The next distribution we're going to look at is a Student's t-distribution. The definition of the generalized student's t-distribution can be divided, starting with the univariate normal distribution that is parametrized by a mean Mu and Tau inverse or the inverse of the variance. Remember that we're parametrized and a little differently here instead of using Sigma or the standard deviation, we're using the precision or Tau here. The conjugate prior to the position is given by the Gamma distribution, which is parameterized by two parameters a and b. We can compute the marginal distribution for x by using the prior of the precision and integrating the dependence of the normal distribution on it's precision over all values of precision from 0 to infinity. What we're doing is essentially saying we have a probability distribution that depends on Mu and Tau, but we want to get rid of the Tau term by integrating all possible values, and we do that as shown here. We're integrating that our Tau from 0 to infinity, and we end up with the term similar to this. The integral above has the following interpretation, a Student's t-distribution is made up of an infinite number of Gaussians with the same mean and differing variances, with the variances varying from 0 to infinity. Now, the following new parameters are defined for this t-distribution in terms of the Gamma distribution or the prior distribution for Tau. We have mu, which is defined as 2 times a, and Lamda equals a or b. Now, the PDF for the generalized Students t-distribution can be read as shown here. This is the generalized form that you're most likely to see. Understand, we have a few parameters here, Mu corresponds to the mean of the distribution, as you can see here, and Lamda corresponds to the position of the distribution, though, is generally not the same thing as the inverse of the variance of that distribution. Mu here is the degrees of freedom and takes values between 0 and infinity. The degrees of freedom corresponds to the number of independent observations minus 1. If the sample size is 8, the distribution used to model [inaudible] would have degrees of freedom set to 7. A value of 1, for degrees of freedom, corresponds to a Cauchy distribution and indicates very heavy tails, while a value of infinity corresponds to a normal distribution or the Student's t-distribution essentially becomes a normal distribution. What does the mean and variance for the distribution look like? Mean, as we already mentioned, is simply Mu. Variance now takes this term given by Mu over Mu minus 2 multiplied by Lamda. Now, if you have a 0 identity distribution, which is the simplest form of t-distribution, it basically becomes something of this one, and our mean is 0 and the variance is given by the term here. We also have an alternative interpretation for the t-distribution, which I will not go over here [inaudible] to read through this explanation. What is an example of a Student's t-distribution? A distribution of test scores for an exam, which is a significant number of outliers, and that would not be appropriate for a normal distribution, but use a Cauchy distribution with a Mu set to 1, we can use that to model extreme observations for rare events. What are some of the conditions for the Student's t-distribution tool? It takes continuous data, it can be used to model continuous data. It is essentially an unbounded distribution similar to the normal distribution, and it is often called and overdispersed normal distribution, or it's a mixture of individual normal distributions with different variances. The code to generate and plot the student's t-distribution looks something like this, it takes as input the degrees of freedom and you can vary that using the slider. The one thing you notice is that it's hard to notice the variations in the probability distribution here as you've changed them, but you can notice minor changes in the tails. A better way to maybe illustrate this is using an overlay of three different degrees of freedom. You notice that as the degrees of freedom increase, essentially this tends more towards normal distribution, and for a degree of freedom set to 1, you have large or flatter tails.
