Penalized maximum likelihood for multivariate Gaussian mixture
Hichem Snoussi
} \def\eabs{
abrege.tex
% TEX 7(ascii) bits
%
% ABREGE.tex
% LaTeX document
% Author: J.I. 
% Date  : Juillet 1996
%
% Raccourcis d'expressions tr\`es usit\'ees
%
% Derniere modif le 10-07-96
%

% Raccourci pour \XS
\def\XS{\xspace}

%%%%%%%%%%%%%%%%%%% Pour definir nouvelle commande de changement de taille
\RequirePackage{ifthen}
\RequirePackage{calc}

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\ifthenelse{#1 = -5}{}{}%
\ifthenelse{#1 = -4}{\tiny}{}%
\ifthenelse{#1 = -3}{\scriptsize}{}%
\ifthenelse{#1 = -2}{\footnotesize}{}%
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\ifthenelse{#1 = 4}{\huge}{}%
\ifthenelse{#1 = 5}{\Huge}{}}

\newboolean{@serif}
\setboolean{@serif}{true}
\def\scad{-5} % scadefaultsize

\newcounter{taille}
\newcommand{\sca}[2][\scad]{\setcounter{taille}{#1}%
  \ifthenelse{\boolean{@serif}}
  {{\taille[\thetaille]\textsc{#2}}}
  {\setcounter{taille}{\value{taille}-1}{\uppercase{\taille[\thetaille]#2}}}}

% Mise en commentaire rapide !
\def\rem#1{}                    % rem{bla bla bla}

% References automatiquement entre parenthese
\def\reff#1{\XS(\ref{#1})}
\def\refs#1{section~\ref{#1}}
\def\refS#1{Section~\ref{#1}}

\newcommand{\clearemptydoublepage}{\newpage{\pagestyle{empty}\cleardoublepage}}  
% Num�ro (pour biblio)
\def\numero{n�}
\def\Numero{N�}

% Majuscules
\def\UP#1{\uppercase{#1}}       % majuscules
\def\up#1{\uppercase{#1}}       % majuscules

% Liste des abreges
\def\adhoc{\textit{ad hoc}\XS}
\def\apost{\textit{a posteriori}\XS}
\def\Apost{\textit{A posteriori}\XS}
\def\APost{\textit{A Posteriori}\XS}
\def\post{\textit{posterior}\XS}
\def\Post{\textit{Posterior}\XS}
\def\afortio{\textit{a fortiori}\XS}
\def\Afortio{\textit{A fortiori}\XS}
\def\aprio{\textit{a priori}\XS}
\def\Aprio{\textit{A priori}\XS}
\def\Cad{C'est-\`a-dire\XS}
\def\cad{c'est-\`a-dire\XS}
\def\coeur{c{\oe}ur\XS}
\def\cf{\textit{cf.}\XS}
\def\cv{\textit{curriculum vit\ae}\XS}
\def\Cv{\textit{Curriculum vit\ae}\XS}
\def\CV{\textit{Curriculum Vit\ae}\XS}
\def\eg{\textit{e.g.,}\XS}
\def\etal{\textit{et al.}\XS}
\def\etc{\textit{etc\dots}\XS}
\def\etcoll{{\textit et coll.}\XS}
\def\idem{\textit{idem}\XS}
\def\ie{\textit{i.e.,}\XS}
\def\IFF{\textit{if and only if}\XS}
\def\iid{i.i.d.\XS}
\def\infine{\textit{in fine}\XS}
\def\insit{\textit{in situ}\XS}
\def\inviv{\textit{in vivo}\XS}
\def\oeuvre{{\oe}uvre\XS}
\def\oeuvres{{\oe}uvres\XS}
\def\recto{\textit{recto}\XS}
\def\Toeplitz{T{\oe}plitz\XS}
\def\vav{vis-\`a-vis\XS}
\def\via{\textit{via}\XS}
\def\vic{\textit{vice-versa}\XS}
\def\verso{\textit{verso}\XS}
%\def\vitae{\textit{vit\ae}\XS}
\def\vs{\textit{vs}\XS}
\def\versus{\textit{versus}\XS}
\def\ssi{\textit{si et seulement si}\XS}
\def\strsen{\textit{stricto sensu}\XS}
\def\wrt{w.r.t.\XS}

\def\ier{$^{\text{\small er}}$\XS}
\def\iere{$^{\text{\small re}}$\XS}
\def\ieme{$^{\text{\small e}}$\XS}

\def\primo{\ensuremath{1^{\oD}}\XS}
\def\secundo{\ensuremath{2^{\oD}}\XS}
\def\tertio{\ensuremath{3^{\oD}}\XS}
\def\quatro{\ensuremath{4^{\oD}}\XS}


\newcommand{\art}[1][\scad]{\sca[#1]{art}\XS}
\newcommand{\cnd}[1][\scad]{\sca[#1]{cnd}\XS}
\newcommand{\ndt}[1][\scad]{\sca[#1]{ndt}\XS}
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\newcommand{\map}[1][\scad]{\sca[#1]{map}\XS}

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\newcommand{\dess}[1][\scad]{\sca[#1]{dess}\XS}
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\newcommand{\eea}[1][\scad]{\sca[#1]{eea}\XS}
\def\EEA{\'Electronique \'Electrotechnique Automatique\XS}
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\def\ATS{Automatique et Traitement du Signal\XS}

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\def\TE{travaux d'�tudes\XS}
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\def\TD{travaux dirig�s\XS}
\newcommand{\tp}[1][\scad]{\sca[#1]{tp}\XS}
\def\TP{travaux pratiques\XS}
%\newcommand{\tdao}[1][\scad]{\sca[#1]{tdao}\XS}
\def\TDAO{travaux dirig�s assist�s sur ordinateur\XS}
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\def\ts{traitement du signal\XS}
\def\TS{Traitement du Signal\XS}

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\newcommand{\meg}[1][\scad]{\sca[#1]{meg}\XS}

