A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information. This probability is derived by adjusting the prior probability using Bayes’ theorem. In straightforward terms, the posterior probability is the probability of event A occurring given that event B has occurred.
Key Takeaways
- Revised Probability: Posterior probability represents a revised estimate taking new evidence into account.
- Bayesian Foundation: It’s calculated by updating prior probability using Bayes’ theorem.
- Conditional Probability: Statistically, it reflects the likelihood of event A happening given that event B has taken place.
The formula for calculating the posterior probability of event A occurring given that event B has occurred is stated leveraging Bayes’ theorem:
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A) \times P(B|A)}{P(B)}
Where:
- A, B = Events
- P(B | A) = The probability of B occurring given that A is true
- P(A) and P(B) = The probabilities of A occurring and B occurring independently of each other
Hence, the posterior probability is the distributing result, P(A|B).
What Does a Posterior Probability Tell You?
Bayes’ theorem finds wide applications across medicine, finance, and economics. In finance, it can refine investment strategies based on emerging market data. The prior probability denotes the initial belief before encountering new evidence, and the posterior modifies this belief with new insights.
Posterior probability distributions offer an enhanced reflection of a data-generating process’s truth due to their inclusive nature of more information than the prior probability. As new information arises, this posterior adjustment allows continuous updates, with posterior probabilities transitioning into new priors for adaptive analysis and decision-making.
Related Terms: prior probability, Bayes’ theorem, conditional probability, likelihood, belief updating.
References
- Data Science Discovery. “Bayes’ Theorem”.
Get ready to put your knowledge to the test with this intriguing quiz!
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## What is meant by Posterior Probability?
- [ ] The probability of an event after a new result has occurred, disregarding prior information.
- [ ] The initial probability of an event before any new evidence is considered.
- [x] The probability of an event occurring after taking into account new evidence.
- [ ] The expected value of a random variable.
## What is the key formula used to calculate Posterior Probability?
- [ ] Black-Scholes formula
- [x] Bayes' Theorem
- [ ] Capital Asset Pricing Model (CAPM)
- [ ] Modigliani-Miller Theorem
## Posterior Probability is most commonly used in which field?
- [ ] Fundamental Analysis
- [ ] Algorithmic Trading
- [ ] Long-term Investment Strategies
- [x] Bayesian Statistics
## Which term describes the probability of event A occurring given that event B has occurred?
- [ ] Marginal Probability
- [ ] Joint Probability
- [ ] Prior Probability
- [x] Posterior Probability
## In Bayesian analysis, what is "updated" to derive the Posterior Probability?
- [ ] Price-Earnings Ratio
- [ ] Market Hypothesis
- [x] Prior Probability with new evidence
- [ ] Discount Rate
## In terms of everyday decision-making, Posterior Probability helps in:
- [ ] Estimating interest rates
- [x] Updating beliefs according to new evidence
- [ ] Projecting company revenue
- [ ] Analyzing market trends
## Which of the following components is NOT used in calculating the Posterior Probability?
- [ ] Likelihood
- [ ] Prior Probability
- [ ] Evidence
- [x] Expected Value
## Posterior Probability is an example of:
- [ ] Frequentist inference
- [x] Bayesian inference
- [ ] Classical hypothesis testing
- [ ] Non-parametric statistics
## What element is essentially required to compute the Posterior Probability?
- [ ] A large dataset sample
- [ ] Identical, independently distributed random variables
- [ ] The current stock index
- [x] Prior knowledge or estimates (Prior Probability)
## If new data contradicts the prior assumptions, how does Posterior Probability react?
- [ ] Ignore the new data
- [ ] Reduce to zero probability
- [ ] Increase the Prior Probability proportionally
- [x] Adjust probabilities according to Bayes' Theorem
