In this segment, we introduce a model with multiple parameters and multiple observations. It is a model that appears in countless real-world applications. But instead of giving you a general and abstract model, we will talk about a specific context that has all the elements of the general model, but it has the advantage of …
Probability – L15.6 Multiple Parameters; Trajectory Estimation Read More »
We now continue our discussion of the model in which we obtain several measurements of an unknown random variable Theta in the presence of additive noise, under the same assumptions as before. Theta and Wi are all independent random variables. And they’re also normal. We have seen that in this case, the posterior distribution of …
We are now ready to move on to a model which is quite interesting and quite realistic. This is a model in which we have an unknown parameter modeled as a random variable that we try to estimate. This is the random variable, Theta. And we have multiple observations of that random variable. Each one …
Probability – L15.4 The Case of Multiple Observations Read More »
Introduction As a preparation for more complex and more difficult models, we will start by looking at the simplest model that there is, that involves a linear relation and normal random variables. Model The specifics of the model are as follows. There’s an unknown parameter modeled as a random variable, Theta, that we wish to …
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 »