\newcommand{\pc}[1][\scad]{\sca[#1]{pc}\XS}
% Operating Systemes 
\newcommand{\linux}[1][\scad]{\sca[#1]{Linux}\XS}
\newcommand{\unix}[1][\scad]{\sca[#1]{Unix}\XS}
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\newcommand{\win}[1][\scad]{\sca[#1]{Windows}\XS}

% Calcul scientifique
\newcommand{\labview}[1][\scad]{\sca[#1]{Labview}\XS}
\newcommand{\matlab}[1][\scad]{\sca[#1]{Matlab}\XS}
\newcommand{\maple}[1][\scad]{\sca[#1]{Maple}\XS}
\newcommand{\mathe}[1][\scad]{\sca[#1]{Mathematica}\XS}
\newcommand{\idl}[1][\scad]{\sca[#1]{Idl}\XS}

% langages
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\newcommand{\pascal}[1][\scad]{\sca[#1]{Pascal}\XS}
\newcommand{\java}[1][\scad]{\sca[#1]{Java}\XS}
\newcommand{\lisp}[1][\scad]{\sca[#1]{Lisp}\XS}
\newcommand{\forts}[1][\scad]{\sca[#1]{Fortran}{\footnotesize 77}\XS}
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\newcommand{\tcsh}[1][\scad]{\sca[#1]{tcsh}\XS}
\newcommand{\bash}[1][\scad]{\sca[#1]{bash}\XS}
\newcommand{\lgc}[1][\scad]{\sca[#1]{C}\XS}
\newcommand{\cpp}[1][\scad]{\sca[#1]{C}{\footnotesize ++}\XS}

\def\crec{charg� de recherches\XS}
\def\Crec{Charg� de recherches\XS}
\def\dr{directeur de recherches\XS}
\def\Dr{Directeur de recherches\XS}


% congres
\newcommand{\gretsi}[1][\scad]{\sca[#1]{gretsi}\XS}
\newcommand{\maxent}[1][\scad]{\sca[#1]{maxent}\XS}
\newcommand{\icip}[1][\scad]{\sca[#1]{icip}\XS}
\newcommand{\icassp}[1][\scad]{\sca[#1]{icassp}\XS}
\newcommand{\spie}[1][\scad]{\sca[#1]{spie}\XS}
\newcommand{\gdr}[1][\scad]{\sca[#1]{gdr--prc~isis}\XS}
\newcommand{\rfia}[1][\scad]{\sca[#1]{rfia}\XS}
\newcommand{\eusipco}[1][\scad]{\sca[#1]{eusipco}\XS}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ajout de famouss.tex
% modif le 25/5/2000 Phiphi
% GPInuts - Famous people - VIP
%
\newcommand{\beaunom}[4][0]{{\taille[#1]#2 \uppercase{#3}}%
{\sca[#1]{#4}}\XS}

%\newcommand{\MA}[1][\scad]{{\taille[#1]Marc \uppercase{A}}% 
%{\sca[#1]{llain}\XS}}
\newcommand{\MA}[1][\scad]{\beaunom[#1]{Marc}{A}{llain}}
\newcommand{\AB}[1][\scad]{\beaunom[#1]{Amandine}{B}{lanc}}
\newcommand{\ABF}[1][\scad]{\beaunom[#1]{Amandine}{B}{lanc~--~Fusco}}
\newcommand{\AH}[1][\scad]{\beaunom[#1]{Alain}{H}{erment}}
\newcommand{\AMD}[1][\scad]{\beaunom[#1]{Ali}{M}{ohammad--Djafari}}
\newcommand{\AR}[1][\scad]{\beaunom[#1]{Andrea}{R}{idolfi}}
\newcommand{\CH}[1][\scad]{\beaunom[#1]{Christian}{H}{einrich}}
\newcommand{\CS}[1][\scad]{\beaunom[#1]{Charles}{S}{oussen}}
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\newcommand{\FC}[1][\scad]{\beaunom[#1]{Fr\'ed\'eric}{C}{hampagnat}}
\newcommand{\FL}[1][\scad]{\beaunom[#1]{Franck}{L}{etimonier}}
\newcommand{\FV}[1][\scad]{\beaunom[#1]{Francisco}{V}{arela}}
\newcommand{\FDU}[1][\scad]{\beaunom[#1]{Fr\'ed\'eric}{D}{ublanchet}}
\newcommand{\GDE}[1][\scad]{\beaunom[#1]{Guy}{D}{emoment}}
\newcommand{\GLB}[1][\scad]{\beaunom[#1]{Guy Le}{B}{esnerais}}
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\newcommand{\JFM}[1][\scad]{\beaunom[#1]{Jean--Fran\c{c}ois}{M}{angin}}
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\newcommand{\JBP}[1][\scad]{\beaunom[#1]{Jean-Baptiste}{P}{oline}}
\newcommand{\MF}[1][\scad]{\beaunom[#1]{Marc}{F}{ayolle}}
\newcommand{\Mimi}[1][\scad]{\beaunom[#1]{Myriam}{F}{iani}}
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\newcommand{\PC}[1][\scad]{\beaunom[#1]{Philippe}{C}{iuciu}}
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\newcommand{\HS}[1][\scad]{\beaunom[#1]{Hichem}{S}{noussi}}

% Acronymes et raccourcis
%
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\def\SHFJ{Service Hospitalier Fr�d�ric Joliot\XS}
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\def\UNAF{Unit\'e de Neuroimagerie Anatomo Foncxtionnelle\XS}
\newcommand{\unaf}[1][\scad]{\sca[#1]{unaf}\XS}


\newcommand{\chu}[1][\scad]{\sca[#1]{chu}\XS}
\def\CHU{Centre Hospitalier Universitaire\XS}
\def\salpe{Salp�tri�re\XS}
\def\hsalpe{H�pital de la Salp�tri�re\XS}
\def\pisalpe{Piti�-Salp�tri�re\XS}
\def\gpisalpe{Groupe Piti�-Salp�tri�re\XS}

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\newcommand{\ensea}[1][\scad]{\sca[#1]{ensea}\XS}


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\def\IPSN{Institut de Protection et de Suret� Nucl�aire\XS}

