The law of likelihood is a principle in statistics that states that the likelihood of a hypothesis being true is directly proportional to its prior probability and the degree to which it explains the observed data. This means that a hypothesis with a higher prior probability and a good explanation of the data is more likely to be true than a hypothesis with a lower prior probability and a poor explanation of the data.

Here are two examples of the law of likelihood in action:

Imagine that you are a detective trying to solve a crime. You have two suspects, John and Mary, and you have gathered some evidence that could point to the culprit. John has a prior probability of 0.3, meaning that there is a 30% chance that he is the culprit based on previous evidence. Mary has a prior probability of 0.7, meaning that there is a 70% chance that she is the culprit based on previous evidence.

After examining the evidence, you determine that the data strongly supports the hypothesis that the culprit is John. Based on the law of likelihood, this means that the likelihood of John being the culprit has increased, and the likelihood of Mary being the culprit has decreased. In other words, the law of likelihood helps you to update your beliefs about the suspects based on new evidence.

Imagine that you are a doctor trying to diagnose a patient with a rare disease. You have two hypotheses about the patient’s condition, Hypothesis A and Hypothesis B. Hypothesis A has a prior probability of 0.1, meaning that there is a 10% chance that it is true based on previous evidence. Hypothesis B has a prior probability of 0.9, meaning that there is a 90% chance that it is true based on previous evidence.

After examining the patient and conducting some tests, you determine that the data strongly supports the hypothesis that the patient has the rare disease. Based on the law of likelihood, this means that the likelihood of Hypothesis A being true has decreased, and the likelihood of Hypothesis B being true has increased. In other words, the law of likelihood helps you to update your beliefs about the patient’s condition based on new evidence.

In both of these examples, the law of likelihood is used to update beliefs about a hypothesis based on new evidence. The principle states that the likelihood of a hypothesis being true is directly proportional to its prior probability and the degree to which it explains the observed data. This means that a hypothesis with a higher prior probability and a good explanation of the data is more likely to be true than a hypothesis with a lower prior probability and a poor explanation of the data.