Tldr
It’s uncommon for two random variables to be truly independent in real-world situations. However, the concept of conditional independence allows us to describe situations where two variables become independent give the knowledge of a third variable.
The Butterfly Effect and Interconnectedness
- In reality, true independence between two distinct random variables is quite rare.
- Think of it like the “butterfly effect”: a single random variable can indirectly influence many others, even if it’s not directly connected to them, by affecting intermediate variables.
What is Conditional Independence?
- This interconnectedness leads us to the concept of conditional independence.
- Even if two random variables might seem independent at first glance, their relationship can drastically change when we consider them under a specific condition.
- Conversely, two variables that appear dependent might actually become independent once we account for a third variable.
(Murphy, 2022, p. 39)
Reference
Murphy, K. P. (2022). Probabilistic Machine Learning: An Introduction. MIT press.