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\def\MDCs{Ma\^\i tres de conf\'erences\XS}
\newcommand{\mdc}[1][\scad]{\sca[#1]{mdc}\XS}

\newcommand{\UND}[1][\scad]{Universit� de Notre Dame, \sca[#1]{usa}\XS}

\def\UPS{Universit\'e de Paris-Sud\XS}
\newcommand{\ups}[1][\scad]{\sca[#1]{ups}\XS}
\newcommand{\upxi}[1][\scad]{\sca[#1]{upxi}\XS}

% Unites Thomson
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% Unite propre de recherche
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% LENA, lena upr640,...
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\def\LENA{Laboratoire d'Enc�phalographie et de Neurophysiologie Appliqu�e\XS}
\newcommand{\lena}[1][\scad]{\sca[#1]{lena}\XS}
% Nous
\def\GsY{Gif--sur--Yvette\XS}
\newcommand{\ESE}[1][\scad]{\sca[#1]{sup\'elec}\XS}

\def\adrshort{\ESE, Plateau de Moulon, 91192 Gif--sur--Yvette Cedex\XS}
%\def\adresse{\ESE, Plateau de Moulon, 91192 Gif--sur--Yvette Cedex, France\XS}
\newcommand{\hiercea}[1][\scad]{\dsv[#1]/\drm[#1]/\shfj[#1]\XS}

\newcommand{\adresse}[1][\scad]{\ESE[#1], Plateau de Moulon, 91192 Gif--sur--Yvette Cedex, France\XS}
\newcommand{\adresscea}[1][\scad]{4 Place du G�n�ral Leclerc, 91406 Orsay Cedex, France\XS}
\newcommand{\tutelle}[1][\scad]{\cnrs[#1]~--~\ESE[#1]~--~\ups[#1]}
%\newcommand{\tutelle}[1][\scad]{\sca[#1]{Cnrs~--~Sup\'elec~--~Ups}}
\def\ALSS{\LSS~(\tutelle)\\\adresse}
\def\ALSSun{\LSS (\tutelle) \adresse}
%


% ---------------------------------------------------- 
% Modele d'en-tete par A. Mohammad-Djafari et J. Idier 
% ---------------------------------------------------- 
%
\def\helvnormal#1{{\fontfamily{phv}\fontsize{11}{14pt}\selectfont #1}}
\def\helvfoot  #1{{\fontfamily{phv}\fontsize{9}{14pt} \selectfont #1}}
\def\helvscript#1{{\fontfamily{phv}\fontsize{8}{14pt} \selectfont #1}}
%
\newcommand{\entete}[5][Nom.Pr\'enom]{
%%%%%%\def\entete#1#2#3#4#5{%
\helvscript{\renewcommand{\arraystretch}{.8}
\begin{tabular}{c} Sup�lec
\\--------\end{tabular} \hfill
\begin{tabular}{c}Centre national de la recherche scientifique          \\
--------\end{tabular} \hfill
\begin{tabular}{c}\UPS  \\--------\end{tabular}} \\[-.4cm]%
%
\begin{center}\helvnormal{\textbf{LABORATOIRE DES SIGNAUX \& SYST\`EMES}}\\%
\helvfoot{Unit\'e mixte de recherche n$^{\rm o}$ 8506\\%
Sup\'elec, plateau de Moulon, 
3 rue Joliot-Curie, 91192 \sca[-3]{Gif--sur--Yvette} Cedex (France)\\%
T\'el\'ephone\,: 01 69 85 17 12 --- 
T\'el\'ecopie\,: 01 69 85 17 65 --- 
Courriel\,: {#1}@lss.supelec.fr}%
\\[0.5cm]\end{center}% 
\begin{tabular}{p{8.5cm}l}%
\helvscript{V/R\'ef.\,: #2} & \helvscript{\sca[-3]{Gif}, le} #5\\%
\helvscript{N/R\'ef.\,: #3} \\%
\helvscript{Objet\,: #4}%
\end{tabular}
} 

\newcommand{\inhead}[5][Name.Surname]{
%%%%%%\def\entete#1#2#3#4#5{%
\helvscript{\renewcommand{\arraystretch}{.8}
\begin{tabular}{c} Sup�lec
\\--------\end{tabular} \hfill
\begin{tabular}{c}Centre national de la recherche scientifique  \\
--------\end{tabular} \hfill
\begin{tabular}{c}\UPS  \\--------\end{tabular}} \\[-.4cm]%
%
\begin{center}\helvnormal{\textbf{LABORATOIRE DES SIGNAUX \& SYST\`EMES}}\\%
\helvfoot{Unit\'e mixte de recherche n$^{\rm o}$ 8506\\%
Sup\'elec, plateau de Moulon, 
3 rue Joliot-Curie, 91192 \sca[-3]{Gif--sur--Yvette} Cedex (France)\\%
Telephone: 01 69 85 17 12 --- 
Fax: 01 69 85 17 65 --- 
E-mail: {#1}@lss.supelec.fr}%
\\[0.5cm]\end{center}% 
\begin{tabular}{p{8.5cm}l}%
\helvscript{Y/Ref.: #2} & \helvscript{\sca[-3]{Gif}, } #5\\%
\helvscript{O/Ref.: #3} \\%
\helvscript{Subject: #4}%
\end{tabular}
} 


abrmath.tex
% TEX 7(ascii) bits
%
% ABRMATH.tex           LaTeX document
% Author: J.I.          Date  : Juillet 1996
% Raccourcis d'expressions tres usitees en math
% Derniere modif importante le 12-08-96


%-- Accolades, parenth�ses, etc --------------------------------------
        
\def\pth#1{\left(#1\right)}                \def\stdpth#1{(#1)}
\def\acc#1{\left\{#1\right\}}              \def\stdacc#1{\{#1\}}
\def\cro#1{\left[#1\right]}                \def\stdcro#1{[#1]}
\def\bars#1{\left|#1\right|}               \def\stdbars#1{|#1|}
\def\norm#1{\left\|#1\right\|}             \def\stdnorm#1{\|#1\|}
\def\scal#1{\left\langle#1\right\rangle}   \def\stdscal#1{\langle#1\rangle}
 
