Posterior Probability Calculator

Posterior probability is a fundamental concept in Bayesian statistics. It represents the updated probability of an event (A) occurring given new evidence (B). The posterior probability is calculated using Bayes' theorem:

Formula

\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]

Where:

• \( P(A) \) is the prior probability (the initial probability of event A before seeing evidence B),
• \( P(B|A) \) is the likelihood (the probability of observing evidence B given that A is true),
• \( P(B) \) is the marginal likelihood (the total probability of observing B under all possible scenarios).

Example Calculation

If the prior probability \( P(A) \) is 0.2, the likelihood \( P(B|A) \) is 0.8, and the marginal likelihood \( P(B) \) is 0.5, then:

\[ P(A|B) = \frac{0.8 \times 0.2}{0.5} = 0.32 \]

Importance of Posterior Probability

Posterior probability allows statisticians and decision-makers to update their beliefs about an event based on new data, leading to more informed predictions and decisions. It has applications in various fields, including medical diagnosis, machine learning, and risk assessment.

Common FAQs

1. What is the difference between prior and posterior probability?

   • The prior probability is the initial belief before new evidence is considered. The posterior probability is the updated belief after considering the new evidence.

2. What is Bayes' theorem used for?

   • Bayes' theorem is used to calculate the posterior probability by updating prior beliefs with new evidence, making it useful for inference and decision-making.

3. Can posterior probability be greater than 1?

   • No, probabilities always lie between 0 and 1, representing the likelihood of an event occurring.