The Solution.

To find the maximum we differentiate Equation (including Equation for $f$_i ) and set the derivatives to zero. This gives two sets of equations, those for the differentials with respect to $p$_j

$\sum$_i=1ⁿd_iA_jif_i- A_ji=0∀j

and those for the differentials with respect to $A$_ji

$d$_ip_jf_i- p_j+ a_jiA_ji-1=0∀i,j

These $m\; x(n+1)$ simultaneous equations are nonlinear and coupled (remembering that the $f$_i that appear in them are functions of the $p$_j and the $A$_ji ). However they can be remarkably simplified. Equations can be rewritten $1-d$_if_i=1 p_j(a_jiA_ji-1)∀i,j

The left hand side depends on i only, so write it as $t$_i . $t$_i= 1-d_if_i

The right hand side then becomes $A$_ji= a_ji1 + p_jt_i

which is a great simplification: for a given set of $p$_j , the $nxm$ unknown quantities $A$_ji are given by the n unknown quantities $t$_i .

The $t$_i are given by Equation . If $d$_i is zero then $t$_i is 1: if not then $d$_i1 - t_i= f_i= ∑_jp_jA_ji= ∑_jp_ja_ji1 + p_jt_i

If these n equations are satisfied, with Equation used to define the $A$_ji , then all the $m\; xn$

Equations are satisfied.

The method adopted by HMCLNL is (for a given set of $p$_j ), to solve equations for the $t$_i , thus giving the $A$_ji via equation . The log likelihood may then be calculated using equation . The maximum of the likelihood may then be found using numerical means - HMCMLL uses MINUIT to perform this maximisation (this is equivalent to solving equations ).

Although there are n equations , to be solved numerically, this does not present a problem. They are not coupled, and each equation clearly has one and only one solution in the `allowed' region for $t$_i , i.e. the region where the $A$_ji are all positive, which lies between $t=-1/p$_max and t=1 ($p$_max being the largest of the $p$_j ). t=0 is a suitable place to start, and Newton's method readily gives a solution. Special considerations apply when there are no events from one or more of the MC sources - more details of the solution can be found in [17].