The method

The question is : Given a set of incomplete and noisy data (say, F_o, with its (F) and ), which map (of a large number of maps consistent with the observed data) is the one that will minimise the probability of misinterpreting it ? Stating the same problem in a different way, we could ask (i) which map (of the set of admissible maps) will only show features for which there is evidence in the data, or, (ii) which map makes the least assumptions about the data (especially the missing data, but also the distribution of errors in the observed).

Clearly, the F_o,exp(i) synthesis is not the map we want : Not only we assume that all missing data have F = 0 (a rather improbable event), but also that F = F_o,,. Gull, S.F. & Daniell, G.J.⁴, suggested that the map we need is the one for which the configurational entropy - m_jlogm_j, where m_j is the density at the grid point j of the map, reaches a maximum. It is easy to see that - m_jlogm_j reaches a maximum when m_j = e^-1,j, that is, when the map has a uniform density, and thus, contain no information. Maximising - m_jlogm_j subject to the constraint that the map is consistent with the observed data, gives the MAXENT map.

The consistency with the observed data is described in terms of the difference between the observed data and those calculated from a trial map, weighted by the standard deviation of the measurement. If F_c, is the calculated value of the datum h, F_o, its observed value and (F) the standard deviation of the observation, then the statistic

possesses a distribution with an expected value equal to the number of data points. Maximising - m_jlogm_j subject to the constraint | F_c, - F_o,/(F)² = n, where n is the number of data points, gives the basic iteration formula :

m = exp{ - 1 + exp(2i)}

Given F_o,, (F) and an positive multiplier , this equation can determine the densities m on a map. The program GraphEnt applies this formula iteratively (starting from a uniform map) until convergence (as judged by the value of ) is achieved. Although this algorithm is neither the most efficient nor the most stable, it is relatively easy to code and it leads (at least in the case of Patterson functions), to the same results as other, more complex algorithms⁵.

Footnotes

... G.J.⁴

Gull, S.F. & Daniell, G.J., (1978), Nature, 272, 686-690.

... algorithms⁵

Skilling, J. & Bryan, R.K., (1984), Mon. Not. R. astr. Soc., 211, 111-124.