Sequential updates

Accumulate evidence using conditional likelihoods and avoid counting the same signal twice.

12 minutes · advanced · Original Poker4Quant lesson

What you will learn

Update sequentially with stated conditional-independence assumptions.

The second likelihood must reflect the first observation

For two pieces of evidence E1 and E2, use P(E2|S,E1) and P(E2|weak,E1) in the second likelihood ratio. Simply reusing marginal likelihoods is justified only under the relevant conditional-independence assumptions.

If E1 and E2 are independent given each state, their likelihood ratios multiply. They need not be independent unconditionally: a shared hidden state can make the observations related in the mixed population. State which version of independence is being assumed.

Repeated reporting is not repeated evidence

Reading the same revealed action twice does not produce two independent observations. After E1 is already known, a perfect duplicate of E1 supplies no additional state discrimination. Its conditional likelihood ratio is one on the compatible histories.

Sequential Bayesian updating is coherent when each step uses the correct information set. This does not validate the likelihood estimates, the selected states or whether an opponent’s behavior stayed stable across observations. Those are model questions beyond algebra.

Worked example

Prior odds of strength are 1:9. Two signals each have likelihood ratio 3 and are conditionally independent given each state. What are posterior odds and probability?

After signal one, odds are 3:9=1:3.
After signal two, odds are 9:9=1:1.
Normalize equal odds into a probability.

The posterior is 50%. Multiplying by the second ratio would be wrong if it merely repeated the first signal without new information.

Common mistakes

Assuming unconditional independence is the needed property.
Reusing an unconditional second likelihood after a history-dependent first action.
Counting a duplicate message as fresh evidence.

Keep these ideas

Condition each likelihood on all previous observations.
Multiply simple ratios only under stated assumptions.
Evidence duplication creates false confidence.