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Conditional probability

As explained in Causal Inference for Data Science

Conditional probability is the probability of an event given that another event or condition is known to occur.

Conditional probability describes how likely an event is after restricting attention to cases where another event or condition is known. In data terms, conditioning on X = a means filtering the rows to those with X = a, then recalculating the distribution of another variable. section 1.5.2

The restriction can change what appears likely. In simulated data, Z = A occurs 3/5 of the time overall but 1/2 of the time among rows where X = a, written P(Z = A) = 3/5 and P(Z = A|X = a) = 1/2. The formal calculation divides the joint probability by the probability of the condition. section 1.5.2

Use conditional probability to:

  • Describe what has been observed among a selected subpopulation, including cases selected by two or more conditions. section 1.5.2
  • Distinguish prediction from intervention: P(R = 1|T = A) describes recovery among patients who received treatment A, whereas P(R = 1|do(T = A)) asks what would happen if everyone received it. section 2.2.4
  • Support supervised learning, whose target for categorical outcomes is a conditional probability, while recognizing that prediction depends on future events resembling past data. section 4.0 section 4.6

Treatment and recovery

Historical data may show P(R = 1|T = A) = 78% because the patients receiving treatment A have a particular mix of kidney-stone sizes. Giving treatment A to everyone changes that mix and produces the intervened recovery rate P(R = 1|do(T = A)) = 83%. The two probabilities answer different questions. section 2.2.4

Table 1.2 Simulated data
X
Y
Z
3
0.03
A
6
–24.08
A
a
7.01
A
–2
–3.00
B
a
10.89
B

Inspect the simulated rows before conditioning, then compare them with the filtered table for X = a to see how a condition changes frequencies. section 1.5.2

Figure 2.20 Differences between the conditional probability, the intervened probability, and the adjustment formula for the cases of an RCT, a confounder, and a mediator. The equal and unequal signs determine whether the adjacent quantities agree. T stands for Treatment, C for Confounder, M for Mediator, and X for Covariates.
figure

Compare conditional probability, intervened probability, and adjustment-formula estimates across randomized trials, confounding, and mediation. section 2.5.10

How Causal Inference for Data Science covers it

How deep it goes
Explains row filtering, notation, frequency changes, multiple conditions, and causal contrasts, while deferring advanced probability derivations.
Examples
Uses simulated X-Y-Z tables, filtered conditions, conditional distributions, and kidney-stone treatment and recovery rates.
What it assumes
Assumes basic statistics but reinforces conditional probabilities from elementary notation and data filtering before applying them to causal inference.
What you can do after
Evaluate/understand observed probabilities, distinguish them from intervention probabilities, and recognize their role in prediction and causal adjustment.
How it connects
Connects conditional expectation, supervised learning, causal inference, interventions, confounding, mediation, and the adjustment formula.
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This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.