Posteriority and Bias in Probability

Posteriority and bias are important concepts in probability theory and statistics.

Posteriority refers to the updated probability distribution that results from incorporating new information into a prior probability distribution. The updated distribution is called the posterior distribution, and it takes into account both the prior information and the new evidence. In Bayesian statistics, the posterior distribution is calculated using Bayes’ theorem, which relates the prior distribution, the likelihood of the data, and the evidence to the posterior distribution.

Bias, on the other hand, refers to a systematic error in the estimation of a quantity or parameter. In probability and statistics, bias can arise in many different ways. For example, a sampling bias can occur if the sample used to estimate a parameter is not representative of the population from which it is drawn. A measurement bias can occur if the measurement device used to collect data systematically overestimates or underestimates the true value.

Posteriority

Posteriority, also known as posterior probability, refers to the likelihood of an event or hypothesis given the available evidence or data. It is a fundamental concept in probability and plays a central role in statistical inference, which is the process of using data to make conclusions about a population or phenomenon.

In statistical inference, we often want to estimate the likelihood of an event or hypothesis based on our collected data. For example, we want to know the probability that a coin is fair based on the results of flipping it a certain number of times. In this case, the probability of the event (the coin being fair) given the data (the results of the flips) is known as the posterior probability.

The posterior probability is calculated using Bayes’ theorem, which states that the posterior probability of an event or hypothesis can be calculated by multiplying the prior probability of the event by the likelihood of the data given the event and then dividing by the overall likelihood of the data. In other words, the posterior probability is a weighted average of the prior probability and the possibility of the data, with the weighting being determined by the overall probability of the data.

One of the key advantages of using posterior probability is that it allows us to incorporate new evidence or data into our analysis as it becomes available. We can continually refine our understanding of the likelihood of different events or hypotheses by updating our posterior probabilities as new data becomes available.

Example

Suppose we have developed a subscription-based software and have data on the usage of the software. Now we want to check what the probability of a particular user churning on the basis of this data is. We have collected data on the usage patterns of many customers, including whether or not they churned.