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T_HOMATE

2. Definitions

Table of contents

4. THOMATE

3. Backtracing

3.1 The theory

The model can be rejected if no distribution satisfies equation (2). That relation can be inverted:

(3)

where, by the Bayes formula

(4)

Using (2) to replace in (4), (3) becomes

(5)

which is of the type that can be solved by iterations using

(6)

To implement this method, one must discretize the possible values of a variable. The domains and are divided into a certain number of cells. If each source variable can take values and if each observable can take values, the dimensions of the corresponding spaces will be

	(7)
	(8)

In other words, once the source variables discretized, the domain is divided into -dimensional cells, to which one can attribute a number from 1 to . The value of the distribution in the cell is noted . The are defined similarly after discertization of the observables.

(2) and (3) are re-written as

	(9)
	(10)

The necessary ingredient for a backtracing analysis are hence:

, constructed directly from the experimental data.
, obtained from the model, or a previous survey. In the case of a model in nuclear physics, it can be obtained by running a Monte-Carlo simulation for each cell of the source space (). This simulation must include the physical process studied and the experimental filter (e.g. detectors efficiencies).

(6) can be re-written as

(11)

This method has been succesfully applied by several authors [4][5][6]

3.3 Notes on the solutions

By using this iterative procedure, the solution depends on the initial guess .
There is not always convergence towards a solution.
In practice, this methods needs a lot of statistics to give good results. As a matter of fact, it tries to reproduce every statistical fluctuation and this could make it fail although the model was valid.
Convergence here means that the Kullback-Leibner

is minimized. It is really this quantity that is used in the backtracing program "rcn" by P. D&eacut;sesquelles, although he gives the norm

in his paper [2]. The Kullback-Leibner is equivalent to the χ² only as the number of events tends to infinity.