Let's take a look at plotting the prior and the posterior. So plotting the prior density. For theta, it looks like this. It's uniform on zero, one. The posterior density. Is a little bit different. We saw that is now, two theta. And so that's looks like this. So graphically, we can see that under the posterior, theta is much more likely to be close to one than to zero because we saw one head. We can look at posterior and prior interval estimates. Under the prior, the probability that theta is between .025 and .975 is .95. So 95% prior probability interval. We can also say probability theta's greater than .05, it's 0.95. Graphically, we can think about that as thinking about integrating regions under the density. But because it's uniformed, it's really simple to compute. We can then think about posterior interval estimates. Under the posterior density we can ask what's the probability the data is between 0.025 and 0.975. This case we have a little integral to do. So we integrate from 0.025 to 0.975, the density of 2 theta d theta. The integral of 2 theta is going to be theta squared. And so this works out to be .975 squared minus .025 squared and this happens to be .95. And so it turns out in this case to be the same as under the prior. You can look at the picture and think about just moving some of the pieces around. If you're chopping off both ends, we still end up with the same amount in the middle. However if we ask the second question under the posterior, what's the property fit that is bigger than point 05. This is now, when we run through the integral, 1 minus .05 squared, and this turns out to be .9975. There's very little chance now that theta is less than .05, having observed ahead. This is a big change now from the prior to the posterior. Now these we're looking at, sort of intervals we would get from the prior and saying what's their posterior probability. In other cases, we may want to ask, what is the posterior interval of interest? What's a interval that contains 95% of posterior probability in some meaningful way? This would be equivalent then to a frequent disconfidence interval in a concept. We can do this in several different ways, 2 million ways that we make Bayesian Posterior Intervals or credible intervals are equal-tailed intervals and highest posterior density intervals. In the case of an equal-tailed interval, we put the equal amount of probability in each tail. So to make a 95% interval we'll put .025 in each tail. To be able to do this, we're going to have to figure out what the quantiles are. So we're going to need some value, q, say the probability Theta is less than q in posterior, given Y equals 1. Here we can get this by integrating from 0 to q to theta, d theta. And again we'll see that's just q squared. So we can say an interval that goes from the square root of .025 to the square root of .975, I'm plugging in those numbers It's going to be .158, 2.987, under the posterior that will have probability .95. This is an equal tailed interval in that the probability that Theta's less than .158. Is the same as the probability that Theta is greater than .987. And we can say under the posterior, there's a 95% probability that Theta is in this interval. Another approach is what we refer to as Highest Posterior Density or HPD. Here we want to ask where in the density is it highest? And so in this case we want to say, well, about here are the highest levels. And so this interval's going to go between these two points. Theoretically this will the shortest possible interval that contains the given probability, in this case a 95% probability. We can see here that it's going to go all the way up to 1 just from the graph. So we're asking, what's the probability Theta is bigger than some value. And looking at the quantiles again, that value's going to be the square root of .05. I should write that this is a posterior. And so we can say the probability Theta is greater than .224 is .95. So this is the shortest possible interval that under the posterior has probability .95 it's Theta going from .224 up to 1. Let's think about what we have here. The posterior distribution describes our understanding of our uncertainty combining our prior beliefs and the data. With a probability density function, so at the end of the day, we can make intervals and talk about probabilities of data being in the interval. This is a much more satisfying approach, than the frequentest approach, where we get confidence intervals. But we can't say a whole lot about the actual parameter relative to this confidence interval. We can only make long run frequency statements about a lot of hypothetical intervals. In this case, we can legitimately say, the posterior probability Theta is bigger than .05, is .9975. Or, we believe there's a 95% probability the Theta is in between .158 and .987. Frequentest can't do this. Bayesians represent uncertainty with probabilities. So the coin itself is a physical quantity. It may have a particular value for Theta. It may be fixed, but because we don't know what that value is, we represent our uncertainty about that value with a distribution. And at the end of the day, we can represent our uncertainty, collect it with the data, and get a posterior distribution and make intuitive statements.