Bayesian Inference
Combine prior and likelihood into a posterior.
Key idea
Combine prior and likelihood into a posterior.
In depth
Combine a prior belief with data using Bayes’ theorem to get a posterior: posterior ∝ likelihood × prior. As data accumulate, the prior matters less. Results are probability statements about parameters.
Worked example
With a Beta(1,1) prior on a coin’s p and 7 heads in 10, the posterior is Beta(8, 4), mean 0.667.
Formula
Real-world uses
- Updating a weather forecast or a medical diagnosis as new test results arrive
Common mistake
Choosing a prior without examining how sensitive the answer is.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Bayesian Inference?
Combine prior and likelihood into a posterior. It is a topic at the expert level in Probability & Statistics.
What is the formula for Bayesian Inference?
The key formula is posterior ∝ likelihood × prior.
Where is Bayesian Inference used in real life?
Real-world examples: Updating a weather forecast or a medical diagnosis as new test results arrive.
What should I learn before Bayesian Inference?
Make sure you are comfortable with Maximum Likelihood Estimation, the previous topic in Probability & Statistics.
What is a common mistake with Bayesian Inference?
Choosing a prior without examining how sensitive the answer is.
How can I practise Bayesian Inference?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Bayesian Inference?
Move on to Bootstrap and Resampling next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.