philentropy
The laws of probability, so true in general, so fallacious in particular.
- Edward Gibbon
Information theory and statistics were beautifully fused by Solomon Kullback
. This fusion allowed to quantify correlations and similarities between random variables
using a more sophisticated toolkit. Modern fields such as machine learning and statistical data science build upon this fusion and the most powerful statistical techniques used today are based on an information theoretic foundation.
The philentropy
aims to follow this tradition and therefore, it implements
the most important information theory measures.
$H(X) = -\sum\limits_{i=1}^n P(x_i) * log_b(P(x_i))$
# define probabilities P(X) Prob <- 1:10/sum(1:10) # Compute Shannon's Entropy H(Prob)
[1] 3.103643
$H(X,Y) = -\sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b(P(x_i, y_j))$
# define the joint distribution P(X,Y) P_xy <- 1:100/sum(1:100) # Compute Shannon's Joint-Entropy JE(P_xy)
[1] 6.372236
$H(Y|X) = \sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b( P(x_i) / P(x_i, y_j) )$
# define the distribution P(X) P_x <- 1:10/sum(1:10) # define the distribution P(Y) P_y <- 1:10/sum(1:10) # Compute Shannon's Joint-Entropy CE(P_x, P_y)
[1] 0
$MI(X,Y) = \sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b( P(x_i, y_j) / ( P(x_i) * P(y_j) )$
# define the distribution P(X) P_x <- 1:10/sum(1:10) # define the distribution P(Y) P_y <- 20:29/sum(20:29) # define the joint-distribution P(X,Y) P_xy <- 1:10/sum(1:10) # Compute Shannon's Joint-Entropy MI(P_x, P_y, P_xy)
[1] 3.311973
$KL(P || Q) = \sum\limits_{i=1}^n P(p_i) * log_2(P(p_i) / P(q_i)) = H(P, Q) - H(P)$
where H(P, Q)
denotes the joint entropy of the probability distributions P
and Q
and H(P)
denotes the entropy of probability distribution P
. In case P = Q
then KL(P, Q) = 0
and in case P != Q
then KL(P, Q) > 0
.
The KL divergence is a non-symmetric measure of the directed divergence between two probability distributions P and Q. It only fulfills the positivity property of a distance metric.
Because of the relation KL(P||Q) = H(P,Q) - H(P)
, the Kullback-Leibler divergence of two probability distributions P
and Q
is also named Cross Entropy
of two probability distributions P
and Q
.
# Kulback-Leibler Divergence between random variables P and Q P <- 1:10/sum(1:10) Q <- 20:29/sum(20:29) x <- rbind(P,Q) # Kulback-Leibler Divergence between P and Q using different log bases KL(x, unit = "log2") # Default KL(x, unit = "log") KL(x, unit = "log10")
# KL(x, unit = "log2") # Default Kulback-Leibler Divergence using unit 'log2'. kullback-leibler 0.1392629 # KL(x, unit = "log") Kulback-Leibler Divergence using unit 'log'. kullback-leibler 0.09652967 # KL(x, unit = "log10") Kulback-Leibler Divergence using unit 'log10'. kullback-leibler 0.0419223
This function computes the Jensen-Shannon Divergence
JSD(P || Q)
between two probability distributions P
and Q
with equal weights π_1
= π_2
= 1/2.
The Jensen-Shannon Divergence JSD(P || Q) between two probability distributions P and Q is defined as:
$JSD(P || Q) = 0.5 * (KL(P || R) + KL(Q || R))$
where R = 0.5 * (P + Q)
denotes the mid-point of the probability vectors P
and Q
, and KL(P || R)
, KL(Q || R)
denote the Kullback-Leibler Divergence
of P
and R
, as well as Q
and R
.
