Suppose a website feature changes how users engage. A single predicted engagement number hides variation across users. A probability distribution represents that variation: it assigns probability values to possible outcomes of a random variable. P(X) denotes the distribution over X; P(X=x) denotes the value assigned to one outcome. For a finite set of discrete outcomes, the distribution can be written as a table. section 2.3.1 section 2.1.2
A three-outcome distribution
Let X have outcomes 1, 2, and 3 with probabilities 0.45, 0.30, and 0.25. Then P(X=1)=0.45, and the three values sum to one. To parameterize another distribution with the same outcomes, choose the first two probabilities; the third is one minus their sum. In pgmpy, DiscreteFactor stores the outcome labels, cardinality, and values, and dist.sample(n=1) generates a one-row pandas DataFrame. section 2.1.2 section 2.1.6 section 2.2.2
Listing 2.1 Implementing a discrete distribution table in pgmpy
from pgmpy.factors.discrete import DiscreteFactor
dist = DiscreteFactor(
variables=["X"], #1
cardinality=[3], #2
values=[.45, .30, .25], #3
state_names= {'X': ['1', '2', '3']} #4
)
print(dist)
The pgmpy listing makes the table concrete: variables names X, cardinality sets its three outcomes, and values supplies their probabilities. section 2.1.2
The distribution needed depends on the question:
- For a population question, a conditional distribution can isolate a subpopulation. P(E|F=True) describes engagement among users exposed to a feature; adding another condition narrows that population further. section 2.3.1
- For simulation, sample outcomes from the specified distribution. A computer uses a seeded pseudo-random number generator to emulate random generation; sampling does not replace specifying the distribution. section 2.2.2
- For a causal question, distinguish an observed action distribution from an intervention. P(X|C) can describe investors’ choices under economic conditions, but asking what would happen if you chose equity calls for intervention-based reasoning. Fixing an intervention target can change distributions of its causal descendants. section 12.1.2 section 7.3
A bar plot can display a discrete distribution; a curve can display a continuous normal distribution. Neither should be confused with a histogram of observed values. Distributions can also cover several variables jointly: an autoregressive language model, for example, assigns probabilities to token sequences by factoring their joint distribution into probabilities conditioned on preceding tokens. That predictive factorization does not, by itself, answer an intervention question. section 2.1.8 section 13.2.1 section 12.1.2
