Hello again, as we continue now into module four, we've started to build some momentum. At this point, we're going to start getting more technical. Don't worry, we won't be doing anything overly intense and we'll keep the mathematics simple, but things will definitely get harder. In this module, we're going to introduce you to the concepts of uncertainty and how we model uncertainty using probability theory. This module serves as an essential primer for the final module where we can take these concepts and merge them into the scientific method to see how we can devise a rigorous research process founded on the principles of uncertainty quantification. So without further ado, let's begin with the basics. Here, we just want to introduce you to the concept of uncertainty. I'm sure you probably already feel like you have a grasp on uncertainty or that you're pretty certain you know what uncertainty is, but we're going to formalize things a bit. Now, we could start by simply going to the dictionary and looking up the definition of uncertainty. And here you would get several different answers that vary from being vague, imprecise, or indefinite to being variable, ambiguous, or unknown. But when we do this, it doesn't feel like we're really understanding uncertainty. We seem to just be putting new words to it, all of this might seem to make us more uncertain. So to help us gain a deeper understanding, we're going to begin by defining two different types of uncertainty, what we call aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty is uncertainty that's due to randomness. There are many instances where events in our world occur randomly. A simple example of this would be the roll of a dice. But they can be much more complicated, such as the wind forces that act on a long span suspension bridge due to turbulence in the atmosphere. An important characteristic of aleatory uncertainty is that because it is random, it cannot be reduced. That is, no matter how much data we collect, we cannot reduce or eliminate the uncertainty. No matter how many times I roll the dice, the next roll will always remain random. Aleatory uncertainty is inherently probabilistic, that means that it's mathematically characterized using probability theory. This type of uncertainty may also be referred to as statistical uncertainty due to its randomness, or irreducible uncertainty given its nature. The other form of uncertainty that we'll discuss here is called epistemic uncertainty. Epistemic uncertainty has a fundamentally different nature. It's uncertainty that results from incomplete information or a lack of knowledge or data. Now you can clearly see why this is relevant here as our research process and the scientific method are designed to improve our knowledge and hence reduce epistemic uncertainty. Epistemic uncertainty therefore deals with things that are knowable, but at present are not known because we don't have the necessary information or the data to fill in the blanks. Now because epistemic uncertainty deals with things that are knowable, it is reducible, that is, we can minimize or even eliminate this uncertainty entirely by collecting more data or making additional observations. But unlike aleatory uncertainty, which deals with observations that are inherently random and therefore is fundamentally linked with probability theory. There's actually no scientific consensus on the correct way to model epistemic uncertainty mathematically. This is an important fact in light of all that we've talked about. The scientific community at large remains tentative about how this uncertainty is best treated. That said, there are several different mathematical treatments that have been proposed. These include Bayesian probability theory, imprecise probabilities, so called fuzzy logic, evidence theory, among others. There are arguments for and against each of these methods, and we should recognize that no method has been universally accepted. Here, we're going to work within the context of Bayesian probability theory because as we'll see, the Bayesian perspective on probabilities is consistent with the scientific method and fits naturally within the research process framework. But we would be remiss if we didn't recognize that there are some criticisms of the approach. These criticisms stem largely from the fact that Bayesian probability theory introduces some subjectivity. And this subjectivity can introduce biases and other challenges, we'll therefore do our best to address these challenges as they arise, that concludes this lesson. In the remainder of this module, we'll continue to set the table of Bayesian probability theory so that we can apply it later.