Posterior Probability: Definition, Formula, and Calculation Methods

An in-depth analysis of posterior probability, its formulation and methods for calculation, and its applications in various fields such as Bayesian statistics, machine learning, and decision making.

What is Posterior Probability?

Posterior probability is the probability of an event occurring after accounting for new evidence or information. It is a key concept in Bayesian statistics, which updates the initial or prior probability with new data to form a more accurate or refined probability.

Mathematically, posterior probability is represented as:

$$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$

where:

• \( P(A|B) \) is the posterior probability of event \( A \) occurring given evidence \( B \),
• \( P(B|A) \) is the likelihood of observing \( B \) given that \( A \) is true,
• \( P(A) \) is the prior probability of \( A \) before observing \( B \),
• \( P(B) \) is the marginal likelihood or the total probability of observing \( B \).

Calculation Methods

Bayes’ Theorem

The calculation of posterior probability primarily relies on Bayes’ theorem. Bayes’ theorem provides a mathematical framework to update the probability estimate for an event based on new information. It combines the prior probability distribution, the likelihood of new evidence, and the marginal likelihood.

Example Calculation

Consider a medical test for a disease:

• Suppose 1% of the population has the disease (\( P(D) = 0.01 \)),
• The probability of a positive test result given that the person has the disease is 99% (\( P(Pos|D) = 0.99 \)),
• The test gives a false positive in 5% of the healthy population (\( P(Pos|NoD) = 0.05 \)).

To find the posterior probability that a person has the disease given a positive test result (\( P(D|Pos) \)):

$$ P(D|Pos) = \frac{P(Pos|D)P(D)}{P(Pos)} $$

Where:

• \( P(Pos) = P(Pos|D)P(D) + P(Pos|NoD)P(NoD) \)

Plugging in the values:

$$ P(Pos) = (0.99 \times 0.01) + (0.05 \times 0.99) = 0.0099 + 0.0495 = 0.0594 $$

Therefore:

$$ P(D|Pos) = \frac{0.99 \times 0.01}{0.0594} \approx 0.1666 $$

The posterior probability is approximately 16.66%.

Applications and Relevance

Bayesian Statistics

Posterior probability is a cornerstone of Bayesian statistics, allowing statisticians to revise estimated probabilities as new data becomes available. This method is especially useful in fields like:

• Machine Learning
• Data Science
• Economics
• Medical Decision-Making

Decision-Making Under Uncertainty

Posterior probabilities assist in improving decision-making processes under uncertainty by continuously integrating new information and refining existing predictions.

Special Considerations

Assumptions in Calculation

It’s essential to understand the assumptions behind the model, such as the independence of events or the accuracy of prior distributions and likelihoods.

Potential Limitations

Limitations include sensitivity to the choice of the prior distribution and the computational complexity in high-dimensional spaces.

Related Terms

• Prior Probability: The initial probability estimate before new evidence is considered (\( P(A) \)).
• Likelihood: The probability of the observed evidence given an event (\( P(B|A) \)).
• Marginal Probability: The total probability of observing the evidence (\( P(B) \)).

FAQs

What is the difference between prior and posterior probability?

Prior probability is the initial assessment before any new evidence is taken into account, while posterior probability is the revised assessment after considering new evidence.

How does posterior probability apply in real-world situations?

It is widely used in fields such as medical diagnostics, finance (e.g., assessing risks or returns), and machine learning (e.g., updating model predictions based on new data).

References

1. Bayes, T. (1763). An Essay towards solving a Problem in the Doctrine of Chances.
2. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis. CRC Press.

Summary

Posterior probability is a fundamental concept in updating the likelihood of events based on new data. It plays a crucial role in areas that require continuous learning and adaptation, making it an invaluable tool in both theoretical and applied statistics.

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