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The
Fisher information
for the M-state distribution,
i.e. M-1 parameters,
T=<T1,...TM-1>,
define TM=1-T1-...-TM-1.
The Fisher, F(T), is the following determinant:
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N/T1+N/TM N/TM ... N/TM
N/TM N/T2+N/TM ... ...
N/TM N/TM ... ...
... ... ... ...
N/TM ... ... N/TM-1+N/TM
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Take a factor of N out of every row:
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= |
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1/T1+1/TM 1/TM ... 1/TM
1/TM 1/T2+1/TM ... 1/TM
1/TM 1/TM ...
...
1/TM 1/TM-1+1/TM
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.NM-1
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Subtract column one from the other rows:
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1/T1+1/TM -1/T1 -1/T1 ... -1/T1
1/TM 1/T2 0 0
1/TM 0 1/T3 ... 0
...
1/TM 0 0 ... 1/TM-1
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.NM-1
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For each row, take a factor of 1/Ti out of row i:
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1+T1/TM -1 -1 ... -1
T2/TM 1 0 ... 0
T3/TM 0 1 ... 0
...
TM-1/TM 0 0 1
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NM-1
.--------
T1...TM-1
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Now add row 2 and row 3 and ... and row M-1 to row 1
which removes the `-1's from row 1 and makes the
top left element into:
(1 + T1/TM + T2/TM + ... + TM-1/TM)
= (TM + T1 + T2 + ... + TM-1)/TM
= 1/TM
Consequently the Fisher is
NM-1/(T1...TM).
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