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Bayes Theorem, or Bayes Rule, is one of the most important concepts used in Bayesian estimation or

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probabilistic estimation, Bayes Theorem allows you to calculate the likelihood or bounds of an unknown

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parameter or event based on some prior information related to that event.

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And this process is called Bayesian inference.

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Bayes theorem is derived from conditional probability.

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So from conditional probability, we can work out the conditional probability of A given B as the joint

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probabilities of A and B over the marginal probability.

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We can also write this equation in the opposite way so we can work out the conditional probability of

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being given a it's just going to be the reverse.

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And now from probability, we know that the joint probability, so the probability of A and B and the

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probability of B and A, they're going to be equal the same thing.

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So joint probabilities are equivalent.

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So then we can rearrange this equation and substitute them into each other.

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We end up with Bayes Theorem.

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So Bayes Theorem, a rule is now displayed on the slide.

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We can see that this first term here, this is a conditional probability and it is the likelihood of

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event A, given the event B occurs the next time over here is another conditional probability.

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It is the likelihood of event B given the event A has occurred.

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The next term is the marginal probability is just the likelihood of event A occurring.

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And this last time the divisor is another marginal probability and it's the likelihood of Event B occurring.

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Now when we talk about these different conditional and marginal probabilities, we give them special

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names.

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When we're talking about Bayes Theorem, first time here.

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Sometimes we can call that the posterior probability.

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So this is the probability, given this information over here, this term here is called the likelihood.

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This next term here is called the primary probability.

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And this term here is called the evidence.

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Now, we'll go over what these terms actually mean in the next sample here.

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So let's look at an example.

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So we might be asked, what is the likelihood of it raining today?

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Well, we can use Bayes Theorem and a bit of information to work this out.

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What we want to do is we can want to use some prior information.

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So if we know that it has rained in the last 18 days and 30 days, well, then we can work out the probability

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of it raining today so we get probability of rain being point six.

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So this is the prior information.

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Now, if we just use this information, we can get one estimate of the likelihood of rain is going to

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be today, but we can include some other information.

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So if we know some other information, we can come up with a better estimate of the likelihood of rain.

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So this is called the prior information.

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This is this term up here.

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Next, we can look at evidence so we can look out the window and we can come to some evaluation and

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some judgment.

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And we're going to say it is a forty eight percent chance of clouds in the morning today.

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So now we can look at including some other information and we can include the likelihood.

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So that's this term up here.

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And in this case, it's going to be the what's the likelihood that is going to be cloudy, given that

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it rains so we can use some prior information so we can look at on the days that is range rain, we

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can sort of say that 68 percent of the time it was also cloudy in the morning.

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So this is its point, six, eight.

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So we can include all these term into using Bayes Theorem and we can come up with the posterior probability,

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what is a probability that it's going to rain today, given that it might be cloudy in the morning?

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So evaluating this equation here, we can work out that there's a point eighty five or eighty five percent

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chance that there's actually going to rain today.

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So you can see here this probability is a lot has included a lot more information than just this probability

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of rain over here.

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So using this information, we can say the probability that it's going to rain today is not point six

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is point eight five.

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So in that case, we better bring an umbrella.

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So this is an example of how we can use Bayes Theorem to include more information on prior information

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into the estimate to come up with an even better estimate of some probability.