In this lecture sequence, we will do a lot with normal random variables. And for this reason, it is useful to start with a simple observation that will allow us later on to move much faster. Recall that a normal random variable with a certain mean and variance has a PDF of this particular form. …
In the previous lecture, we went through examples of inference involving some of the variations of the Bayes rule. One case that we did not consider was the case where the unknown random variable and the observation are both continuous. In this lecture, we will focus exclusively on an important model of this kind. And …
In the context of the problem of estimating the unknown bias of a coin, we encountered this distribution, which is known as a Beta distribution. It’s a probability density for a random variable, Theta, that takes values in the interval from 0 to 1. So this formula is valid for thetas in this range. And …
In this lecture sequence, we introduced quite a few new concepts and went through a fair number of examples. So for this reason, it is useful now to just take stock and summarize the key ideas and concepts. The starting point in a Bayesian inference problem is the following. There’s an unknown parameter, Theta, and …
We will now continue with the problem of inferring the unknown bias of a certain coin for which we have a certain prior distribution and of which we observe the number of heads in n independent coin tosses. We have already seen that if we assume a uniform prior, the posterior takes this particular form, …
Probability – L14.9 Inferring the Unknown Bias of a Coin – Point Estimates Read More »
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 »