# Information Theory

## Information Theory measures in `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` package aims to follow this tradition and therefore, in addition to a comprehensive catalog of distance measures it also implements the most important information theory measures.

### Shannon’s Entropy H(X)

> \(H(X) = -\sum\limits_{i=1}^n P(x_i) * log_b(P(x_i))\)

    # define probabilities P(X)
    Prob <- 1:10/sum(1:10)
    # Compute Shannon's Entropy
    H(Prob)

    [1] 3.103643

### Shannon’s Joint-Entropy H(X,Y)

> \(H(X,Y) = -\sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b(P(x_i, y_j))\)

    # 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

### Shannon’s Conditional-Entropy H(X | Y)

> \(H(Y|X) = \sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b( P(x_i) / P(x_i, y_j) )\)

    # 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

### Mutual Information I(X,Y)

> \(MI(X,Y) = \sum\limits_{i=1}^n\sum\limits_{j=1}^m P(x_i, y_j) * log_b( P(x_i, y_j) / ( P(x_i) * P(y_j) )\)

    # 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

### Kullback-Leibler Divergence

> \(KL(P || Q) = \sum\limits_{i=1}^n P(p_i) * log_2(P(p_i) / P(q_i)) = H(P, Q) - H(P)\)

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

### Jensen-Shannon Divergence

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

#### Properties of the `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.

### Generalized Jensen-Shannon Divergence

The generalized Jensen-Shannon Divergence \(gJSD_{\pi_1,...,\pi_n}(P_1, ..., P_n)\) enables distance comparisons between multiple probability distributions \(P_1,...,P_n\):

> \(gJSD_{\pi_1,...,\pi_n}(P_1, ..., P_n) = H(\sum_{i = 1}^n \pi_i*P_i) - \sum_{i = 1}^n \pi_i*H(P_i)\)

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