\def\bigpth#1{\bigl(#1\bigr)}              \def\biggpth#1{\biggl(#1\biggr)}
\def\bigacc#1{\bigl\{#1\bigr\}}            \def\biggacc#1{\biggl\{#1\biggr\}}
\def\bigcro#1{\bigl[#1\bigr]}              \def\biggcro#1{\biggl[#1\biggr]}
\def\bigbars#1{\bigl|#1\bigr|}             \def\biggbars#1{\biggl|#1\biggr|}
\def\bignorm#1{\bigl\|#1\bigr\|}           \def\biggnorm#1{\biggl\|#1\biggr\|}
\def\bigscal#1{\bigl\langle#1\bigr\rangle} \def\biggscal#1{\biggl\langle#1\biggr\rangle}

\def\Bigpth#1{\Bigl(#1\Bigr)}              \def\Biggpth#1{\Biggl(#1\Biggr)}
\def\Bigacc#1{\Bigl\{#1\Bigr\}}            \def\Biggacc#1{\Biggl\{#1\Biggr\}}
\def\Bigcro#1{\Bigl[#1\Bigr]}              \def\Biggcro#1{\Biggl[#1\Biggr]}
\def\Bigbars#1{\Bigl|#1\Bigr|}             \def\Biggbars#1{\Biggl|#1\Biggr|}
\def\Bignorm#1{\Bigl\|#1\Bigr\|}           \def\Biggnorm#1{\Biggl\|#1\Biggr\|}
\def\Bigscal#1{\Bigl\langle#1\Bigr\rangle} \def\Biggscal#1{\Biggl\langle#1\Biggr\rangle}


% Quelques fonctions classiques + Arguments entre [] ----------------
%
% Laisser les \mathrm entre {}, sinon �a d�conne dans le style "ieeetran". 
%
\def\diag{{\mathrm{diag}}}              \def\Diag#1{{\mathrm{diag}}\bigcro{#1}} \def\Diagold#1{{\mathrm{diag}}\cro{#1}}
\def\tr{{\mathrm{tr}}\,}                \def\Tr#1{{\mathrm{tr}}\bigcro{#1}}                     \def\Trold#1{{\mathrm{tr}}\cro{#1}}
\def\rg{{\mathrm{rg}}\,}                \def\Rg#1{{\mathrm{rg}}\bigcro{#1}}                     \def\Rgold#1{{\mathrm{rg}}\cro{#1}}
\def\esp{{\mathrm{E}}\,}                \def\Esp#1{{\mathrm{E}}\bigcro{#1}}                     \def\Espold#1{{\mathrm{E}}\cro{#1}} 
\def\var{{\mathrm{var}}\,}              \def\Var#1{{\mathrm{var}}\bigcro{#1}}           \def\Varold#1{{\mathrm{var}}\cro{#1}}
\def\cov{{\mathrm{Cov}}\,}              \def\Cov#1{{\mathrm{Cov}}\bigcro{#1}}           \def\Covold#1{{\mathrm{Cov}}\cro{#1}}

\def\Cos#1{\cos\cro{#1}}                %\def\Cosold#1{\cos\cro{#1}}
\def\Sin#1{\sin\cro{#1}}                %\def\Sinold#1{\sin\cro{#1}}
\def\Exp#1{\exp\cro{#1}}                %\def\Expold#1{\exp\cro{#1}}
\def\Log#1{\log\cro{#1}}                %\def\Logold#1{\log\cro{#1}}
\def\Ln#1{\ln\cro{#1}}                  
\def\Det#1{\det\bigcro{#1}}     %\def\Detold#1{\det\cro{#1}}

\def\cond{\textrm{Cond}\,}              \def\Cond#1{\cond\bigpth{#1}}

\def\sinc{{\mathrm{sinc}}\,}    \def\Sinc#1{{\mathrm{sinc}}\bigcro{#1}} \def\Sincold#1{{\mathrm{sinc}}\cro{#1}}
\def\rang{{\mathrm{rang}}\,}    \def\Rang#1{\rang\bigcro{#1}}                                   \def\Rangold#1{\rang\cro{#1}}
\def\ker{\textrm{Ker}\,}                \def\Ker#1{\ker\bigcro{#1}}                                     \def\Kerold#1{\ker\cro{#1}}
\def\img{\textrm{Im}\,}                 \def\Img#1{\img\bigcro{#1}}                                     \def\Imgold#1{\img\cro{#1}}
\def\vect{{\mathrm{Vect}}\,}    \def\Vect#1{\vect\bigcro{#1}}                                   \def\Vectold#1{\vect\cro{#1}}
\def\sgn{{\mathrm{sgn}}}                \def\Sgn#1{\sgn\bigcro{#1}}                                     \def\Sgnold#1{\sgn\cro{#1}}

% Signe proba
\def\Pr{\mathop{\textrm{Pr}}}

% Variante pour \Re:
\def\reel{{\mathrm{Re}}}                \def\Reel#1{{\mathrm{Re}}\cro{#1}}
\def\card{{\mathrm{Card}}}              \def\Card#1{{\mathrm{Card}}\cro{#1}}

\def\sh{{\mathrm{sh}}}                % sin hyperbolique en FR
\def\ch{{\mathrm{ch}}}                % cos hyperbolique en FR
\def\th{{\mathrm{th}}}                % th hyperbolique en FR
\def\coth{{\mathrm{coth}}}            % cot hyperbolique en FR
\def\coeffbin#1#2{\pth{\setlength{\arraycolsep}{.1em}\barr{c}#1\\#2\earr}}
% Remarque : \exist \binom{}{}

% Op�rateurs sur champs
\def\Div{{\mathrm{div}}}                                                                                        % divergence
\def\Rotv{\overrightarrow{\mathop{{\mathrm{rot}}}}}             % rotationnel avec fleche
\def\Gradv{\overrightarrow{\mathop{{\mathrm{grad}}}}}           % gradient avec fleche
%-- Textes (if, si,...) droit en math ---------------------------

