, the probability of you addressing the individual in question as female and the person also happening to be a woman is your prior probability. 

4. Prior and Posterior Probability

The prior probability of a random event will be updated as soon as new data or information becomes available to produce a more precise measure of an outcome. That revised probability is the posterior probability and calculated using Bayes’ theorem, which is discussed above.

If we are curious about the probability of any occurrence of which we have prior conclusions, we call it the prior probability. This event will be deemed say, ‘A’ and its probability will be P(A). If there is a second event that affects P(A) outcome, we will deem it event B. Then, we want to find out the probability of event A occurring, considering that event B has already occurred. This is where the posterior probability comes into the picture.

5. Posterior Probability in Machine Learning

Bayes theorem is a fundamental and important theorem in machine learning because of its capability to analyze various theories given some perceptible data. With this analysis, we can more accurately predict various potential outcomes on unseen data (Do check out the Black Swan Paradox, if interested). Since an agent can only observe the world via the present data, it is important to bring to light any rational hypotheses from that existing data and any prior knowledge.

With the option of posterior, an agent can arrive at the accuracy of its hypothesis since it is possible to determine how precise the data is, given its hypothesis and all prior observations. The posterior probability is introduced to students very early on, in class 12 itself, because today, it has a wide range of practical applicability. It is used in various high power fields such as finance, medicine, economics, weather forecasting etc. It is also used in spam filtering, medical diagnosis, news categorization and determining environmental damage. 


The main purpose of using a tool like the posterior probability is to update a previous understanding we had about a certain event or fact, new information with regards to the same is obtained. We’re able to give rise to posterior probabilities of events occurring. This helps us gather a more accurate understanding of the ways of the world and helps us make more accurate predictions about future events.

In the real world, people bump into new information constantly. This new information enables us to recondition pre-conceived notions. Philosophically and statistically speaking, that is something we should all be open to.

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