Updating beliefs based on new evidence. How?

Bayes' Theorem is a way to update our beliefs about something based on new evidence. Here's how it works in simple terms:

Imagine you have a belief or a guess about how likely something is to happen. This is your prior probability.

Now, let's say you get some new information or evidence. Bayes' Theorem helps you figure out how this new evidence should change your original guess. Here’s the process:

1. Prior Probability: This is what you think the chances are before you see any new evidence.
2. Likelihood: This is how likely it is to see this new evidence if your initial guess were correct.
3. Evidence: This is the probability of seeing this evidence under any circumstances, not just if your guess is correct.
4. Posterior Probability: After considering the new evidence, this is your updated belief or guess.

The formula looks like this:

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

• P(A|B) is the probability of A given that B has occurred (your updated belief).
• P(B|A) is the probability of B given that A is true (likelihood).
• P(A) is the initial probability of A (prior probability).
• P(B) is the total probability of B occurring (evidence).

Here’s a practical example:

• Suppose you're trying to guess if it will rain tomorrow. Your prior probability might be 30% based on the weather forecast.
• Then, you see dark clouds in the sky (new evidence).
• You know from past experience that when it rains, there's an 80% chance of seeing dark clouds (P(B|A)), but dark clouds can appear even if it doesn't rain, say 20% of the time (P(B)).
• Using Bayes' Theorem, you calculate how this new evidence changes your prediction about rain:
  
  
  • P(A|B) = (0.80 * 0.30) / 0.20 = 1.2 (but since probabilities can't exceed 1, this indicates you've significantly increased your belief in rain - you'd need to normalize this with all possible outcomes).

So, your updated chance of rain tomorrow, considering the dark clouds, would be much higher than 30%.

In essence, Bayes' Theorem is about refining your predictions with new data, making your guesses smarter and more accurate over time.