# Jensen-Shannon Divergence between P and Q P <- 1:10/sum(1:10) Q <- 20:29/sum(20:29) x <- rbind(P,Q) # Jensen-Shannon Divergence between P and Q using different log bases JSD(x, unit = "log2") # Default JSD(x, unit = "log") JSD(x, unit = "log10")
# JSD(x, unit = "log2") # Default Jensen-Shannon Divergence using unit 'log2'. jensen-shannon 0.03792749 # JSD(x, unit = "log") Jensen-Shannon Divergence using unit 'log'. jensen-shannon 0.02628933 # JSD(x, unit = "log10") Jensen-Shannon Divergence using unit 'log10'. jensen-shannon 0.01141731
Alternatively, users can specify count data.
# Jensen-Shannon Divergence Divergence between count vectors P.count and Q.count P.count <- 1:10 Q.count <- 20:29 x.count <- rbind(P.count, Q.count) JSD(x, est.prob = "empirical")
Jensen-Shannon Divergence using unit 'log2'. jensen-shannon 0.03792749
Or users can compute distances based on a probability matrix
# Example: Distance Matrix using JSD-Distance Prob <- rbind(1:10/sum(1:10), 20:29/sum(20:29), 30:39/sum(30:39)) # compute the KL matrix of a given probability matrix JSDMatrix <- JSD(Prob) JSDMatrix
v1 v2 v3 v1 0.00000000 0.0379274917 0.0435852218 v2 0.03792749 0.0000000000 0.0002120578 v3 0.04358522 0.0002120578 0.0000000000
Jensen-Shannon Divergence
:JSD is non-negative.
JSD is a symmetric measure JSD(P || Q) = JSD(Q || P).
JSD = 0, if and only if P = Q.
The generalized Jensen-Shannon Divergence $gJSD_{\pi_1,...,\pi_n}(P_1, ..., P_n)$ enables distance comparisons between multiple probability distributions $P_1,...,P_n$:
$gJSD_{\pi_1,...,\pi_n}(P_1, ..., P_n) = H(\sum_{i = 1}^n \pi_iP_i) - \sum_{i = 1}^n \pi_iH(P_i)$
where $\pi_1,...,\pi_n$ denote the weights selected for the probability vectors $P_1,...,P_n$ and $H(P_i)$ denotes the Shannon Entropy of probability vector $P_i$.
# generate example probability matrix for comparing three probability functions Prob <- rbind(1:10/sum(1:10), 20:29/sum(20:29), 30:39/sum(30:39)) # compute the Generalized JSD comparing the PS probability matrix gJSD(Prob)
#> No weights were specified ('weights = NULL'), thus equal weights for all #> distributions will be calculated and applied. #> Metric: 'gJSD'; unit = 'log2'; comparing: 3 vectors (v1, ... , v3). #> Weights: v1 = 0.333333333333333, v2 = 0.333333333333333, v3 = 0.333333333333333 [1] 0.03512892
As you can see, the gJSD
function prints out the exact number of vectors that were used to compute the generalized JSD. By default, the weights are uniformly distributed (weights = NULL
).
Users can also specify non-uniformly distributed weights via specifying the weights
argument:
# define probability matrix Prob <- rbind(1:10/sum(1:10), 20:29/sum(20:29), 30:39/sum(30:39)) # compute generalized JSD with custom weights gJSD(Prob, weights = c(0.5, 0.25, 0.25))
#> Metric: 'gJSD'; unit = 'log2'; comparing: 3 vectors (v1, ... , v3). #> Weights: v1 = 0.5, v2 = 0.25, v3 = 0.25 [1] 0.04081969
Finally, users can use the argument est.prob
to empirically estimate probability vectors when they wish to specify count vectors as input:
P.count <- 1:10 Q.count <- 20:29 R.count <- 30:39 x.count <- rbind(P.count, Q.count, R.count) gJSD(x.count, est.prob = "empirical")
#> No weights were specified ('weights = NULL'), thus equal weights for all distributions will be calculated and applied. #> Metric: 'gJSD'; unit = 'log2'; comparing: 3 vectors (v1, ... , v3). #> Weights: v1 = 0.333333333333333, v2 = 0.333333333333333, v3 = 0.333333333333333 [1] 0.03512892
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