Conditional and Unconditional Dependencies in Causal Inference: In Plain English
While reading some causal inference literature, I found it might be confusing to have some terms correct. Terms like “conditioning” and “dependency” often trip up me, frankly to say.
So, I thought maybe I can break down these concepts using simple examples, making it easy to grasp how they work and their importance in causal analysis. In fact I did that for my own self on my notes and I thought it might be useful for others, hopefully 😊!
What confused me is having these 4 terms write quickly when reading a topic about causal inferences. These terms come up when they try to describe two variables (maybe 3):
Unconditionally Independent, Unconditionally Dependent, Conditionally Independent and Conditionally Dependent!!
I also ended up with a truth table, like the one that we used to create in Boolean Algebra 😊.
Unconditionally Independent
That is the easiest to get! When two variables, X and Y, are unconditionally independent, there is no direct or indirect relationship between them, regardless of any other variables.
Example:
- X: Number of books in a library
- Y: Distance from the Earth to the moon
X and Y are unconditionally independent because the number of books in a library has no connection to the distance from the Earth to the moon, as the best of my knowledge, at least. These two variables are completely unrelated in any context.
Unconditionally Dependent
This is also easy to understand! So, when two variables, X and Y, are unconditionally dependent, it means they have a direct relationship without considering any other variables.
Example:
- X: Amount of time studied
- Y: Exam score
In this scenario, X and Y are unconditionally dependent because the amount of time studied directly impacts the exam score, of course you can argue, but let us just think about what our parents told us when we were kids! More study time generally leads to better exam scores.
Conditionally Independent
No, this is the tricky one, because we consider other variables!! So, two variables, X and Y, are conditionally independent given a third variable, Z, if knowing Z makes X and Y independent of each other. In other words, once you know Z, X provides no additional information about Y, and vice versa. BTW, this one is very important in causal inference!
Example:
- X: Number of umbrellas sold
- Y: Number of raincoats sold
- Z: Weather condition (rainy or not)
In this case, X and Y are conditionally independent given Z. If we know the weather condition (Z), knowing the number of umbrellas sold (X) doesn’t tell us anything new about the number of raincoats sold (Y) because both are directly influenced by whether it’s raining or not.
Conditionally Dependent
Two variables, X and Y, are conditionally dependent if their relationship depends on a third variable, Z. This means that the connection between X and Y is influenced by the value of Z.
Example:
- X: Number of ice creams sold
- Y: Number of people at the beach
- Z: Temperature
Here, X and Y are conditionally dependent on Z. On a hot day (when Z is high), both ice cream sales and beach attendance increase. Thus, the number of ice creams sold and the number of people at the beach are related, but this relationship hinges on the temperature.
Truth Table
To further clarify, here’s a truth table I created summarizing the different types of dependencies and independencies between variables X, Y, and Z, along with short explanations from the examples:
Summary of Differences
- Unconditionally Dependent: X and Y are directly related without any other variables.
- Conditionally Dependent: X and Y’s relationship depends on another variable, Z.
- Conditionally Independent: Given Z, X and Y have no relationship.
- Unconditionally Independent: X and Y have no relationship at all, in any context.
Hope that helps in nderstanding these distinctions which I think it is important for anyone delving into the causal inference literature and its graphical models.
