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##

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*_{j}log*m*_{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*_{j}log*m*_{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*_{j}log*m*_{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*_{j}log*m*_{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 :
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 algorithms^{5}.

#### Footnotes

- ... G.J.
^{4}
- Gull, S.F. & Daniell, G.J., (1978),
* Nature*, ** 272**,
686-690.
- ... algorithms
^{5}
- Skilling, J. & Bryan, R.K., (1984),
* Mon. Not. R. astr. Soc.*, ** 211**, 111-124.

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NMG, Nov 2002