Bayes’ Theorem How to interpret Bayes rule

Created	@April 23, 2022
Tags	Basic Concepts

Q6: What is Bayes’ Theorem? How is it useful in a machine learning context?

Answer: Bayes’ Theorem gives you the posterior probability of an event given what is known as prior knowledge.

Mathematically, it’s expressed as the true positive rate of a condition sample divided by the sum of the false positive rate of the population and the true positive rate of a condition. Say you had a 60% chance of actually having the flu after a flu test, but out of people who had the flu, the test will be false 50% of the time, and the overall population only has a 5% chance of having the flu. Would you actually have a 60% chance of having the flu after having a positive test?

Bayes’ Theorem says no. It says that you have a (.6 * 0.05) (True Positive Rate of a Condition Sample) / (.6*0.05)(True Positive Rate of a Condition Sample) + (.5*0.95) (False Positive Rate of a Population) = 0.0594 or 5.94% chance of getting a flu.

Bayes’ Theorem is the basis behind a branch of machine learning that most notably includes the Naive Bayes classifier. That’s something important to consider when you’re faced with machine learning interview questions.

https://www.zhihu.com/question/27670909

Bayes' Theorem is a fundamental principle in probability theory and statistics that describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It's named after Thomas Bayes, an 18th-century British statistician and philosopher. Bayes' Theorem provides a way to update the probability estimates for a hypothesis as more evidence or information becomes available.

Mathematical Formulation of Bayes' Theorem

Bayes' Theorem is mathematically expressed as:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Where:

$P(A|B)$ is the posterior probability of $A$ given $B$, i.e., the probability of event $A$ happening given that $B$ is true.

$P(B|A)$ is the likelihood, i.e., the probability of observing $B$ given $A$.

$P(A)$ is the prior probability of $A$, i.e., the initial degree of belief in $A$.

$P(B)$ is the marginal probability of $B$, i.e., the total probability of observing $B$ under all possible circumstances.

Understanding the Components

Prior ($P(A)$): This is what you initially believed before you had any specific evidence.

Likelihood ($P(B|A)$): This is the probability of observing the evidence assuming the hypothesis is true.

Marginal Probability ($P(B)$): This accounts for the total probability of observing the evidence under all hypotheses.

Posterior ($P(A|B)$): This is the updated belief after taking the evidence into account.

Example Application of Bayes' Theorem

Suppose you have a disease that affects 1% of the population ($P(A) = 0.01$), and there is a test for this disease that has a 95% chance of correctly identifying a diseased person as positive ($P(B|A) = 0.95$) but also has a 5% chance of incorrectly identifying a healthy person as positive ($P(B|\neg A) = 0.05$). If a person tests positive, what is the probability they actually have the disease?

To find out, we need to calculate $P(A|B)$, the probability that the person has the disease given that they tested positive. We also need $P(B)$, the total probability of testing positive, which can be calculated as:

\[
P(B) = P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)
\]

Given:

$P(B|A) = 0.95$

$P(A) = 0.01$

$ $P(B|\neg A$ ) = 0.05$

$ $P(\neg A$ ) = 0.99$

The calculation proceeds as follows:

$P(B) = 0.95 \times 0.01 + 0.05 \times 0.99$

$P(A|B) = \frac{0.95 \times 0.01}{P(B)}$

Let's compute this:

P_A = 0.01
P_B_given_A = 0.95
P_B_given_not_A = 0.05
P_not_A = 0.99

P_B = P_B_given_A * P_A + P_B_given_not_A * P_not_A
P_A_given_B = (P_B_given_A * P_A) / P_B
P_A_given_B

This calculation will give us the posterior probability, which is the updated probability of having the disease after considering the test result.

Importance of Bayes' Theorem

Bayes' Theorem is crucial across various fields, including:

Statistics and Data Analysis: For updating the probability estimates as new data becomes available.

Machine Learning: In Bayesian inference, to update the model's parameters' probabilities as more data is observed.

Medical Field: To determine the probability of a disease given the test results.

Decision Making: In business and economics, to make informed decisions under uncertainty by updating the probabilities with new evidence.

Bayes' Theorem elegantly combines prior knowledge with new evidence to make statistical inferences more grounded and informed.

Example: Disease Diagnosis

Suppose we're dealing with a disease that affects 1% of a population. A test for the disease is 99% accurate, meaning:

If you have the disease, the test will correctly identify you as positive 99% of the time.

If you don't have the disease, the test will correctly identify you as negative 99% of the time.

Now, if a person is tested and receives a positive result, what is the probability that they actually have the disease?

Let's define our events:

$A$: Having the disease.

$B$: Testing positive for the disease.

We know that:

$P(A) = 0.01$ (1% of the population has the disease)

$P(A') = 0.99$ (99% of the population does not have the disease)

$P(B|A) = 0.99$ (If you have the disease, there's a 99% chance the test is positive)

$P(B|A') = 0.01$ (If you don't have the disease, there's a 1% chance the test is still positive)

We want to find $P(A|B)$, the probability of having the disease given that you've tested positive.

First, we need $P(B)$, the total probability of testing positive, which can be found using the law of total probability:

$P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')$

Let's calculate this:

$P(B) = 0.99 \cdot 0.01 + 0.01 \cdot 0.99 = 0.0099 + 0.0099 = 0.0198$

Now, we can apply Bayes' Theorem to find $P(A|B)$:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.99 \cdot 0.01}{0.0198} = \frac{0.0099}{0.0198}$

Let's calculate the final probability:

P(A|B) = 0.5

So, despite the high accuracy of the test, if an individual tests positive, there's still only a 50% chance that they actually have the disease. This counterintuitive result highlights the importance of considering the base rate (prior probability) when interpreting test results.