### M-State (2)

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>Fisher>

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:

 ``` | | | | | | | ``` ```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 ``` ```| | | | | | | ```
Take a factor of N out of every row:
 ``` | | | = | | | | ``` ```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 ``` ```| | | | | | | ``` ```.NM-1 ```
Subtract column one from the other rows:
 ``` | | | = | | | | ``` ```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 ``` ```| | | | | | | ``` ```.NM-1 ```
For each row, take a factor of 1/Ti out of row i:
 ``` | | | = | | | | ``` ```1+T1/TM -1 -1 ... -1 T2/TM 1 0 ... 0 T3/TM 0 1 ... 0 ... TM-1/TM 0 0 1 ``` ```| | | | | | | ``` ``` 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).

 Coding Ockham's Razor, L. Allison, Springer A Practical Introduction to Denotational Semantics, L. Allison, CUP

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 © L. Allison   http://www.allisons.org/ll/   (or as otherwise indicated), Faculty of Information Technology (Clayton), Monash University, Australia 3800 (6/'05 was School of Computer Science and Software Engineering, Fac. Info. Tech., Monash University, was Department of Computer Science, Fac. Comp. & Info. Tech., '89 was Department of Computer Science, Fac. Sci., '68-'71 was Department of Information Science, Fac. Sci.) Created with "vi (Linux + Solaris)",  charset=iso-8859-1,  fetched Tuesday, 16-Jul-2024 12:52:59 AEST.