Normalization:

\( \sum_i p(H_i|\ldots) = 1 \).

Marginalization:

\( \sum_i p(A,H_i|I) = \sum_i p(H_i|A,I) p(A|I) = p(A|I) \).

Marginalization (continuum limit):

\( \int dx p(A,H(x)|I) = p(A|I) \).

In the above, \( H_i \) is an exclusive and exhaustive list of hypotheses. For example,let’s imagine that there are five candidates in a presidential election; then \( H_1 \) could be the proposition that the first candidate will win, and so on. The probability that \( A \) is true, for example that unemployment will be lower in a year’s time (given all relevant information \( I \), but irrespective of whoever becomes president) is then given by \( \sum_i p(A,H_i|I) \).

In the continuum limit of propositions we must understand \( p(\ldots) \) as a pdf (probability density function).

Marginalization is a very powerful device in data analysis because it enables us to deal with nuisance parameters; that is, quantities which necessarily enter the analysis but are of no intrinsic interest. The unwanted background signal present in many experimental measurements are examples of nuisance parameters.