Comparing splits using information theory

Martin R. Smith

2020-09-17

To understand the information-based metrics implemented in ‘TreeDist’, it is useful to recall some basic concepts of information theory (see MacKay (2003) for an introduction).

Shannon information

Information is usually measured in bits. One bit is the amount of information generated by tossing a fair coin: to record the outcome of a coin toss, I must record either a H or a T, and with each of the two symbols equally likely, there is no way to compress the results of multiple tosses.

The Shannon (1948) information content of an outcome \(x\) is defined to be \(h(x) = -\log_2{P(x)}\), which simplifies to \(\log_2{n}\) when all \(n\) outcomes are equiprobable. Thus, the information content of a fair coin toss is \(\log_2{2} = 1\textrm{ bit}\); the information content of rolling a six-sided die is \(\log_2{6} \approx 2.58\textrm{ bits}\); and the information content of selecting at random any one of the 105 unrooted binary six-leaf trees is \(\log_2{105} \approx 6.71\textrm{ bits}\).

Application to splits

The split \(S_1 =\) AB|CDEF is found in 15 of the 105 six-leaf trees; as such, the probability that a randomly drawn tree contains \(S_1\) is \(P(S_1) = \frac{15}{105}\), and the information content \(h(S_1) = -\log_2{\frac{15}{105}} \approx 2.81\textrm{ bits}\). Steel & Penny (2006) dub this quantity the phylogenetic information content.

Likewise, the split \(S_2 =\) ABC|DEF occurs in nine six-leaf trees, so \(h(S_2) = -\log_2{\frac{9}{105}} \approx 3.54\textrm{ bits}\). Three six-leaf trees contain both splits, so in combination the splits deliver \(h(S_1,S_2) = -\log_2{\frac{3}{105}} \approx 5.13\textrm{ bits}\) of information.

Because \(h(S_1,S_2) < h(S_1) + h(S_2)\), some of the information in \(S_1\) is also present in \(S_2\). The information in common between \(S_1\) and \(S_2\) is \(h_{shared}(S_1, S_2) = h(S_1) + h(S_2) - h(S_1,S_2) \approx 1.22\textrm{ bits}\). The information unique to \(S_1\) and \(S_2\) is \(h_{different}(S_1,S_2) = 2h(S_1,S_2) - h(S_1) - h(S_2) \approx 3.91\textrm{ bits}\).

These quantities can be calculated using functions in the ‘TreeTools’ package.

library('TreeTools', quietly = TRUE, warn.conflicts = FALSE)
library('TreeDist')
treesMatchingSplit <- c(
  AB.CDEF = TreesMatchingSplit(2, 4),
  ABC.DEF = TreesMatchingSplit(3, 3)
)
treesMatchingSplit
## AB.CDEF ABC.DEF 
##      15       9
proportionMatchingSplit <- treesMatchingSplit / NUnrooted(6)
proportionMatchingSplit
##    AB.CDEF    ABC.DEF 
## 0.14285714 0.08571429
splitInformation <- -log2(proportionMatchingSplit)
splitInformation
##  AB.CDEF  ABC.DEF 
## 2.807355 3.544321
treesMatchingBoth <- TreesConsistentWithTwoSplits(6, 2, 3)
combinedInformation <- -log2(treesMatchingBoth / NUnrooted(6))

sharedInformation <- sum(splitInformation) - combinedInformation
sharedInformation
## [1] 1.222392
# Or more concisely:
SplitSharedInformation(n = 6, 2, 3)
## [1] 1.222392

Entropy

Entropy is the average information content of each outcome, weighted by its probability: \(\sum{-p \log_2(p)}\). Where all \(n\) outcomes are equiprobable, this simplifies to \(\log_2{n}\).

Consider a case in which Jane rolls a dice, and makes two true statements about the outcome \(x\):

\(S_1\): “Is the roll even?”.

\(S_2\): “Is the roll greater than 3?”

