General Report on Tunny
24W Page 132
It is found that the entries in the table are not sensitive to the exact value of ζ assumed. (R1 p 49, R2 p 1)
The sense in which this type of convergence is accurate is that if the xi's are a correct measure of the odds of the characters of one wheel (measured in 'pips' of 10log10ζ decibans each), then the formula gives the odds of the characters of the other wheel accurately, if the numbers Θij are regarded as evidence which is independent of the xi's. Clearly the last assumption is not really accurate, but this does not prevent accurate convergence from being theoretically more satisfactory than crude convergence. In fact crude convergence is the limiting case of accurate scoring when ζ→∞, if the pippages of the characters of the wheel taken through exceed those in the cells of the rectangle.
The precise interpretation of the pippages of the characters of a wheel, as the result of a crude convergence is that they are proportioned to the decibanages assuming the pattern of the other wheel to be certain. In practice the relationship between the pippages and the true decibanage (assuming the patterns to be substantially correct) is not linear (see R3 p 132).
When a message contains a lot of 9's (representing letters missed) there is a modification that can be made to crude convergence. The modification was given in R4 p 39, but it was seldom used.
(b) Proof of accurate scoring formula.
The formula for accurate scoring is of exactly the same form as that for 'scoring one column of the rectangle against another'. Given two columns of the rectangle we may be interested in the question of whether the two corresponding characters of the wheel are the same or different.
Let us suppose that the pippages of the two columns are respectively
| Θ1, Θ2. | |
| Θ1', Θ2'. |
Then the factor in favour of the two columns being the same (i.e. the two corresponding characters of the wheel being the same) is
|
Where ƒ0(Θi, Θi') is the factor accruing from one pair of corresponding cells. Let us then consider the following problem: Given two characters where odds of being dots are ζΘ, ζΘ', what are the odds ζπ that the two
Back to General Report on Tunny. Contents.