MPE (Most Probable Explanation) vs. MAP (Maximum A Posteriori)

quangngoc

MPE (Most Probable Explanation) and MAP (Maximum A Posteriori) are concepts used in probabilistic graphical models and Bayesian inference to find the most likely explanation or configuration of variables given evidence. They are related but have distinct differences:

1. MPE (Most Probable Explanation):

Definition: MPE seeks to find the most likely configuration of variables in a probabilistic model given observed evidence. In other words, it finds the explanation that maximizes the joint probability of the variables in the context of the observed evidence.
Objective: To determine the most probable state of the variables, typically conditioned on observed evidence, by finding the configuration that maximizes (P(X|E)), where (X) represents the variables of interest, and (E) represents the evidence.
Use Case: MPE is often used in scenarios where you want to find the single most likely explanation or state of the system based on available evidence. It's commonly employed in tasks like inference in Bayesian networks.
Mathematical Formulation:
[ \text{MPE} = \arg\max_{X} P(X|E) ]

2. MAP (Maximum A Posteriori):

Definition: MAP also aims to find the most likely configuration of variables given evidence, but it considers not only the likelihood of the evidence but also incorporates prior beliefs about the variables. It finds the configuration that maximizes the posterior probability.
Objective: To determine the most probable state of the variables by finding the configuration that maximizes (P(X|E) \cdot P(X)), where (P(X|E)) is the likelihood of the evidence, and (P(X)) represents the prior probability distribution over the variables.
Use Case: MAP is often used when you have prior knowledge or beliefs about the variables of interest. It combines this prior information with observed evidence to find the most likely configuration.
Mathematical Formulation:
[ \text{MAP} = \arg\max_{X} P(X|E) \cdot P(X) ]

Key Differences:

Consideration of Prior Information: The main difference is that MAP incorporates prior beliefs (prior probabilities) about the variables, while MPE does not. MAP balances the likelihood of the evidence with prior knowledge, making it more robust when prior information is available.
Objective Function: MPE maximizes the conditional probability (P(X|E)), while MAP maximizes the posterior probability (P(X|E) \cdot P(X)). This means that MAP explicitly accounts for both the likelihood of the evidence and the prior distribution.
Use Cases: MPE is suitable when you have no prior information about the variables and want to find the most probable explanation solely based on observed evidence. MAP is used when you have prior knowledge or beliefs and want to incorporate them into the inference process.

In summary, both MPE and MAP aim to find the most likely explanation given evidence, but MAP introduces the concept of prior probabilities, making it a more flexible approach when prior information is available. The choice between MPE and MAP depends on the specific problem and the availability of prior knowledge or beliefs.

quangngoc

Give an example of when they would produce different results.

Let's consider an example where MPE (Most Probable Explanation) and MAP (Maximum A Posteriori) would produce different results. Suppose we have a medical diagnostic scenario where we want to determine whether a patient has a rare disease, "Disease X," based on the results of two diagnostic tests: Test A and Test B.

Here's the setup:

Prior Belief (Prior Probability): We have prior knowledge that Disease X is extremely rare in the population, so our prior belief is that the probability of a randomly selected person having Disease X, denoted as (P(Disease\ X)), is very low, say 0.01% (0.0001).
Diagnostic Tests:
- Test A: The sensitivity of Test A (true positive rate) is relatively high, around 95%, but it also has a false positive rate of 10%.
- Test B: The sensitivity of Test B is lower, around 80%, but it has a very low false positive rate of 1%.

Now, let's consider two scenarios:

Scenario 1: MPE (Most Probable Explanation)

In this scenario, we want to find the single most likely explanation (whether the patient has Disease X) based solely on the results of the diagnostic tests.

Patient's Test A result: Positive
Patient's Test B result: Positive

Using MPE, we calculate the joint probability:

[P(Disease\ X | Test\ A\ positive \cap Test\ B\ positive) \propto P(Test\ A\ positive | Disease\ X) \cdot P(Test\ B\ positive | Disease\ X) \cdot P(Disease\ X)]

Since Test A has a relatively high sensitivity and Test B also indicates positivity, MPE may conclude that the patient likely has Disease X.

Scenario 2: MAP (Maximum A Posteriori)

In this scenario, we want to incorporate our prior belief (prior probability) about the rarity of Disease X into the inference process.

Patient's Test A result: Positive
Patient's Test B result: Positive

Using MAP, we calculate the posterior probability:

[P(Disease\ X | Test\ A\ positive \cap Test\ B\ positive) \propto P(Test\ A\ positive | Disease\ X) \cdot P(Test\ B\ positive | Disease\ X) \cdot P(Disease\ X)]

However, MAP also considers the prior probability (P(Disease\ X)), which is very low due to the rarity of the disease. Therefore, even though both tests are positive, MAP may conclude that the patient is more likely to be a false positive, given the low prior probability of having Disease X.

Result:

In this example, MPE and MAP would produce different results. MPE, which only considers the likelihood of the evidence (test results), may suggest that the patient likely has the disease. In contrast, MAP, which incorporates prior beliefs, would consider the rarity of the disease and may conclude that the patient is more likely to be a false positive, especially if the prior probability is significantly low. This illustrates how the consideration of prior information can lead to different results between MPE and MAP.