Bayes’ Theorem

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Bayes’ Theorem

Bayes’ Theorem helps us figure out how likely something is, given new information. It tells us how to update our beliefs when we get new evidence.

  • Probability: This is just a measure of how likely something is to happen. It’s a number between 0 and 1, where:
    • 0: Impossible.
    • 1: Certain.
  • Hypothesis: This is something you’re trying to figure out the likelihood of. For example, maybe you’re trying to figure out whether it’s going to rain today.
  • Evidence: This is new information that might help you figure out the probability of the hypothesis being true. For example, you might see dark clouds in the sky, which could be evidence that it’s going to rain.

The Bayes Formula

Bayes’ Theorem looks like this:

 P(A|B) = P(B|A) * P(A) / P(B)

Where:

  • P(A|B): The posterior probability – the updated probability of A being true, after you have seen the evidence B.
  • P(B|A): The likelihood – the probability of seeing the evidence B, given that A is true.
  • P(A): The prior probability – your initial belief about A before seeing the evidence.
  • P(B): The marginal probability – the total probability of seeing the evidence B, no matter what the situation is.

Example

Let’s say you’re trying to figure out whether someone is carrying an umbrella (let’s call this the hypothesis, or A).

  • You know that on rainy days, people often carry umbrellas (the likelihood, P(B|A)).
  • You also know that there’s a 30% chance of rain today (the prior probability, P(A)).
  • Finally, you know that 50% of people with umbrellas carry them on any given day (the probability of seeing someone with an umbrella, P(B)).

Bayes’ Theorem lets you take all this information and calculate the updated probability that someone is carrying an umbrella given that it is raining.

Bayes’ Theorem helps us adjust our beliefs about something based on new information. So, it’s a great way to make decisions in situations where things are uncertain. For example, in medicine, it helps doctors update the probability that a patient has a certain disease, based on new test results.

Bayes’ Theorem is like a tool that helps you adjust your expectations (or beliefs) when you get new evidence. It’s all about updating your “guess” based on how probable something is, given what you already know.