Random Variable

random variables takes on different values with different probabilities. Each value a random variable take on is an event.

For instance, here’s a random variable representing a die: \(X\). It can takes on the following values, with the following probabilities:

\begin{align} P(X=1) = \frac{1}{6}\\ P(X=2) = \frac{1}{6}\\ \dots \\ P(X=6) = \frac{1}{6} \end{align}

where each assignment \(X=k\) is what we refer to above as an event.

The set of assignments of a random variable and their associated probability is called a distribution: distributions “assigns probabilities to outcomes.” When we say a certain random variable \(X\) is “distributed” following a distribution \(D\), we say \(X \sim D\). Semantically, we say \(X\) is a \(D\) random variable.

After a conversation with an LM https://chatgpt.com/share/697143db-c3e0-8000-b56c-07cf7ca43795 the following proposal was generated.

Hello everyone! It’s April, which means we are ready again for a new unit. Let’s dive in.

You know what’s better than one neural network? TWO!!! Multi-modal approaches—making two neural networks interact for a certain result—dominate many of the current edge of neural network research. In this unit, we are going to introduce one such approach, Generative Adversarial Networks (GAN), but leave you with some food for thought for other possibilities for what training multiple networks together can do.

Woof. As I begin to write this I should add that this unit is going to be conceptually dense. Though we are teaching one particular algorithm (incidentally, named, REINFORCE), the world of reinforcement learning is build by one, if not many, very advanced treatments in maths.

So if anything, I would focus on getting the conceptual flavor of how these problems are formulated and discuses. If you can be along for the mathematical and algorithmic journey, then even better — but by no means required or expected… There’s still lots for all of us to learn together.

Welcome back! I think, over the last few days, we have been hyping up convolutional neural networks enough such that you are probably ready to dive right in. So… Let’s, uh, motivate it first!

Why do we use a CNN?

Let’s think of a toy problem to play with. Given a pattern made using two colours (let’s name them a and b, or perhaps black and white), let’s classify whether it is the “zebra” pattern" or the “checkerboard” pattern.