Fitting with Poisson random variables1
Recall that the normal fitting procedure is to choose parameters of a fitting function
F(xi;c) that will minimize the chi-square:
N
 
2
n
i
 F  xi ; c  
2
(46)
F  xi ; c 
The probability of seeing an entire Poisson data set is
i 1


N
P ni  ; F  xi  ; c    e
Define
 F  xi ;c 
ni !
i 1
2
 Poisson
 2ln P ni  ; F  xi  ; c  
F  xi ; c 
ni
(50)
(51)
So that maximizing Eq. (50) is equivalent to minimizing
 N

2
 Poisson
 2    F  xi ; c   ni ln  F  xi ; c    ln  ni ! 
(52)
 i 1

The term with ni! is independent of the fitting constants and can thus be dropped. Thus
minimizing 2Poisson is equivalent to minimizing
N
 P2 ,m  2 F  xi ; c   ni ln  F  xi ; c  
(53)
i 1
Bob Coldwell speculation
N F x ; c 

 P2 ,m
 i 
ni
 2
(54)
1 
0
cm
cm 
F  xi ; c  
i 1
This is slightly different from minimizing the usual 2 in Eq. (46)
2
2
N F x ; c  n  F  x ; c 
ni  F  xi ; c   


 i   i

i
0
 2

(55)
cm
cm  F  xi ; c 
F 2  xi ; c  
i 1


Simplify, but only a bit
2
2
N F x ; c 
ni  F  xi ; c   

 i 
ni

0
 2
1

(56)
cm
cm
F 2  xi ; c  
 F  xi ; c 
i 1


The second term coming from the derivative of the error in Eq (46) is wrong. – Careful –
it lowers the chi-square of (46) by maximizing i in
N
 
2
i 1
n
i
 F  xi ; c  
 i2
2
(57)
This is heavily based on Bob Deserio’s same named section in Deserio’s UFlab Poisson Statistics The
use of F(x,c) goes beyond the original.
1
Notice, that Eq (57) with i independent of c – but somehow equal to F does minimize for
the correct value of F. This says that the current method for a Poisson situation requires.
1.
Determine a best estimate for (xi).
2.
With (xi) fixed, determine F(xi,c) by minimizing Eq. (57)
3.
Re-determine (xi)  iterate.
This procedure will minimize Eq (57) with de-facto  i2  F  xi , c  , but at the end Eq (56)
will be
2
N F x ; c
 i   ni  F  xi ; c  

 2
cm
cm
F 2  xi ; c 
i 1
2
(58)
In the case that F = , this term is
2
2
N
 ni   

2
(59)
 2
 2 n2   2
2



i 1
This was shown in Derivation of the Poisson distribution.doc#SigmaP to be equal to 
implying that
2

2
(60)



So that

2 
2
  2 (61)
2


Using
2
2
N
 ni   

2
 2
 2
2



i 1

2
0
 2   

  m   
 m     
n
 2
2



2
 2
2  2   
 2   
2  2   
2

2


(62)

(63)

1

2
or
m   
1

This is not quite in agreement with Bob Deserio’s results of
Poisson = 2.9
Gauss = 2 Compare predicted 2.6 from the above.