Marginal Probability

The marginal probability is the probability of an event happening without regard of any other event.

$p(x)=\int p(x,y)dy$

Discrete case $p(x)={\sum }_{y}p(x,y)={\sum }_{y}p(x|y)p(y)$

From DL Book: Sometimes we know the probability distribution over a set of variables and we want to know the probability distribution over just a subset of them. The probability distribution over the subset is known as the marginal probability distribution.

Use of "Marginal" in different context

There’s the

• Marginal effect/value → “What happens if $X$ changes a little?”
• Marginal probability/distribution → “What remains if I ignore $Y$?”
• Marginal improvement/difference → “Small or insignificant.”

• The stats and calculus meanings are different concepts that happen to share the same word.

When reading, the noun after marginal usually tells you which meaning is intended:

• marginal cost/effect/benefit → calculus/econ
• marginal distribution/probability → statistics
• marginal improvement/difference → everyday English

Related

• Marginal Distribution
• Joint Probability
• Prior
• Likelihood