Bayes' rule
Bayes' Rule helps us figure out the probability of something happening based on certain information or evidence we have.
Imagine you have a bag with some green and some blue marbles inside. You want to know the probability of picking a green marble from the bag.
Bayes' Rule says that you need to take into account two things: the prior probability and the evidence. The prior probability is what you know about the situation before you get any evidence. In this case, the prior probability would be the ratio of green marbles to blue marbles in the bag. Let's say there are 3 green marbles and 7 blue marbles, so the prior probability would be 3/10, or 0.3.
Now, let's say you shake the bag and you hear a different sound coming from the marbles. You reach in and pull out a marble, and it happens to be green. This is the evidence, and it gives you more information that you can use to update your probability.
Bayes' Rule says that you need to multiply the prior probability by the likelihood of the evidence, and then divide by the sum of all possible outcomes. The likelihood of the evidence is how likely it is to observe that evidence if the hypothesis (in this case, the bag has more green marbles than blue marbles) is true. In this case, the likelihood of the evidence would be the probability of picking a green marble given that there are more green marbles than blue marbles in the bag. This is simply the number of green marbles divided by the total number of marbles, or 3/10.
The denominator (the sum of all possible outcomes) is just a normalization factor that ensures the probabilities add up to 1. In this case, it would be the probability of picking a green marble plus the probability of picking a blue marble: 3/10 + 7/10 = 1.
So, using Bayes' Rule, we have:
P(green | evidence) = P(evidence | green) * P(green) / [P(evidence | green) * P(green) + P(evidence | blue) * P(blue)]
where P(green | evidence) is the probability of picking a green marble given the evidence, P(evidence | green) is the likelihood of the evidence (picking a green marble) given that there are more green marbles than blue marbles in the bag, P(green) is the prior probability of picking a green marble, P(blue) is the prior probability of picking a blue marble, and P(evidence | blue) is the likelihood of the evidence (picking a green marble) given that there are more blue marbles than green marbles in the bag.
As you can see, Bayes' Rule helps us update our prior beliefs (in this case, the prior probability of picking a green marble) based on new evidence. This can be applied to many different situations, such as determining the probability of a medical test being accurate given a person's symptoms, or the probability of a stock going up based on news about the company.
Imagine you have a bag with some green and some blue marbles inside. You want to know the probability of picking a green marble from the bag.
Bayes' Rule says that you need to take into account two things: the prior probability and the evidence. The prior probability is what you know about the situation before you get any evidence. In this case, the prior probability would be the ratio of green marbles to blue marbles in the bag. Let's say there are 3 green marbles and 7 blue marbles, so the prior probability would be 3/10, or 0.3.
Now, let's say you shake the bag and you hear a different sound coming from the marbles. You reach in and pull out a marble, and it happens to be green. This is the evidence, and it gives you more information that you can use to update your probability.
Bayes' Rule says that you need to multiply the prior probability by the likelihood of the evidence, and then divide by the sum of all possible outcomes. The likelihood of the evidence is how likely it is to observe that evidence if the hypothesis (in this case, the bag has more green marbles than blue marbles) is true. In this case, the likelihood of the evidence would be the probability of picking a green marble given that there are more green marbles than blue marbles in the bag. This is simply the number of green marbles divided by the total number of marbles, or 3/10.
The denominator (the sum of all possible outcomes) is just a normalization factor that ensures the probabilities add up to 1. In this case, it would be the probability of picking a green marble plus the probability of picking a blue marble: 3/10 + 7/10 = 1.
So, using Bayes' Rule, we have:
P(green | evidence) = P(evidence | green) * P(green) / [P(evidence | green) * P(green) + P(evidence | blue) * P(blue)]
where P(green | evidence) is the probability of picking a green marble given the evidence, P(evidence | green) is the likelihood of the evidence (picking a green marble) given that there are more green marbles than blue marbles in the bag, P(green) is the prior probability of picking a green marble, P(blue) is the prior probability of picking a blue marble, and P(evidence | blue) is the likelihood of the evidence (picking a green marble) given that there are more blue marbles than green marbles in the bag.
As you can see, Bayes' Rule helps us update our prior beliefs (in this case, the prior probability of picking a green marble) based on new evidence. This can be applied to many different situations, such as determining the probability of a medical test being accurate given a person's symptoms, or the probability of a stock going up based on news about the company.
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