Chain rule and conditional probabilities

Suppose we have two random variables $X$ and $Y$ with joint distribution $\mathbb{P}(X, Y)$. We understand $\mathbb{P}(X=x, Y=y)$ as the probability that $X$ has outcome $x$ and $Y$ has outcome $y$. Intuitively, we can also understand that joint probability as follows: my belief that will take on outcome $x$ and $y$ takes on outcome can be ‘decomposed’ into my belief that $X$ will take on $x$ in general, and then that $Y$ takes on value $y$ given that $X$ has taken on value $x$. For example, my belief in me studying well and passing the exam is simply my belief in me studying well in general combined with my belief that if I study well, I will pass the exam. Formally, we write this as

$$\mathbb{P}(X=x, Y=y) = \mathbb{P}(X=x) \cdot \mathbb{P}(Y=y \mid X=x).$$ The opposite reasoning also goes, so $$\mathbb{P}(X=x, Y=y) = \mathbb{P}(Y=y) \cdot \mathbb{P}(X=x \mid Y=y).$$ This is called the chain rule of probability theory (or the general multiplication rule), and this rule you will use over and over when doing probability calculus. At this point, it becomes a bit convoluted to write $X=x$ every time, so we might write this also as $$\mathbb{P}(X, Y) = \mathbb{P}(X) \cdot \mathbb{P}(Y \mid X),$$ to mean that $$\mathbb{P}(X=x, Y=y) = \mathbb{P}(X=x) \cdot \mathbb{P}(Y=y \mid X=x)\quad\text{for all }x\text{ and }y.$$ Moreover, this rule generalizes, e.g. $$\mathbb{P}(X, Y, Z) = \mathbb{P}(X) \cdot \mathbb{P}(Y, Z \mid X)$$ and thus $$\mathbb{P}(X, Y, Z) = \mathbb{P}(X) \cdot \mathbb{P}(Y \mid X) \cdot \mathbb{P}(Z \mid X, Y).$$ In its most general glory, it would say that $$\boxed{\mathbb{P}(X_1, \cdots, X_n) = \prod_{k=1}^n \mathbb{P}(X_k \mid X_1, \cdots, X_{k-1})}.$$ This also gives us a way to define conditional probabilities directly. Since $$\mathbb{P}(X, Y) = \mathbb{P}(X) \cdot \mathbb{P}(Y \mid X),$$ we know that $$\boxed{\mathbb{P}(Y \mid X) = \frac{\mathbb{P}(X, Y)}{\mathbb{P}(X)}}$$ given that $\mathbb{P}(X)\neq 0$.

Marginalization and marginal probabilities

Another very important fact of probability theory is marginalization. It says that if we have two random variables $X$ and $Y$, we can understand $\mathbb{P}(X=x)$ also as a ‘summed out’ probabilities of the joint distribution of $X$ and $Y$, i.e.

$$\boxed{\mathbb{P}(X=x) = \sum_{y} \mathbb{P}(X=x, Y=y)}.$$ Equivalently, we can say that $$\boxed{\mathbb{P}(X=x) = \sum_{y} \mathbb{P}(X=x \mid Y=y) \cdot \mathbb{P}(Y=y)}.$$ This definition makes intuitive sense: my believe in $X$ taking on outcome $x$ in general, is simply a combined version of $X$ taking on $x$ given that $Y=y$, weighted by my belief in $Y=y$. Here, we say that $\mathbb{P}(X)$ is a marginal probability.

Bayes' rule