multiplicativecombining

Multiplicative combining refers to the fusion of multiple sources of evidence by multiplying their contributions to form a single combined score or distribution. It is commonly used in statistics, machine learning, and sensor fusion to integrate information under the assumption that sources provide independent evidence about the same state or hypothesis.

In probabilistic terms, if each source i provides a likelihood Li(x) for a state x, and the sources are conditionally independent given x, the joint likelihood is proportional to the product over all sources: ∏i Li(x). When combined with a prior p0(x), Bayes’ rule yields a posterior p(x|D) that is proportional to p0(x) times the product of the likelihoods: p(x|D) ∝ p0(x) ∏i Li(x). The product of experts model is a prominent instance of multiplicative combining, where the outputs of multiple expert distributions are multiplied to form a joint distribution, often producing sharper and more constrained predictions.

Practical considerations include normalization to obtain a proper probability distribution and the potential for numerical underflow when products are small. Working in log space, where products become sums, is a common solution. The approach relies on a (often strong) assumption of independence among sources; when this is violated, results may be biased. Weighted variants exist, such as p(x) ∝ p0(x) ∏i Li(x)Wi, to modulate the influence of each source.

Applications span sensor fusion, ensemble learning, and information retrieval, among others. Related concepts include Bayes’ theorem, product of experts, and log-linear models.