M-State (2)

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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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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
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
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
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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