The joint entropy of \(S_1\) and \(S_2\) is the entropy of the association matrix that considers each possible outcome:

  \(S_1: x\) odd \(S_1: x\) even
\(S_2: x \le 3\) \(x \in {1, 3}; p = \frac{2}{6}\) \(x = 2; p = \frac{1}{6}\)
\(S_2: x > 3\) \(x = 5; p = \frac{1}{6}\) \(x \in {4, 6}; p = \frac{2}{6}\)

\(\begin{aligned} H(S_1, S_2) = \frac{2}{3}\log_2{\frac{2}{3}} + \frac{1}{3}\log_2{\frac{1}{3}} + \frac{1}{3}\log_2{\frac{1}{3}} + \frac{2}{3}\log_2{\frac{2}{3}} \approx 1.84 \textrm{ bits} \end{aligned}\)

Note that this less than the \(\log_2{6} \approx 2.58\textrm{ bits}\) we require to determine the exact value of the roll: knowledge of \(S_1\) and \(S_2\) is not guaranteed to be sufficient to unambiguously identify \(x\).

The mutual information between \(S_1\) and \(S_2\) describes how much knowledge of \(S_1\) reduces our uncertainty in \(S_2\) (or vice versa). So if we learn that \(S_1\) is ‘even’, we become a little more confident that \(S_2\) is ‘greater than three’.

The mutual information \(I(S_1;S_2)\) corresponds to the sum of the individual entropies, minus the joint entropy:

\[\begin{aligned} I(S_1;S_2) = H(S_1) + H(S_2) - H(S_1, S_2) \end{aligned}\]

The entropy distance, also termed the variation of information (Meilă, 2007), is the information that \(S_1\) and \(S_2\) do not have in common:

\[\begin{aligned} H_D(S_1, S_2) = H(S_1, S_2) - I(S_1;S_2) = 2H(S_1, S_2) - H(S_1) - H(S_2) \end{aligned}\]

Application to splits

Let split \(S\) divides \(n\) leaves into two partitions \(A\) and \(B\). The probability that a randomly chosen leaf \(x\) is in partition \(k\) is \(P(x \in k) = \frac{|k|}{n}\). \(S\) thus corresponds to a random variable with entropy \(H(S) = -\frac{|A|}{n} \log_2{\frac{|A|}{n}} - \frac{|B|}{n}\log_2{\frac{|B|}{n}}\) (Meilă, 2007).

The entropy of a split corresponds to the average number of bits necessary to transmit the partition label for a given leaf.

The joint entropy of two splits, \(S_1\) and \(S_2\), corresponds to the entropy of the association matrix of probabilities that a randomly selected leaf belongs to each pair of partitions:

  \(S_1: x \in A_1\) \(S_1: x \in B_1\)
\(S_2: x \in A_2\) \(P(A_1,A_2) = \frac{|A_1 \cap A_2|}{n}\) \(P(B_1,A_2) = \frac{|B_1 \cap A_2|}{n}\)
\(S_2: x \in B_2\) \(P(A_1,B_2) = \frac{|A_1 \cap B_2|}{n}\) \(P(B_1,B_2) = \frac{|B_1 \cap B_2|}{n}\)

\(H(S_1, S_2) = P(A_1,A_2) \log_2 {P(A_1,A_2)} + P(B_1,A_2) \log_2 {P(B_1,A_2)}\)

\(+ P(A_1,B_2)\log_2{P(A_1,B_2)} + P(B_1,B_2)\log_2{P(B_1,B_2)}\)

These values can then be substituted into the definitions of mutual information and entropy distance given above.

As \(S_1\) and \(S_2\) become more different, the disposition of \(S_1\) gives less information about the configuration of \(S_2\), and the mutual information decreases accordingly.

References

MacKay, D. J. C. (2003). Information theory, inference, and learning algorithms. Cambridge: Cambridge University Press. Retrieved from https://www.inference.org.uk/itprnn/book.pdf

Meilă, M. (2007). Comparing clusterings—an information based distance. Journal of Multivariate Analysis, 98(5), 873–895. doi:10.1016/j.jmva.2006.11.013

Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423, 623–656.

Steel, M. A., & Penny, D. (2006). Maximum parsimony and the phylogenetic information in multistate characters. In V. A. Albert (Ed.), Parsimony, phylogeny, and genomics (pp. 163–178). Oxford: Oxford University Press.