\def\IF{\text{if\:}}             \def\SI{\text{si\:}}
\def\If{\text{If\:}}             \def\Si{\text{Si\:}}
\def\AND{\text{and\:}}           \def\ET{\text{et\:}}
\def\OR{\text{or\:}}             \def\OU{\text{ou\:}}
\def\THEN{\text{then\:}}         \def\ALORS{\text{alors\:}}
                                 \def\DOU{\text{d'o\`u\:}}
\def\WHERE{\text{where\:}}       \def\Ou{\text{o\`u\:}}
\def\WHEN{\text{when\:}}         \def\QUAND{\text{quand\:}}
\def\FOR{\text{for\:}}           \def\POUR{\text{pour\:}}
\def\FORALL{\text{for all\:}}    \def\POURTOUT{\text{pour tout\:}}
\def\ST{\text{s.t.\:}}           \def\SC{\text{s.c.\:}}
\def\SUBJTO{\text{subject to\:}} \def\SOUSC{\text{sous contraintes\:}}
\def\OTHERWISE{\text{otherwise}} \def\SINON{\text{sinon}}
\def\WITH{\text{with\:}}         \def\AVEC{\text{avec\:}}
\def\IN{\text{in\:}}             \def\DANS{\text{dans\:}}

%-- Triture param�tres tableaux... -----------------------------

\def\arrayp{\renewcommand{\arraystretch}{.7}\setlength{\arraycolsep}{2pt}}
\def\tabp{\renewcommand{\arraystretch}{.7}\setlength{\tabcolsep}{2pt}}

%-- Environnement encadr\'e ------------------------------------
% Ces d\'efinitons introduisent un nouvel environnement appele "fminipage". Il 
% permet d'encadrer une partie de texte.

\newsavebox{\fminibox}
\newlength{\fminilength}
\newenvironment{fminipage}[1][\linewidth]
  {\setlength{\fminilength}{#1}%-2\fboxsep-2\fboxrule}%
   \begin{lrbox}{\fminibox}\begin{minipage}{\fminilength}}
  {\end{minipage}\end{lrbox}\noindent\fbox{\usebox{\fminibox}}}

%-- Divers : "^{-1}", "dag", "," ... %---------------------------

\def\M{^{-1}} \def\T{^\tD} \def\+{^\dagger}
\def\I{\,|\,}           % "sachant" bien espac\'e pour les formules
\def\J{\mathop{\,;\,}}  % "point virgule" bien espac\'e pour les formules.
\def\w{,\thinspace}
\def\W{,\thickspace}
\def\ldotsv{,\,\ldots,\,}
\def\V{,}               % Virgule pour les nombres decimaux
%\def\V{.}              % Point pour les anglo-saxons
\def\e#1{.10^{#1}}      % Notation scientifique a la francaise
%\def\e#1{\rm{e}^{#1}}  % Notation scientifique anglo-saxonne

\def\egdef{\stackrel{\Delta}{=}}
\def\nequiv{\not\kern-.05em\equiv}
\def\egal{\kern-.5em=\kern-.5em}        % Moins d'espace autour de "="
\def\propt{\kern-.2em\propto\kern-.2em} % Idem
\def\dans{\in\!}                        % Trop d'espace apres "appartient a"
\def\pourtt{\forall\,}                  % Pas assez d'espace
\def\wh#1{\widehat{#1}}                 % Sombrero !
\def\wt#1{\widetilde{#1}} 

%\def\arg{\mathop{\mathrm{arg}}}
\def\argmax{\mathop{\mathrm{arg\,max}}} % Mieux que \def\argmax{\arg\max}
\def\argmin{\mathop{\mathrm{arg\,min}}} % car l'indice est reparti
\def\EL{\mathrm{EL}}
\def\Vect{\mathop{\text{Vect}}}

\def\froc#1#2{{#1/#2}}                  % Frac en toc
\def\fric#1#2{\frac1{#2}#1}
\def\fracds#1#2{\frac{\displaystyle#1}{\displaystyle#2}}
\def\diff#1#2{{\frac{d#1}{d#2}}}
\def\dd{\,d}                            % doit etre en italique en anglais
%\def\dd{\,\dD}                         % doit etre droit en francais
\def\derpar#1#2{{\frac{\partial #1}{\partial #2}}}
\def\derpor#1#2{{\froc{\partial #1}{\partial #2}}}
\def\parsec#1#2#3{{\frac{\partial^2 #1}{\partial #2\,\partial #3}}}
\def\parsecd#1#2{{\frac{\partial^2 #1}{{\partial #2}^2}}}
\def\porsec#1#2#3{{\froc{\partial^2 #1}{\partial #2\,\partial #3}}}
\def\porsecd#1#2{{\froc{\partial^2 #1}{{\partial #2}^2}}}
\def\pornd#1#2#3{{\froc{\partial^{#3} #1}{{\partial #2}^{#3}}}}
\def\intdouble{\int\kern-0.3em\int}
\def\inttriple{\int\kern-0.3em\int\kern-0.3em\int}
\def\prods{\mathop{\text{\footnotesize$\displaystyle\prod$}}}

% Pour mettre un \No{} au-dessus d'une lettre (\TM, 02/97) (package{amsmath.sty})
\def\rond#1{\overset{\kern-0.33em~_\circ}{#1}}
\def\rondit[#1]#2{\overset{\kern#1~_\circ}{#2}}

% Pour mettre un texte dans un rond : (utilise pstricks ; \JI, 08/97)
\def\incirc#1{\pscirclebox[framesep=1pt]{\scriptsize#1}}
\def\incircp#1{\pscirclebox[framesep=1pt]{\tiny#1}}
\def\outcirc#1{\raisebox{-3pt}{\huge$\times$}\hspace*{-.45cm}\text{\incirc{#1}}}




alphabet.tex
% Document LaTeX        Auteur : Guy Le Besnerais       Date : 25/05/94
% Modification : J.-F. Gio      (15/09/95)
%                               J.I. (format Latex 2e, taille adaptative 22/03/99)
%                                    (fonte blackboard 27/08/99)

% Objet : Definitions compactes d'alphabets pour ecriture en Math
% b pour bold italique : \Ab
% v pour bold droit : \Av
% D pour droit : \AD
% c pour cursif : \Ac
% bb pour blackboard : \Abb
% Capitale pour capitale : \Lambda

