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Hello everyone, let me introduce myself. My name is Damien Scalia, I am a professor at the Université Libre de Bruxelles, I specialize in international criminal law and international crimes. I would first like to thank Amah and Lilianne, who kindly invited me to participate in your seminar. Today we are going to make a rather theoretical and short presentation of the issues at stake in this matter mainly concerning... This is why I was called or why I was invited here: it is mainly concerning the crimes of the French army or the crimes France committed in Algeria. We notice that these crimes are of two kinds. First, war crimes or war crimes against humanity, we will see them, since the designation is important, committed before the decolonization, before 1962. And also crimes committed after 1962, since we are also going to look today at the French nuclear tests committed in the Sahara which started before 1962, but ended after 1962. I will come back to this in a few mom...

In this segment, we introduce the subject of least mean squares estimation. But as a warm-up, we will start with a very simple special case. Suppose that we have some random variable that we wish to estimate and that we have the probability distribution of this random variable-- a probability mass function if it's discrete or a probability density function if it's continuous. As a concrete instance, suppose that our random variable is uniformly distributed over a certain interval. We would like to estimate this random variable. We're interested in a point estimate. However, we look at the very special case where there are no observations available. All that we have is this probability distribution. How do we estimate this random variable? Well, we can use the rules that we have already developed, for example, the maximum a posteriori probability rule, what would it do? In this case, since there are no observations, the posterior distribution of Theta is the sa...

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 being concrete. And we can also visualize the results. The model is as follows. Somebody is holding a ball and throws it upwards. This ball is going to follow a certain trajectory. What kind of trajectory is it? According to Newton's laws, we know that it's going to be described by a quadratic function of time. So here's a plot of such a quadratic function, where this is the time axis. And this variable here, x, is the vertical displacement of the ball. The ball initially is it a certain location, at a certain height-- theta 0. It is thrown upwards. And it starts moving with some initial velocity, theta 1. But because of the gravitational fo...

We now introduce normal random variables, which are also often called Gaussian random variables. Normal random variables are perhaps the most important ones in probability theory. They play a key role in the theory of the subject, as we will see later in this class in the context of the central limit theorem. They're also prevalent in applications for two reasons. They have some nice analytical properties, and they're are also the most common model of random noise. In general, they are a good model of noise or randomness whenever that noise is due to the addition of many small independent noise terms, and this is a very common situation in the real world. We define normal random variables by specifying their PDFs, and we start with the simplest case of the so-called standard normal. The standard normal is indicated with this shorthand notation, and we will see shortly why this notation is being used. It is defined in terms of a PDF. This PDF is defined for all val...

We now introduce a new of random variable, the exponential random variable. It has a probability density function that is determined by a single parameter lambda, which is a positive number. And the form of the PDF is as shown here. Note that the PDF is equal to 0 when x is negative, which means that negative values of X will not occur. They have zero probability. And so our random variable is a non-negative random variable. The shape of the PDF is as shown in this diagram. It's 0 for negative values, and then for positive values, it starts off, it starts off at a value equal to lambda. This is because if you plug in x equal to 0 in this expression, you get lambda times e to the 0, which leaves you just with lambda. So it starts off with lambda, and then it decays at the rate of lambda. Notice that when lambda is small, the initial value of the PDF is small. But then the decay rate is also small, so that the PDF extends over a large range of x's. On the other hand...

In this segment, we introduce the concept of continuous random variables and their characterization in terms of probability density functions, or PDFs for short. Let us first go back to discrete random variables. A discrete random variable takes values in a discrete set. There is a total of one unit of probability assigned to the possible values. And the PMF tells us exactly how much of this probability is assigned to each value. So we can think of the bars in the PMF as point masses with positive weight that sit on top of each possible numerical value. And we can calculate the probability that the random variable falls inside an interval by adding all the masses that sit on top of that interval. So for example, if we're looking at the interval from a to b, the probability of this interval is equal to the sum of the probabilities of these three masses that fall inside this interval. On the other hand, a continuous random variable will be taking values over a continuou...

In this segment we introduce a simple but powerful tool, the basic counting principle, which we will be using over and over to deal with counting problems. Let me describe the idea through a simple example. You wake up in the morning and you find that you have in your closet 4 shirts, 3 ties, and 2 jackets. In how many different ways can you get dressed today? To answer this question, let us think of the process of getting dressed as consisting of three steps, three stages. You first choose a shirt, let's say this one, and you have 4 choices of shirts. But each shirt can be used together with 1 of the 3 available ties to make 3 different shirt-tie combinations. But since we had 4 choices for the shirt, this means that we have 4 times 3, equals 12, shirt-tie combinations. Finally, you choose a jacket. Each shirt-tie combination can go together with either jacket, and so the fact that you have 2 jackets available doubles the number of options that you have, leading to 2...

In this video, we'll introduce a method that is similar to CART called Random Forests. This method was designed to improve the prediction accuracy of CART and works by building a large number of CART trees. Unfortunately, this makes the method less interpretable than CART, so often you need to decide if you value the interpretability or the increase in accuracy more. To make a prediction for a new observation, each tree in the forest votes on the outcome and we pick the outcome that receives the majority of the votes. So how does Random Forests build many CART trees? We can't just run CART multiple times because it would create the same tree every time. To prevent this, Random Forests only allows each tree to split on a random subset of the available independent variables. And each tree is built from what we call a bagged or bootstrapped sample of the data. This just means that the data used as the training data for each tree is selected randomly with replacement....