M-State (2)

home1 home2
 Bib
 Algorithms
 Bioinfo
 FP
 Logic
 MML
 Prog.Lang
and the
 Book

MML
 Discrete
  <m-state(1)<
  m-state(2)
  >Integers>

  >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

Linux
 Ubuntu
free op. sys.
OpenOffice
free office suite
The GIMP
~ free photoshop
Firefox
web browser

© 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.