\typeout{\space}
\typeout{\space\space\space\space Fichier 'alphabet.tex' -- LSS 1994/95/99}
\typeout{\space\space\space\space (necessite les packages xspace, amsmath, bbm)}
\typeout{\space}

\RequirePackage{amsmath}
\RequirePackage{xspace}
\RequirePackage{bbm}

%-------------------------------------------

% DEFINITIONS PRELIMINAIRES 

\def\XS{\xspace}
\DeclareMathAlphabet{\mathb}{OML}{cmm}{b}{it}
\def\sbm#1{\ensuremath{\mathb{#1}}}                % Style gras italique (necessite amsmath)           
%
\def\sbmm#1{\ensuremath{\boldsymbol{#1}}}          % Style gras italique (necessite amsmath)           
\def\sdm#1{\ensuremath{\mathrm{#1}}}               % Style droit en math
\def\sbv#1{\ensuremath{\mathbf{#1}}}               % Style gras droit
\def\scu#1{\ensuremath{\mathcal{#1\XS}}}           % Style cursif
\def\scb#1{\ensuremath{\boldsymbol{\mathcal{#1}}}} % Style gras cursif
\def\sbl#1{\ensuremath{\mathbbm{#1}}}              % Style blackboard (necessite bbm)

%-------------------------------------------

% ALPHABET GRAS ITALIQUE, taille adaptative

\def\Ab{{\sbm{A}}\XS}  \def\ab{{\sbm{a}}\XS}
\def\Bb{{\sbm{B}}\XS}  \def\bb{{\sbm{b}}\XS}
\def\Cb{{\sbm{C}}\XS}  \def\cb{{\sbm{c}}\XS}
\def\Db{{\sbm{D}}\XS}  \def\db{{\sbm{d}}\XS}
\def\Eb{{\sbm{E}}\XS}  \def\eb{{\sbm{e}}\XS}
\def\Fb{{\sbm{F}}\XS}  \def\fb{{\sbm{f}}\XS}
\def\Gb{{\sbm{G}}\XS}  \def\gb{{\sbm{g}}\XS}
\def\Hb{{\sbm{H}}\XS}  \def\hb{{\sbm{h}}\XS}
\def\Ib{{\sbm{I}}\XS}  \def\ib{{\sbm{i}}\XS}
\def\Jb{{\sbm{J}}\XS}  \def\jb{{\sbm{j}}\XS}
\def\Kb{{\sbm{K}}\XS}  \def\kb{{\sbm{k}}\XS}
\def\Lb{{\sbm{L}}\XS}  \def\lb{{\sbm{l}}\XS}
\def\Mb{{\sbm{M}}\XS}  \def\mb{{\sbm{m}}\XS}
\def\Nb{{\sbm{N}}\XS}  \def\nb{{\sbm{n}}\XS}
\def\Ob{{\sbm{O}}\XS}  \def\ob{{\sbm{o}}\XS}
\def\Pb{{\sbm{P}}\XS}  \def\pb{{\sbm{p}}\XS}
\def\Qb{{\sbm{Q}}\XS}  \def\qb{{\sbm{q}}\XS}
\def\Rb{{\sbm{R}}\XS}  \def\rb{{\sbm{r}}\XS}
\def\Sb{{\sbm{S}}\XS}  \def\sb{{\sbm{s}}\XS}
\def\Tb{{\sbm{T}}\XS}  \def\tb{{\sbm{t}}\XS}
\def\Ub{{\sbm{U}}\XS}  \def\ub{{\sbm{u}}\XS}
\def\Vb{{\sbm{V}}\XS}  \def\vb{{\sbm{v}}\XS}
\def\Wb{{\sbm{W}}\XS}  \def\wb{{\sbm{w}}\XS}
\def\Xb{{\sbm{X}}\XS}  \def\xb{{\sbm{x}}\XS}
\def\Yb{{\sbm{Y}}\XS}  \def\yb{{\sbm{y}}\XS}
\def\Zb{{\sbm{Z}}\XS}  \def\zb{{\sbm{z}}\XS}


% ALPHABET CURSIF MAJUSCULE, taille adaptative
% Cursif,               % Cursif gras

\def\Ac{{\scu{A}}\XS}   \def\Acb{{\scb{A}}\XS}
\def\Bc{{\scu{B}}\XS}   \def\Bcb{{\scb{B}}\XS}
\def\Cc{{\scu{C}}\XS}   \def\Ccb{{\scb{C}}\XS}
\def\Dc{{\scu{D}}\XS}   \def\Dcb{{\scb{D}}\XS}
\def\Ec{{\scu{E}}\XS}   \def\Ecb{{\scb{E}}\XS}
\def\Fc{{\scu{F}}\XS}   \def\Fcb{{\scb{F}}\XS}
\def\Gc{{\scu{G}}\XS}   \def\Gcb{{\scb{G}}\XS}
\def\Hc{{\scu{H}}\XS}   \def\Hcb{{\scb{H}}\XS}
\def\Ic{{\scu{I}}\XS}   \def\Icb{{\scb{I}}\XS}
\def\Jc{{\scu{J}}\XS}   \def\Jcb{{\scb{J}}\XS}
\def\Kc{{\scu{K}}\XS}   \def\Kcb{{\scb{K}}\XS}
\def\Lc{{\scu{L}}\XS}   \def\Lcb{{\scb{L}}\XS}
\def\Mc{{\scu{M}}\XS}   \def\Mcb{{\scb{M}}\XS}
\def\Nc{{\scu{N}}\XS}   \def\Ncb{{\scb{N}}\XS}
\def\Oc{{\scu{O}}\XS}   \def\Ocb{{\scb{O}}\XS}
\def\Pc{{\scu{P}}\XS}   \def\Pcb{{\scb{P}}\XS}
\def\Qc{{\scu{Q}}\XS}   \def\Qcb{{\scb{Q}}\XS}
\def\Rc{{\scu{R}}\XS}   \def\Rcb{{\scb{R}}\XS}
\def\Sc{{\scu{S}}\XS}   \def\Scb{{\scb{S}}\XS}
\def\Tc{{\scu{T}}\XS}   \def\Tcb{{\scb{T}}\XS}
\def\Uc{{\scu{U}}\XS}   \def\Ucb{{\scb{U}}\XS}
\def\Vc{{\scu{V}}\XS}   \def\Vcb{{\scb{V}}\XS}
\def\Wc{{\scu{W}}\XS}   \def\Wcb{{\scb{W}}\XS}
\def\Xc{{\scu{X}}\XS}   \def\Xcb{{\scb{X}}\XS}
\def\Yc{{\scu{Y}}\XS}   \def\Ycb{{\scb{Y}}\XS}
\def\Zc{{\scu{Z}}\XS}   \def\Zcb{{\scb{Z}}\XS}


