Bayesian Updating, explained.
Bayesian updating revises the probability of a hypothesis using the likelihood of observed evidence under that hypothesis and its alternatives.
Why it happens
Evidence is informative through a comparison: blue is more likely in urn A than in urn B. Multiplying prior odds by that likelihood ratio produces posterior odds, which become the starting odds for the next independent observation.
Bayesian updating combines a prior belief with the relative likelihood of the observed evidence under competing hypotheses.
Read the result
Advance the number of draws and inspect the observed color sequence. Blue favors A and orange favors B. Changing the prior can change both the sampled hidden urn and the update, so compare runs with the stated settings in mind.
A worked example
A blue draw followed by orange
Starting odds for A are one to one. A blue draw multiplies them by 7/3, making probability 70%.
An orange draw multiplies those odds by 3/7, bringing them back to one to one.
The evidence balances under these particular known likelihoods; repeated or dependent evidence would require a different model.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The two urn hypotheses start with the selected prior. A seeded hidden urn generates draws with replacement. Blue has likelihood ratio 7/3, orange 3/7; posterior odds are multiplied by the appropriate ratio.
Where this idea is useful
A practical use
Several independent observations can move an initial estimate substantially, while contradictory observations can move it back.
A common misconception
“A posterior probability proves a hypothesis true.”
It expresses uncertainty under the prior and likelihood assumptions. Changing those assumptions can change the result.
What this explanation leaves out
- Likelihoods and conditional independence are stipulated. Repeatedly counting the same evidence would overstate the update.
Why draw with replacement?
It keeps each urn’s composition fixed and makes the repeated observations conditionally independent given the hidden urn.
What assumptions make a new observation genuinely additional evidence?
Further reading
Explore the original research or the teaching reference behind this experiment.