Log-Likelihood Ratio

This interactive page lets you explore how changes in prior and posterior beliefs affect the Log-Likelihood Ratio (LLR).

What is the LLR?

The log-likelihood ratio (LLR) is a standard tool used to describe how beliefs change when new information arrives. It measures both the direction and the strength of belief updating.

How is the LLR constructed?

Suppose the outcome of an exam is binary: a student either passes or fails. Let the prior probability of passing be \( \mathbb{P}(\text{Passed}) = \pi \). Since the outcome is binary, the probability of failing is \( \mathbb{P}(\text{Failed}) = 1 - \pi \).

Now suppose the student receives a signal about their performance. The signal is informative but not perfect. Conditional on the true outcome, it is generated with probabilities \( \mathbb{P}(\text{Signal} \mid \text{Passed}) \) and \( \mathbb{P}(\text{Signal} \mid \text{Failed}) \).

Using Bayes’ rule, the posterior probability of passing is:

Similarly, the posterior probability of failing is:

Taking the ratio of posterior odds and then taking logs gives:

This leads to the definition of the log-likelihood ratio (LLR):

Interpretation of LLR

• If LLR > 0, the signal provides evidence in favor of passing.
• If LLR < 0, the signal provides evidence in favor of failing.
• Larger absolute values of LLR correspond to stronger evidence.

Example

Suppose your prior belief of passing is 50%. After receiving a positive signal, your belief increases to 75%. The LLR is then:

This positive value means the signal supports passing. If instead your posterior dropped to 25%, the LLR would be negative, indicating evidence against passing.