Welcome everyone to the final module. The previous module was the most challenging and you made it through. Trust me, it's not going to get any harder from here. What we're going to do next is pull all of this together. We're going to show that the mathematical framework for uncertainty quantification that we put forward in the previous section is nothing more than a mathematical statement of the scientific method. It's only going to get better from there. In the next lessons, we're going to show how this mathematical framework fits cleanly into the research process. Let's get to it. Before we do though, let's take a minute to understand this term that I've used a few times now. Uncertainty quantification, or UQ. Uncertainty quantification is the field that I personally conduct research in. It's my job to build models for uncertainty and leverage those models in scientific investigations of various types to better understand the confidence that we have in engineering systems. But what is uncertainty quantification? Simply stated, UQ is the application of the scientific method for the mathematical treatment of uncertainty. The goal of UQ is to build scientifically validated mathematical models for uncertainty. Bayes' rule is one of the fundamental mathematical laws that we use to quantify uncertainty. When we apply Bayes' rule, we're conducting uncertainty quantification. Now, I mentioned that UQ is the application of the scientific method for the treatment of uncertainty. Let's go back and review our scientific method. Remember these four steps. First we make observations, then we make a hypothesis. Then we make more observations to test our hypothesis, and then we refine and continue. But now, if we look at Bayes' rule for uncertainty quantification, we can actually see these steps come to life in a clear mathematical form. Let's quickly review Bayes' rule. Remember that we define two events. H_1 is the event that we're trying to better understand. In other words, we're trying to learn the probability of H_1. In order to do so, we make an observation of event D. Then Bayes' rule states that the probability of H_1 given that we've observed D, is the probability of D given H_1 times the probability of H_1 divided by the total probability of D. Now remember we broke this into various terms and we're going to go through each of those terms and how they relate to the scientific method next. According to the scientific method, the first thing we do is make some observations. These observations define the problem that we're trying to solve. These observations lead us to make a hypothesis. This hypothesis can be expressed through our prior probability P of H_1. Recall that our prior probability states our current knowledge or belief about the probability of the event H_1 that we're trying to better understand. The statement of the prior is in effect a hypothesis. It's saying, given my current state of knowledge, I hypothesize that the probability of H_1 takes some value. Remember, there is some subjectivity here, but that's not inconsistent with the scientific method. Hypotheses can be subjective, but they must be rational and we must allow them to be rejected if we're wrong, which Bayes' rule will allow us to do. Next, we make some observations. These observations come in the form of the event D that we observe. Remember, Bayes' rule helps us to quantify the probability of event H_1 conditioned on observing event D. These observations serve to act on or update our prior. Another way to look at Bayes' rule is to simply rewrite it as the ratio of the likelihood over the total probability times the prior. If we rewrite it this way, we see that the first term serves to act on the prior. This is the effect that the data has on the prior. If we look closely at this term, we see that if this term is greater than one, that is, if observing D is more likely to occur, given that H_1 occurs, then it would be if we hadn't observed H_1. Then we'll increase the probability of H_1 or will strengthen our prior. On the other hand, if this term is less than one, this means that D is less likely to occur when we observe H_1, than it would be if we hadn't observed H_1. In which case, it will serve to decrease the probability of H_1 or weaken our prior. Strengthening the prior serves as evidence in favor of the hypothesis and leads to an increased posterior probability. Weakening the prior serves to act against our hypothesis and leads to a decreased posterior probability. Remember that I said that Bayes' rule allows us to reject our hypothesis. If the evidence is overwhelmingly against the hypothesis, it can drive the posterior probability to zero, thus rejecting the hypothesis. We can then use this posterior probability to refine and restart. Because we could take this posterior probability and use it as a new prior probability. Collecting additional data, applying Bayes' rule again and continuing iteratively. Let's go back and look at our example from the previous lesson using this interpretation. I first started by making some observations. These observations are simply that I have two dice. One of them has six sides and one of them has 10 sides. Next, I formulated a hypothesis. This hypothesis stated that it was equally likely that I rolled each of the dice. I expressed this hypothesis in terms of the probability of rolling the six-sided die by saying that the probability of event D_1 was 0.5. Then I made some additional observations, the third step of my scientific method. In particular, in this example, I rolled a four. I then applied Bayes' rule to refine my hypothesis. I saw that rolling a four increased the probability of my hypothesis. That is, it strengthened the hypothesis that I was rolling a six-sided die from probability 0.5 to probability 0.625. That is, given that I rolled a four, I now have more evidence that I'm rolling a six-sided die than I previously did. Alternatively, I could have stated my hypothesis in terms of the probability of rolling the 10-sided die. In that event, rolling a four would have weakened my hypothesis, reducing my prior probability from 0.5 to 0 375. In other words, observing a four is evidence that acts against my hypothesis. It tells me that my prior probability was too high and it is actually less likely than that, that I'm rolling the 10-sided die. This concludes today's lesson. On your own spend some time with this example. Change the prior probabilities, change what you rolled. Try adding several rolls, see what happens. This will help you to build your intuition. Images from these slides have been taken from the sources listed here.