entropy: entropy.Dirichlet – R documentation

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entropy.Dirichlet

Dirichlet Prior Bayesian Estimators of Entropy, Mutual Information and Other Related Quantities

Description

freqs.Dirichlet computes the Bayesian estimates of the bin frequencies using the Dirichlet-multinomial pseudocount model.

entropy.Dirichlet estimates the Shannon entropy H of the random variable Y from the corresponding observed counts y by plug-in of Bayesian estimates of the bin frequencies using the Dirichlet-multinomial pseudocount model.

KL.Dirichlet computes a Bayesian estimate of the Kullback-Leibler (KL) divergence from counts y1 and y2.

chi2.Dirichlet computes a Bayesian version of the chi-squared divergence from counts y1 and y2.

mi.Dirichlet computes a Bayesian estimate of mutual information of two random variables.

chi2indep.Dirichlet computes a Bayesian version of the chi-squared divergence of independence from a table of counts y2d.

Usage

freqs.Dirichlet(y, a)
entropy.Dirichlet(y, a, unit=c("log", "log2", "log10"))
KL.Dirichlet(y1, y2, a1, a2, unit=c("log", "log2", "log10"))
chi2.Dirichlet(y1, y2, a1, a2, unit=c("log", "log2", "log10"))
mi.Dirichlet(y2d, a, unit=c("log", "log2", "log10"))
chi2indep.Dirichlet(y2d, a, unit=c("log", "log2", "log10"))

Arguments

`y`	vector of counts.
`y1`	vector of counts.
`y2`	vector of counts.
`y2d`	matrix of counts.
`a`	pseudocount per bin.
`a1`	pseudocount per bin for first random variable.
`a2`	pseudocount per bin for second random variable.
`unit`	the unit in which entropy is measured. The default is "nats" (natural units). For computing entropy in "bits" set `unit="log2"`.

Details

The Dirichlet-multinomial pseudocount entropy estimator is a Bayesian plug-in estimator: in the definition of the Shannon entropy the bin probabilities are replaced by the respective Bayesian estimates of the frequencies, using a model with a Dirichlet prior and a multinomial likelihood.

The parameter a is a parameter of the Dirichlet prior, and in effect specifies the pseudocount per bin. Popular choices of a are:

a=0:maximum likelihood estimator (see entropy.empirical)
a=1/2:Jeffreys' prior; Krichevsky-Trovimov (1991) entropy estimator
a=1:Laplace's prior
a=1/length(y):Schurmann-Grassberger (1996) entropy estimator
a=sqrt(sum(y))/length(y):minimax prior

The pseudocount a can also be a vector so that for each bin an individual pseudocount is added.

Value

freqs.Dirichlet returns the Bayesian estimates of the frequencies .

entropy.Dirichlet returns the Bayesian estimate of the Shannon entropy.

KL.Dirichlet returns the Bayesian estimate of the KL divergence.

chi2.Dirichlet returns the Bayesian version of the chi-squared divergence.

mi.Dirichlet returns the Bayesian estimate of the mutual information.

chi2indep.Dirichlet returns the Bayesian version of the chi-squared divergence of independence.

Author(s)

Korbinian Strimmer (http://www.strimmerlab.org).

References

Agresti, A., and D. B. Hitchcock. 2005. Bayesian inference for categorical data analysis. Stat. Methods. Appl. 14:297–330.

Krichevsky, R. E., and V. K. Trofimov. 1981. The performance of universal encoding. IEEE Trans. Inf. Theory 27: 199-207.

Schurmann, T., and P. Grassberger. 1996. Entropy estimation of symbol sequences. Chaos 6:41-427.

Examples

# load entropy library 
library("entropy")


# a single variable

# observed counts for each bin
y = c(4, 2, 3, 0, 2, 4, 0, 0, 2, 1, 1)  

# Dirichlet estimate of frequencies with a=1/2
freqs.Dirichlet(y, a=1/2)

# Dirichlet estimate of entropy with a=0
entropy.Dirichlet(y, a=0)

# identical to empirical estimate
entropy.empirical(y)

# Dirichlet estimate with a=1/2 (Jeffreys' prior)
entropy.Dirichlet(y, a=1/2)

# Dirichlet estimate with a=1 (Laplace prior)
entropy.Dirichlet(y, a=1)

# Dirichlet estimate with a=1/length(y)
entropy.Dirichlet(y, a=1/length(y))

# Dirichlet estimate with a=sqrt(sum(y))/length(y)
entropy.Dirichlet(y, a=sqrt(sum(y))/length(y))


# example with two variables

# observed counts for two random variables
y1 = c(4, 2, 3, 1, 10, 4)
y2 = c(2, 3, 7, 1, 4, 3)

# Bayesian estimate of Kullback-Leibler divergence (a=1/6)
KL.Dirichlet(y1, y2, a1=1/6, a2=1/6)

# half of the corresponding chi-squared divergence
0.5*chi2.Dirichlet(y1, y2, a1=1/6, a2=1/6)


## joint distribution example

# contingency table with counts for two discrete variables
y2d = rbind( c(1,2,3), c(6,5,4) )

# Bayesian estimate of mutual information (a=1/6)
mi.Dirichlet(y2d, a=1/6)

# half of the Bayesian chi-squared divergence of independence
0.5*chi2indep.Dirichlet(y2d, a=1/6)

entropy

Estimation of Entropy, Mutual Information and Related Quantities

v1.3.0

GPL (>= 3)

Authors

Jean Hausser and Korbinian Strimmer

Initial release

2021-04-25