% ALPHABET DROIT EN MATHS, taille adaptative

\def\AD{{\sdm{A}}\XS}  \def\aD{{\sdm{a}}\XS}
\def\BD{{\sdm{B}}\XS}  \def\bD{{\sdm{b}}\XS}
\def\CD{{\sdm{C}}\XS}  \def\cD{{\sdm{c}}\XS}
\def\DD{{\sdm{D}}\XS}  \def\dD{{\sdm{d}}\XS}
\def\ED{{\sdm{E}}\XS}  \def\eD{{\sdm{e}}\XS}
\def\FD{{\sdm{F}}\XS}  \def\fD{{\sdm{f}}\XS}
\def\GD{{\sdm{G}}\XS}  \def\gD{{\sdm{g}}\XS}
\def\HD{{\sdm{H}}\XS}  \def\hD{{\sdm{h}}\XS}
\def\ID{{\sdm{I}}\XS}  \def\iD{{\sdm{i}}\XS}
\def\JD{{\sdm{J}}\XS}  \def\jD{{\sdm{j}}\XS}
\def\KD{{\sdm{K}}\XS}  \def\kD{{\sdm{k}}\XS}
\def\LD{{\sdm{L}}\XS}  \def\lD{{\sdm{l}}\XS}
\def\MD{{\sdm{M}}\XS}  \def\mD{{\sdm{m}}\XS}
\def\ND{{\sdm{N}}\XS}  \def\nD{{\sdm{n}}\XS}
\def\OD{{\sdm{O}}\XS}  \def\oD{{\sdm{o}}\XS}
\def\PD{{\sdm{P}}\XS}  \def\pD{{\sdm{p}}\XS}
\def\QD{{\sdm{Q}}\XS}  \def\qD{{\sdm{q}}\XS}
\def\RD{{\sdm{R}}\XS}  \def\rD{{\sdm{r}}\XS}
\def\SD{{\sdm{S}}\XS}  \def\sD{{\sdm{s}}\XS}
\def\TD{{\sdm{T}}\XS}  \def\tD{{\sdm{t}}\XS}
\def\UD{{\sdm{U}}\XS}  \def\uD{{\sdm{u}}\XS}
\def\VD{{\sdm{V}}\XS}  \def\vD{{\sdm{D}}\XS}
\def\WD{{\sdm{W}}\XS}  \def\wD{{\sdm{w}}\XS}
\def\XD{{\sdm{X}}\XS}  \def\xD{{\sdm{x}}\XS}
\def\YD{{\sdm{Y}}\XS}  \def\yD{{\sdm{y}}\XS}
\def\ZD{{\sdm{Z}}\XS}  \def\zD{{\sdm{z}}\XS}


% ALPHABET GRAS DROIT, taille adaptative

\def\Av{{\sbv{A}}\XS}  \def\av{{\sbv{a}}\XS}
\def\Bv{{\sbv{B}}\XS}  \def\bv{{\sbv{b}}\XS}
\def\Cv{{\sbv{C}}\XS}  \def\cv{{\sbv{c}}\XS}
\def\Dv{{\sbv{D}}\XS}  \def\dv{{\sbv{d}}\XS}
\def\Ev{{\sbv{E}}\XS}  \def\ev{{\sbv{e}}\XS}
\def\Fv{{\sbv{F}}\XS}  \def\fv{{\sbv{f}}\XS}
\def\Gv{{\sbv{G}}\XS}  \def\gv{{\sbv{g}}\XS}
\def\Hv{{\sbv{H}}\XS}  \def\hv{{\sbv{h}}\XS}
\def\Iv{{\sbv{I}}\XS}  \def\iv{{\sbv{i}}\XS}
\def\Jv{{\sbv{J}}\XS}  \def\jv{{\sbv{j}}\XS}
\def\Kv{{\sbv{K}}\XS}  \def\kv{{\sbv{k}}\XS}
\def\Lv{{\sbv{L}}\XS}  \def\lv{{\sbv{l}}\XS}
\def\Mv{{\sbv{M}}\XS}  \def\mv{{\sbv{m}}\XS}
\def\Nv{{\sbv{N}}\XS}  \def\nv{{\sbv{n}}\XS}
\def\Ov{{\sbv{O}}\XS}  \def\ov{{\sbv{o}}\XS}
\def\Pv{{\sbv{P}}\XS}  \def\pv{{\sbv{p}}\XS}
\def\Qv{{\sbv{Q}}\XS}  \def\qv{{\sbv{q}}\XS}
\def\Rv{{\sbv{R}}\XS}  \def\rv{{\sbv{r}}\XS}
\def\Sv{{\sbv{S}}\XS}  \def\sv{{\sbv{s}}\XS}
\def\Tv{{\sbv{T}}\XS}  \def\tv{{\sbv{t}}\XS}
\def\Uv{{\sbv{U}}\XS}  \def\uv{{\sbv{u}}\XS}
\def\Vv{{\sbv{V}}\XS}  \def\vv{{\sbv{v}}\XS}
\def\Wv{{\sbv{W}}\XS}  \def\wv{{\sbv{w}}\XS}
\def\Xv{{\sbv{X}}\XS}  \def\xv{{\sbv{x}}\XS}
\def\Yv{{\sbv{Y}}\XS}  \def\yv{{\sbv{y}}\XS}
\def\Zv{{\sbv{Z}}\XS}  \def\zv{{\sbv{z}}\XS}


