A **normalizing constant** ensures that a probability density function has a probability of 1. The constant can take on various guises: it could be a scalar value, an equation, or even a function. As such, there isn’t a “one size fits all” constant; every probability distribution that doesn’t sum to 1 is going to have an individual normalization constant.

In many cases, the *nonnormalized* density function is known, but the normalization constant is not. In simple PDFs, the constant may be calculated relatively easy. In other cases, especially when the constant is a complicated function of the parameter space, it may be impossible to calculate.

## Normalization Constant Examples

In a beta distribution, the function in the denominator of the pdf acts as a normalizing constant.

The normalizing constant in the Von Mises Fisher distribution is found by integrating polar coordinates.

In the Kent distribution, the normalizing constant is an equation:

For Bayes’ rule, written as [1]:

The denominator is the normalizing constant ensuring the posterior distribution adds up to 1; it can be calculated by summing the numerator over all possible values of the random variable. Calculation of the normalization becomes much more difficult for procedures like *Bayes factor model selection* or *Bayesian model averaging*; these are notoriously difficult to calculate and usually involve high dimensional integrals that are impossible to solve analytically [2].

## References

[1] Murphy, K. In Symbols. Retrieved November 16, 2021 from: https://www.cs.ubc.ca/~murphyk/Bayes/bayesrule.html

[2] Gronau, Q. et al. bridgesampling: An R Package for Estimating Normalizing Constants. Retrieved November 16, 2021 from: https://cran.r-project.org/web/packages/bridgesampling/vignettes/bridgesampling_paper.pdf

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