Now let's take a look at the Bayesian approach. An advantage of the Bayesian approach is that it allows you to easily incorporate prior information, when you know something in advance of the looking at the data. This is difficult to do under the Frequentist paradigm. In this case, we're talking about your brother. You probably know him pretty well. Because you've been around him a bunch, and this isn't the first time he's come to you with a coin, you may have some information on this that you want to incorporate into the problem. So suppose you think that before you've looked at the coin, there's a 60% probability that this is the loaded coin. This case, we put this into our prior, our prior is that the probability the coin is loaded is.0.6. We can now update our prior with the data to get our posterior beliefs, and we do this using base theorem. F of theta given x, is going to be f of x given theta, times f of theta, over all possibilities for theta of f of x given theta, f of theta. In this case, the only two possibilities for theta are that it's fair or loaded. So this likelihood is the same as what we saw before, but this time we'll be adding in the prior. One half to the fifth, times the prior probability that the coin is fair. Plus 0.7 to the x, 0.3 to the 5- x, times the prior probability it's loaded. If we plug in our observed data, x equals 2, this will work out to be 0.0125 that it equals fair, plus 0.0079. Theta equals loaded over 0.0125 plus 0.0079. Simplifying this, we get 0.612 theta equals fair, plus 0.388 Theta equals loaded. As you can see in the calculation here, we have the likelihood times the prior in the numerator, and in the denominator, we have a normalizing constant, so that when we divide by this, we'll get answers that add up to one. These numbers match exactly in this case, because it's a very simple problem. But this is the concept that goes on, what's in the denominator here is always a normalizing constant, so that in the end of the day, we get probabilities. And so we can say at the end of the day here, the probability theta equals loaded, given x equals 2. So our posterior probability that this is the loaded coin works out to be 0.388. Isn't that a much more satisfying answer, under the Bayesian approach, where we get a probability, and we can interpret this probability. We can say, what is the probability that we think this coin is loaded after we've observed some data? This is a much more intuitive answer than we get under the Frequentist paradigm. We can also examine what would happen under different choices of prior. We did this calculation with the prior probability of 0.6 for the coin being loaded, but we might have a different idea, we want to use a different probability. We can use anything between zero and one. Or maybe somebody else is coming in with a different perspective, and they want to see what their answer is, using a different prior. One possibility is that we could say the probability of the coin is loaded, and prior is one half. This might represent an attempt to say, we're not sure what's going on, it's equally likely that it's the loaded coin or the fair coin, not that these are necessarily equally likely outcomes. Taking a prior probability of one half, we can put it through Bayesian machinery, and that will lead us to get a posterior probability, theta equals loaded, given we observe two data points. That works out to be 0.297. We might have a different idea. We might think, because this is our brother and we know him well, that the prior probability it's going to be the loaded coin is 0.9. He really likes to use the loaded coin. If this is the case, then when we go through the calculations, we'll get a posterior probability that this is a loaded coin of 0.792. In this case, the Bayesian approach is inherently subjective. It represents your own personal perspective, and this is an important part of the paradigm. If you have a different perspective, you will get different answers, and that's okay. It's all done in a mathematically vigorous framework, and it's all mathematically consistent and coherent. In the end, we get results that are interpretable. It makes sense, we want to ask, what is the probability the coin is loaded after we've looked at some data, incorporating our beliefs about our past experiences with our brother? This way we get intuitive answers, we can say the probability it's loaded is 0.3, or the probability it's loaded is 0.8. Under the Frequentist paradigm, we can't get a good answer for what is the probability the coin is loaded. The machinery just is not set up that way. It's not possible to get a good confidence interval. What would it even mean to have a confidence interval for, is the coin loaded? The Frequentist approach has a number of buried assumptions. We need to make a choice, which is inherently subjective about what is our reference population? About what is our likelihood? These are assumptions that are inherently subjective, that are buried deeper into the Frequentist paradigm, and so there's an attempt to pretend that everything is objective. Under the Bayesian paradigm, we're explicit that this is a subjective and personal approach. But we can also be explicit about what all of our assumptions are, and then see how our answers depend on our assumptions.