What is Causality?

Defining Causal Inference

• Causal inference involves estimating the impact of events on a given outcome of interest
• It involves determining the independent, actual effect of a phenomenon that is part of a larger system
• Causal inference attempts to observe the response of an effect variable when a cause of the effect variable is changed

Correlation is not Causation

• When the rooster crows, the sun rises soon afterwards
• We know the rooster didn’t cause the sun to rise
• If the rooster had been eaten by a cat, the sun still would have risen
• In other words, a rooster's crow is correlated with the sun rising
• But, a rooster's crow doesn't cause the sun to rise

Comparing Prediction with Causation

• Making predictions requires strict boundaries, where the data used to train the model doesn't really change

  • So, it can be useful for translating from english to portuguese
  • It can be useful for recognizing faces
  • It can be useful for classifying sentiments

• Determining causality requires us to answer what if questions

  • For example, what if we change the price of our product?
  • What if I do a low fat diet, instead of a low sugar diet?
  • Or, how does revenue change if we make modifications to our customer line?

Describing Potential Outcomes

• Theoretically, each observation has two potential outcomes
• In reality, each observation only has one actual outcome
• In other words, actual outcomes are realized outcomes

• Whereas, potential outcomes are hypothetical random variables

  • In other words, they are usually estimated using predictions
• Potential outcomes are defined by $Y^{t}$
• Note, each $i^{th}$ observation has two separate potential outcomes
• And, $t$ refers to an actual, known control group or treatment group
• Whereas, $Y$ refers to an unknown, potential outcome for each group

Defining Notation for Causal Data

• Causal data usually follows a specific type of notation
• $i$ represents a given observation

• $t$ represents the actual and known group that has observation $i$

  • $t=0$ when the $i^{th}$ customer is in the control group
  • $t=1$ when the $i^{th}$ customer is in the treatment group

• $Y$ represents the unknown variable being measured in the study

  • $Y^{0}$ represents $Y$ of only customers in the control group
  • $Y^{1}$ represents $Y$ of only customers in the treatment group

• $\delta$ represents the unit-specific treatment effect

  • This can be used to measure a causal effect

$$Y_{i} = t_{i} Y_{i}^{1} + (1+t_{i})Y_{i}^{0}$$ $$\delta_{i} = Y_{i}^{1} - Y_{i}^{0}$$

Describing the Struggle with Causality

• The fundamental problem with causal inference is that we can never observe the same unit with and without treatment

  • In other words, we're wanting to compute $\delta_{i}$
  • But, we can only observe either $Y_{i}^{1}$ or $Y_{i}^{0}$
  • Thus, it's impossible to be certain abound causal effects
• As a result, we can't know individual treatement effects $\delta_{i}$
• Instead, we estimate the treatment of the overall group as an average

References

• Course on Measuring Causal Effects
• Causal Inference Textbook
• Python Causality Handbook
• Comprehensive Causal Inference Textbook