Graduate · Beta–Bernoulli updating · 28 of 650
Beta–Bernoulli updating · Prior shape α=4; Prior shape β=4
Use the first six fixed observations: 1, 1, 0, 1, 0, 0. Find the posterior shape parameters and the predictive probability of success on the next trial. Givens: Prior shape α=4; Prior shape β=4.
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01 · Make a prediction
Your practice question
The unknown Bernoulli success probability has prior Beta(α=4,β=4); trials are conditionally independent given that probability. Use the first six fixed observations: 1, 1, 0, 1, 0, 0. Find the posterior shape parameters and the predictive probability of success on the next trial.
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A starting point
There are three successes and three failures. Add success and failure counts to the corresponding prior shape parameters.
Work through the reasoning
Step 1
Identify the model and target
The unknown Bernoulli success probability has prior Beta(α=4,β=4); trials are conditionally independent given that probability. Use the first six fixed observations: 1, 1, 0, 1, 0, 0. The governing relation is p|data ~ Beta(α+s,β+n−s); predictive P(success)=(α+s)/(α+β+n). There are three successes and three failures. Add success and failure counts to the corresponding prior shape parameters.
Step 2
Substitute and calculate
Among n=6 observations there are 3 successes and 3 failures. Add these to prior shapes (4,4) to obtain Beta(7,7); the predictive probability is 7/(7+7)=0.5.
Step 3
Check the mathematical meaning
The posterior is Beta(7,7). Its mean and one-step predictive success probability are (4+3)/(4+4+6)=0.5; this is not a point mass of the continuous posterior density. Observations included: 6; Posterior α: 7; Posterior β: 7; Predictive success probability: 0.5. Decimal values are rounded, so use unrounded intermediate values.
The answer
Among n=6 observations there are 3 successes and 3 failures. Add these to prior shapes (4,4) to obtain Beta(7,7); the predictive probability is 7/(7+7)=0.5. The posterior is Beta(7,7). Its mean and one-step predictive success probability are (4+3)/(4+4+6)=0.5; this is not a point mass of the continuous posterior density. Animation check: Observations included: 6; Posterior α: 7; Posterior β: 7; Predictive success probability: 0.5. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
A Beta prior describes uncertainty about an unknown Bernoulli probability. Multiplying by a conditionally independent Bernoulli likelihood adds success and failure counts to the two shape parameters. The posterior mean predicts one additional outcome but is not the probability mass at that one exact parameter value. The unknown Bernoulli success probability has prior Beta(α=4,β=4); trials are conditionally independent given that probability. Use the first six fixed observations: 1, 1, 0, 1, 0, 0. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
p|data ~ Beta(α+s,β+n−s); predictive P(success)=(α+s)/(α+β+n)
03 · Reflect and transfer
Explain what changes and why.
With the same balanced evidence, how does a stronger prior preserve more of its original preference?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.