Bayes' Theorem and Conditional Probability

Understanding how beliefs are updated in the light of new evidence is a central idea in Bayesian statistics. This process is formalized by Bayes' theorem, which is built upon the concept of conditional probability. To see how Bayes' theorem arises, start with the definition of conditional probability: the probability of event A given event B is written as $P(A|B)$ and is defined as the probability of both A and B happening, divided by the probability of B:

P(A|B) = P(A \cap B) / P(B)

Similarly, the probability of B given A is:

P(B|A) = P(A \cap B) / P(A)

Both $P(A|B) \times P(B)$ and $P(B|A) \times P(A)$ are equal to $P(A \cap B)$ . Setting these equal gives:

P(A|B) \times P(B) = P(B|A) \times P(A)

Solving for $P(A|B)$ yields Bayes' theorem:

P(A|B) = [P(B|A) \times P(A)] / P(B)

Each term in Bayes' theorem has a specific interpretation:

$P(A|B)$ is the posterior probability: your updated belief about A after observing B;
$P(A)$ is the prior probability: your belief about A before seeing B;
$P(B|A)$ is the likelihood: how probable B is, assuming A is true;
$P(B)$ is the marginal probability: the overall probability of observing B under all possible scenarios.

This formula allows you to update your beliefs (the prior) in light of new evidence (the likelihood) to obtain a revised belief (the posterior).

Medical Testing

Imagine testing for a rare disease. Even if a test is highly accurate, the probability that a person actually has the disease given a positive test result (posterior) depends on the disease's overall rarity (prior) and the test's accuracy (likelihood). Bayes' theorem helps calculate the true chance a patient is sick after a positive result.

Spam Filtering

Email spam filters use Bayes' theorem to estimate the probability that a message is spam, given the presence of certain words. The prior is the base rate of spam, the likelihood is how often the word appears in spam versus non-spam, and the posterior is the updated probability that the email is spam.

Fault Diagnosis in Engineering

When a machine shows an error signal, engineers use Bayes' theorem to update the probability of different faults being the cause, based on prior failure rates and the likelihood of the signal given each fault.

Legal Reasoning

In court, the probability of guilt (posterior) can be updated as new evidence is presented, considering both the prior odds and the likelihood of the evidence under guilt or innocence.

Weather Forecasting

Meteorologists update the probability of rain (posterior) as new data (such as satellite images) arrives, factoring in prior climate data and the likelihood of observing the data if rain is imminent.

1. Which of the following best describes the "likelihood" term in Bayes' theorem?

2. If P(A|B) is the probability of A given B, and P(B|A) is the probability of B given A, which of the following is true in the context of Bayes' theorem?

3. When you observe new evidence B, how does it affect your belief about hypothesis A in Bayesian inference?

Which of the following best describes the "likelihood" term in Bayes' theorem?

Select the correct answer

The probability of the hypothesis after considering the evidence

The overall probability of the evidence

The probability of the evidence assuming the hypothesis is true

The initial belief about the hypothesis before seeing any evidence

If P(A|B) is the probability of A given B, and P(B|A) is the probability of B given A, which of the following is true in the context of Bayes' theorem?

Select the correct answer

$P(A|B)$ is always greater than $P(B|A)$ .

$P(A|B)$ and $P(B|A)$ are generally different, and Bayes' theorem relates them using the prior and marginal probabilities.

$P(B|A)$ is always greater than $P(A|B)$ .

$P(A|B)$ and $P(B|A)$ are always equal.

When you observe new evidence B, how does it affect your belief about hypothesis A in Bayesian inference?

Select the correct answer

It replaces your prior belief about A with the likelihood of B.

It only affects the marginal probability of B, not your belief about A.

It updates your prior belief about A to form a posterior belief, using the likelihood of B given A.

It has no effect on your belief about A.

Tutto è chiaro?

Come possiamo migliorarlo?

Grazie per i tuoi commenti!

Sezione 1. Capitolo 2

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Can you give an example of how Bayes' theorem is used in practice?

What is the difference between prior and posterior probability?

How do you calculate the marginal probability $$P(B)$$ in Bayes' theorem?

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Bayes' Theorem and Conditional Probability

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P(A|B) = P(A \cap B) / P(B)

Similarly, the probability of B given A is:

P(B|A) = P(A \cap B) / P(A)

Both $P(A|B) \times P(B)$ and $P(B|A) \times P(A)$ are equal to $P(A \cap B)$ . Setting these equal gives:

P(A|B) \times P(B) = P(B|A) \times P(A)

Solving for $P(A|B)$ yields Bayes' theorem:

P(A|B) = [P(B|A) \times P(A)] / P(B)

Each term in Bayes' theorem has a specific interpretation:

$P(A|B)$ is the posterior probability: your updated belief about A after observing B;
$P(A)$ is the prior probability: your belief about A before seeing B;
$P(B|A)$ is the likelihood: how probable B is, assuming A is true;
$P(B)$ is the marginal probability: the overall probability of observing B under all possible scenarios.

This formula allows you to update your beliefs (the prior) in light of new evidence (the likelihood) to obtain a revised belief (the posterior).

Medical Testing

Spam Filtering

Fault Diagnosis in Engineering

Legal Reasoning

In court, the probability of guilt (posterior) can be updated as new evidence is presented, considering both the prior odds and the likelihood of the evidence under guilt or innocence.

Weather Forecasting

1. Which of the following best describes the "likelihood" term in Bayes' theorem?

2. If P(A|B) is the probability of A given B, and P(B|A) is the probability of B given A, which of the following is true in the context of Bayes' theorem?

3. When you observe new evidence B, how does it affect your belief about hypothesis A in Bayesian inference?

Which of the following best describes the "likelihood" term in Bayes' theorem?

Select the correct answer

The probability of the hypothesis after considering the evidence

The overall probability of the evidence

The probability of the evidence assuming the hypothesis is true

The initial belief about the hypothesis before seeing any evidence

If P(A|B) is the probability of A given B, and P(B|A) is the probability of B given A, which of the following is true in the context of Bayes' theorem?

Select the correct answer

$P(A|B)$ is always greater than $P(B|A)$ .

$P(A|B)$ and $P(B|A)$ are generally different, and Bayes' theorem relates them using the prior and marginal probabilities.

$P(B|A)$ is always greater than $P(A|B)$ .

$P(A|B)$ and $P(B|A)$ are always equal.

When you observe new evidence B, how does it affect your belief about hypothesis A in Bayesian inference?

Select the correct answer

It replaces your prior belief about A with the likelihood of B.

It only affects the marginal probability of B, not your belief about A.

It updates your prior belief about A to form a posterior belief, using the likelihood of B given A.

It has no effect on your belief about A.

Tutto è chiaro?

Come possiamo migliorarlo?

Grazie per i tuoi commenti!

Sezione 1. Capitolo 2