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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Combine a Beta prior with a Bernoulli likelihood.
- Justify the conclusion "Posterior is Beta(6,4), with mean 6/10=0.6" using the stated assumptions.
Before you start
Bayes' rule, densities, and Beta distributions.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
With prior p~Beta(2,3), observe four successes and one failure. Find the posterior.
Why this math matters
Combine a Beta prior with a Bernoulli likelihood. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Trials are conditionally independent given p.
- The Beta prior is the stated modeling choice.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Update a probability distribution over an unknown probability
PausedQuestion: Start with the question. Paused.
Question
Start with the question
With prior p~Beta(2,3), observe four successes and one failure. Find the posterior.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
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Build the model
Prior density is proportional to p¹(1−p)²
Beta parameters determine the exponents before observing data.
Work through the mathematics
Multiply by likelihood p⁴(1−p)¹
Bayes' rule combines prior and sample evidence for the same parameter.
Check the conclusion
Posterior is Beta(6,4), with mean 6/10=0.6
Successes and failures add to the corresponding shape parameters, then normalization supplies a probability density.
The result
Posterior is Beta(6,4), with mean 6/10=0.6
Successes and failures add to the corresponding shape parameters, then normalization supplies a probability density.
Common mistakes to catch
- The posterior mean is not the same as the maximum-likelihood estimate.
- Prior parameters should not be mistaken for observed data counts without explanation.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the posterior predictive success probability for one new trial?
Show a hint
Average p under the posterior.
Reveal answer and explanation
0.6
The predictive probability is the posterior mean of p.
Practice 2
Does a continuous prior place positive probability on one exact p value?
Show a hint
A singleton has zero integral under a density.
Reveal answer and explanation
No
Credible intervals concern ranges, while point values have density rather than positive point mass.
Take the idea with you
Compare how different priors affect a small-sample update and diminish in relative influence as data accumulate.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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