When you want to reason about uncertainty in the real world, you often start by describing the relationships between variables using probability. A straightforward way to represent all possible dependencies is with a tabular joint distribution. This means you list every possible combination of variable values, along with the probability of each. However, this approach quickly becomes impractical as the number of variables grows. The size of the table increases exponentially with each additional variable, making storage, computation, and even understanding the relationships nearly impossible for anything but the simplest systems.

Suppose you have just three binary variables. The joint table already needs eight entries. With ten binary variables, you need 1,024 entries. If you want to model more realistic scenarios — like medical diagnoses involving dozens of symptoms and diseases — the joint table would be astronomically large. More importantly, this approach does not take advantage of any structure or independence that might exist between variables. It treats every variable as potentially dependent on every other, even when many relationships are much simpler.

Probabilistic graphical models offer a solution to these challenges. They use graphs — collections of nodes and edges — to represent variables and their dependencies in a compact, structured way. Instead of listing every probability, you can use the graph to encode which variables are directly related and which are conditionally independent, dramatically reducing the complexity of your model. This structure lets you represent real-world systems with many variables in a way that is both efficient and interpretable.

Graphs are powerful tools for describing relationships. In probabilistic graphical models, each node in the graph represents a random variable, and edges describe the direct dependencies between them. This visual and mathematical language makes it easier to see how information flows through your system, identify independent variables, and simplify computations.

For example, in a medical diagnosis problem, you might have nodes for diseases and symptoms. Edges connect diseases to the symptoms they cause. If two symptoms are conditionally independent given a disease, the graph will show this by not connecting them directly. This makes it much easier to both see and compute the relevant probabilities, compared to sifting through a huge joint table. The graph encodes the essential structure of your problem, helping you reason about uncertainty in a principled and scalable way.