Profile parsimony

Application of Occam’s Razor

This interpretation has two applications for phylogenetics. The first is that we may wish to prefer trees (which are depictions of phylogenetic hypotheses) that concentrate homoplasy in characters that we believe to be prone to convergent evolution. This view calls for character weighting, that is, assigning less weight to changes in characters that are believed to be less phylogenetically reliable. Such characters may be identified by successive approximations (Farris, 1969), by comparing the pattern of their tokens with that of other characters, by expert judgement, or by other a priori means.

The second application argues that each additional case of homoplasy beyond the first is successively less surprising: the first observation of homoplasy (rather than some prior intuition) taught us that the character was not entirely reliable, making a second homoplasy less unexpected; the second observed homoplasy, in turn, makes us less surprised by the third. This approach calls for a step weighting approach, in which each additional step in a given character receives less penalty than the last. (Mathematically, this can be expressed in as if it were a character weighting strategy; but I feel that its a posteriori nature conveys a subtly different motivation and justifies a separate treatment.)

This raises the question of how each step beyond the first ought to be penalized. Ultimately, any concave function (in which each step is penalized by a positive amount that is smaller than the penalty applied to the previous step) is consistent with this philosophy (Arias & Miranda-Esquivel, 2004). The most widely used step weighting approach is Goloboff’s (1993) implied weighting, where the total cost associated with a character is expressed as e / (e + k), where e is the number of homoplasies within a character, and k is an arbitrary constant. As e tends to infinity, this approach tends to equal weights; as k tends to zero, it tends to clique analysis (in which a character is either homologous or ignored) (Farris, 1983). The most appropriate value for k may depend on the number of taxa, the number and distribution of observed states, and other factors (Goloboff, Carpenter, Arias, & Esquivel, 2008; Goloboff, Torres, & Arias, 2018) (a more detailed treatment will be provided in a revision of this document). Moreover, some adjustment must be made for ‘missing’ data, i.e. ambiguous tokens, which reduce the opportunity to observe homoplasy (Goloboff, 2014). Implied weighting is described as an approximation (Goloboff, 1993), and I am not aware of a straightforward interpretation of the ‘fit’ score, or a principled definition of the nature of the quantity that is being approximated.

An information theoretic basis

I argue that the quantity that we should seek to minimise is the “surprise” of a tree. Information theory is the science of quantifying unexpectedness. 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 equally likely. Thus, the outcome of a fair coin toss delivers \(\log_2{2} = 1\textrm{ bit}\) of information; the outcome of rolling a fair six-sided die contains \(\log_2{6} \approx 2.58\textrm{ bits}\) of information; and the outcome of selecting at random one of the 105 unrooted binary six-leaf trees is \(\log_2{105} \approx 6.71\textrm{ bits}\).

Unlikely outcomes are more surprising, and thus contain more information than likely outcomes. The information content of rolling a twelve on two fair six-sided dice is \(-\log_2{\frac{1}{36}} \approx 5.16\textrm{ bits}\), whereas a seven, which could be produced by six of the 36 possible rolls (1 & 6, 2 & 5, …), is less surprising, and thus contains less information: \(-\log_2{\frac{6}{36}} \approx 2.58\textrm{ bits}\). An additional 2.58 bits of information would be required to establish whether which of the rolls 1 & 6, 2 & 5, … occurred.

Now consider two competing explanations for an event: (i), three consecutive rolls of two dice each produced a seven; (ii), two consecutive rolls of two dice each produced a twelve. The former event corresponds to \(3 \times 2.58 = 7.75\textrm{ bits}\) of information, so is less surprising than the latter, which represents \(2 \times 5.16 = 10.34\textrm{ bits}\), despite involving an additional roll of the dice.

How do we measure the “surprise” associated with additional steps in a character on a phylogenetic tree? Consider a character with the states 0 0 0 1 1 1. In the most parsimonious situation in which the character contains a single step on a tree, it is compatible with nine of the 105 labelled six-leaf trees, and thus represents \(-log_2\frac{9}{105} = 3.54\textrm{ bits}\) of phylogenetic information. If we are told that the character contains exactly two steps, it can occur on 63 trees, so yields \(-log_2\frac{63}{105} = 0.74\textrm{ bits}\). (The number of trees with m extra steps can be calculated using theorem 1 of Carter, Hendy, Penny, Székely, & Wormald (1990), implemented in the function Carter1().) Learning that a second step occurred meant that 3.54 − 0.74 = 2.81 bits of information we had previously attributed to common ancestry instead represent a signature of homoplasy; this quantity measures our degree of ‘surprise’.

If we subsequently learn that the character contains three steps, then it can occur on any six-leaf tree, and contains no phylogenetic information. We are less surprised by this third step, which attributes the remaining 0.74 bits of information to factors other than common ancestry, because the second step had already reduced the amount of phylogenetic information we expected the character to hold.