Hex, Bugs and More Physics | Emre S. Tasci » Blog Archive » Likelihood of Gaussian(s)

Likelihood of Gaussian(s) – Scrap Notes

December 3, 2007 Posted by Emre S. Tasci

Given a set of N data x, $Formula: % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WG4baacaGL7bGaayzFaaWaa0baaSqaaiaadMgacqGH9aqpcaaIXaaa % baGaamOtaaaaaaa!3CCA! \[ {\left\{ x \right\}_{i = 1}^N } \]$ , the optimal parameters for a Gaussian Probability Distribution Function defined as:

$Formula: % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 % gadaWadaqaaiGaccfadaqadaqaamaaeiaabaWaaiWaaeaacaWG4baa % caGL7bGaayzFaaWaa0baaSqaaiaadMgacqGH9aqpcaaIXaaabaGaam % OtaaaaaOGaayjcSdGaeqiVd0Maaiilaiabeo8aZbGaayjkaiaawMca % aaGaay5waiaaw2faaiabg2da9iabgkHiTiaad6eaciGGSbGaaiOBam % aabmaabaWaaOaaaeaacaaIYaGaeqiWdahaleqaaOGaeq4WdmhacaGL % OaGaayzkaaGaeyOeI0YaaSGbaeaadaWadaqaaiaad6eadaqadaqaai % abeY7aTjabgkHiTiqadIhagaqeaaGaayjkaiaawMcaamaaCaaaleqa % baGaaGOmaaaakiabgUcaRiaadofaaiaawUfacaGLDbaaaeaacaaIYa % Gaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaaaaa!627A! \[ \ln \left[ {\operatorname{P} \left( {\left. {\left\{ x \right\}_{i = 1}^N } \right|\mu ,\sigma } \right)} \right] = - N\ln \left( {\sqrt {2\pi } \sigma } \right) - {{\left[ {N\left( {\mu - \bar x} \right)^2 + S} \right]} \mathord{\left/ {\vphantom {{\left[ {N\left( {\mu - \bar x} \right)^2 + S} \right]} {2\sigma ^2 }}} \right. \kern-\nulldelimiterspace} {2\sigma ^2 }} \]$

are:

with the definitions

Let’s see this in an example:

Define the data set mx:
mx={1,7,9,10,15}

Calculate the optimal mu and sigma:
dN=Length[mx];
mu=Sum[mx[[i]]/dN,{i,1,dN}];
sig =Sqrt[Sum[(mx[[i]]-mu)^2,{i,1,dN}]/dN];
Print["mu = ",N[mu]];
Print["sigma = ",N[sig]];

Now, let’s see this Gaussian Distribution Function:
<<Statistics`NormalDistribution`
ndist=NormalDistribution[mu,sig];

<<Graphics`MultipleListPlot`
MultipleListPlot[Table[{x,PDF[NormalDistribution[mu,sig],x]}, {x,0,20,.04}],Table[{mx[[i]], PDF[NormalDistribution[mu,sig],mx[[i]]]},{i,5}], {PlotRange->{Automatic,{0,.1}},PlotJoined->{False,False}, SymbolStyle->{GrayLevel[.8],GrayLevel[0]}}]