Entropy in Voynichese

It has often been observed that Voynich characters have relatively low entropy (c.f. this discussion on René Zandbergen's site). This is a serious problem for the proposal I made in my last post, where I suggested that page 81R of the Voynich Manuscript might contain a poem in Latin dactylic hexameter.

Suppose you calculate the bits of information conveyed by a character c of a text T using a formula like the following:

S_c = (ln(f_T) - ln(f_c)) / ln(2)

where

S_c is the number of bits conveyed by the single character c

f_T is the number of characters in the text

f_c is the number of times the character c appears in the text

Using this formula we find that the lines on 81R carry, on average, 121.4 bits of information. In contrast, lines of the Aeneid carry an average of 156.1 bits of information. This is a real problem, which becomes even more severe if you look at the incremental information conveyed by the second character in a pair. That is, for a character c appearing immediately after a character b:

P_bc = (ln(f_b) - ln(f_bc)) / ln(2)

where

P_bc is the number of bits conveyed by character c when it appears in the pair bc

f_b is the number of times the character b appears in the text

f_bc is the number of times the sequence bc appears in the text, which may also be expressed as the number of times that the character c appears immediately after b.

This second approach to measuring information tells us, for example, that the character "u" in a Latin text conveys no additional information when it follows "q". Since the total frequency of "qu" is the same as the total frequency of "u", the numerator is zero, and total bits likewise is zero.

When you apply this measure to the lines on 81R and the Aeneid, the average amount of information conveyed the lines of 81R drops to 66.9 bits, while the information conveyed in the average line of the Aeneid drops only to 128.5 bits.

This is a serious challenge to the idea that the plaintext on 81R is a Latin poem in dactylic hexameter, because it suggests that these lines simply don't contain enough information to encode such a poem. In my next post I will look at historically and culturally plausible enciphering schemes that could produce this effect.