We will now go through an example that involves a continuous unknown parameter, the unknown bias of a coin and discrete observations, namely, the number of heads that are observed in a sequence of coin flips. This is an example that we will start in some detail now, and we will also revisit later on. …
Probability – L14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution Read More »
In the next variation we consider, all random variables are continuous. For this case, we do have a Bayes rule, once more. And we have worked [out] quite a few examples. So there’s no point, again, in going through a detailed example. Let us just discuss some of the issues. One question is when do …
Probability – L14.7 Continuous Parameter, Continuous Observation Read More »
In the next variation that we consider, the random variable Theta is still discrete. So it might, for example, represent a number of alternative hypothesis. But now our observation is continuous. Of course, we do have a variation of the Bayes rule that’s applicable to this situation. The only difference from the previous version of …
Probability – L14.6 Discrete Parameter, Continuous Observation Read More »
Let us now discuss in some more detail what it takes to carry out Bayesian inference, when both random variables are discrete. The unknown parameter, Theta, is a random variable that takes values in the discrete set. And we can think of these values as alternative hypotheses. In this case, we know how to do …
Probability – L14.5 Discrete Parameter, Discrete Observation Read More »
We can finally go ahead and introduce the basic elements of the Bayesian inference framework. There is an unknown quantity, which we treat as a random variable, and this is what’s special and why we call this the Bayesian inference framework. This is in contrast to other frameworks in which the unknown quantity theta is …
Probability – L14.4 The Bayesian Inference Framework Read More »
Before we dive into the heart of the subject, I want to make a few comments on the different problem types that show up in the field of inference. You can think of a general distinction between model building versus making inferences about unobserved variables. We said a little earlier that one of the main …
Before we get going with our discussion of inference methods, it is worth looking at the big picture for some perspective. So far, we have concentrated on ways to analyze probability models. This part of the picture. If our model has been selected in a careful way, then it should also be relevant to the …
Probability – L4.2 Overview of Some Application Domains Read More »
In this lecture, we start our systematic study of Bayesian inference. We will first talk a little bit about the big picture, about inference in general, the huge range of possible applications, and the different types of problems that one may encounter. For example, we have hypothesis testing problems in which we are trying to …
In this segment, we point out and discuss some important but also intuitive properties of conditional expectations. The first property is the one that is written up here. What is the intuitive meaning? If you condition on Y, then the value of Y is known, and so g of Y is also known. There’s no …
Probability – S13.1 Conditional Expectation Properties Read More »
We now continue the study of the sum of a random number of independent random variables. We already figured out what is the expected value of this sum, and we found a fairly simple answer. When it comes to the variance, however, it’s pretty hard to guess what the answer will be, and it turns …
Probability – L13.11 Variance of the Sum of a Random Number of Random Variables Read More »