Wald Sequential Likelihood Ratio

Worked Example

Consider comparing two simple Bernoulli hypotheses: H0 where p = 1/3 and H1 where p = 2/3. The likelihood ratio (LR) updates sequentially with each observation in a sequence. Let's trace the sequence "11" with an evidence exponent of 2 (meaning boundaries are 2² = 4 and 2^-2 = 1/4).

1. Start with an initial LR of 1.
2. First observation is "1". Under H1, P(X=1|H1) = 2/3. Under H0, P(X=1|H0) = 1/3. The likelihood factor is (2/3) / (1/3) = 2. The new LR is 1 × 2 = 2.
3. Second observation is "1". The likelihood factor is again 2. The new LR is 2 × 2 = 4.
4. The sequence hits the upper boundary (2² = 4), so we stop and Support H1.

Experiment

You can interact with the Wald Sequential Likelihood Ratio module to visualize this process:

1. Set the Sequence to 00.
2. Observe that the LR halves at each step: it drops from 1 to 1/2, then to 1/4.
3. With the default Evidence exponent set to 2, the sequence hits the lower boundary (1/4), and the Decision updates to Support H0.
4. Now, change the Sequence to 1010. The LR alternates between 2 and 1, never reaching either boundary. The status remains Continue.

Limitations

This model demonstrates a bounded sequential test for simple IID Bernoulli hypotheses. A finite sequence that hits neither boundary remains "Continue" and is not evidence for accepting H0. The chosen boundaries are strictly evidence thresholds (2^h and 2^-h) and do not claim to achieve exact nominal type-I or type-II error probabilities, which would require analyzing overshoot or randomizing at the boundaries.

References

For further reading on the Likelihood Ratio and Sequential Probability Ratio Test, refer to the following educational resource:

• UC Berkeley Statistics 159/259 course notes - Supports likelihood definition, Bernoulli per-observation factors, and first-boundary stopping.