% Nouveau (08/99) : ALPHABET BLACKBOARD, taille adaptative

\def\Abb{{\sbl{A}}\XS}  \def\abb{{\sbl{a}}\XS}
\def\Bbb{{\sbl{B}}\XS}  \def\bbb{{\sbl{b}}\XS}
\def\Cbb{{\sbl{C}}\XS}  \def\cbb{{\sbl{c}}\XS}
\def\Dbb{{\sbl{D}}\XS}  \def\dbb{{\sbl{d}}\XS}
\def\Ebb{{\sbl{E}}\XS}  \def\ebb{{\sbl{e}}\XS}
\def\Fbb{{\sbl{F}}\XS}  \def\fbb{{\sbl{f}}\XS}
\def\Gbb{{\sbl{G}}\XS}  \def\gbb{{\sbl{g}}\XS}
\def\Hbb{{\sbl{H}}\XS}  \def\hbb{{\sbl{h}}\XS}
\def\Ibb{{\sbl{I}}\XS}  \def\ibb{{\sbl{i}}\XS}
\def\Jbb{{\sbl{J}}\XS}  \def\jbb{{\sbl{j}}\XS}
\def\Kbb{{\sbl{K}}\XS}  \def\kbb{{\sbl{k}}\XS}
\def\Lbb{{\sbl{L}}\XS}  \def\lbb{{\sbl{l}}\XS}
\def\Mbb{{\sbl{M}}\XS}  \def\mbb{{\sbl{m}}\XS}
\def\Nbb{{\sbl{N}}\XS}  \def\nbb{{\sbl{n}}\XS}
\def\Obb{{\sbl{O}}\XS}  \def\obb{{\sbl{o}}\XS}
\def\Pbb{{\sbl{P}}\XS}  \def\pbb{{\sbl{p}}\XS}
\def\Qbb{{\sbl{Q}}\XS}  \def\qbb{{\sbl{q}}\XS}
\def\Rbb{{\sbl{R}}\XS}  \def\rbb{{\sbl{r}}\XS}
\def\Sbb{{\sbl{S}}\XS}  \def\sbb{{\sbl{s}}\XS}
\def\Tbb{{\sbl{T}}\XS}  \def\tbb{{\sbl{t}}\XS}
\def\Ubb{{\sbl{U}}\XS}  \def\ubb{{\sbl{u}}\XS}
\def\Vbb{{\sbl{V}}\XS}  \def\vbb{{\sbl{v}}\XS}
\def\Wbb{{\sbl{W}}\XS}  \def\wbb{{\sbl{w}}\XS}
\def\Xbb{{\sbl{X}}\XS}  \def\xbb{{\sbl{x}}\XS}
\def\Ybb{{\sbl{Y}}\XS}  \def\ybb{{\sbl{y}}\XS}
\def\Zbb{{\sbl{Z}}\XS}  \def\zbb{{\sbl{z}}\XS}


% ALPHABET GREC GRAS italique, taille adaptative

\def\alphab      {{\sbmm{\alpha}}\XS}
\def\betab       {{\sbmm{\beta}}\XS}
\def\gammab      {{\sbmm{\gamma}}\XS}      \def\Gammab    {{\sbmm{\Gamma}}\XS}
\def\deltab      {{\sbmm{\delta}}\XS}      \def\Deltab    {{\sbmm{\Delta}}\XS}
\def\epsilonb    {{\sbmm{\epsilon}}\XS}
\def\varepsilonb {{\sbmm{\varepsilon}}\XS}
\def\etab        {{\sbmm{\eta}}\XS}
\def\thetab      {{\sbmm{\theta}}\XS}      \def\Thetab    {{\sbmm{\Theta}}\XS}
\def\varthetab   {{\sbmm{\vartheta}}\XS}
\def\iotab       {{\sbmm{\iota}}\XS}
\def\kappab      {{\sbmm{\kappa}}\XS}
\def\varkappab   {{\sbmm{\varkappa}}\XS}
\def\lambdab     {{\sbmm{\lambda}}\XS}     \def\Lambdab   {{\sbmm{\Lambda}}\XS}
\def\mub         {{\sbmm{\mu}}\XS}
\def\nub         {{\sbmm{\nu}}\XS}
\def\xib         {{\sbmm{\xi}}\XS}         \def\Xib       {{\sbmm{\Xi}}\XS}
\def\pib         {{\sbmm{\pi}}\XS}         \def\Pib       {{\sbmm{\Pi}}\XS}
\def\varpib      {{\sbmm{\varpi}}\XS}
\def\rhob        {{\sbmm{\rho}}\XS}
\def\varrhob     {{\sbmm{\varrho}}\XS}
\def\sigmab      {{\sbmm{\sigma}}\XS}      \def\Sigmab    {{\sbmm{\Sigma}}\XS}
\def\varsigmab   {{\sbmm{\varsigma}}\XS}   \def\Varsigmab {{\sbmm{\Varsigma}}\XS}
\def\phib        {{\sbmm{\phi}}\XS}        \def\Phib      {{\sbmm{\Phi}}\XS}
\def\varphib     {{\sbmm{\varphi}}\XS}
\def\chib        {{\sbmm{\chi}}\XS}
\def\psib        {{\sbmm{\psi}}\XS}        \def\Psib      {{\sbmm{\Psi}}\XS}
\def\omegab      {{\sbmm{\omega}}\XS}      \def\Omegab    {{\sbmm{\Omega}}\XS}
\def\taub        {{\sbmm{\tau}}\XS}
\def\upsilonb    {{\sbmm{\upsilon}}\XS}   \def\Upsilonb  {{\sbmm{\Upsilon}}\XS}
\def\zetab       {{\sbmm{\zeta}}\XS}


% CHIFFRES
% Gras et Blackboard, taille adaptative

\def\zerob   {{\sbv{0}}\XS}        \def\zerobb   {{\sbl{O}}\XS}
\def\unb     {{\sbv{1}}\XS}        \def\unbb     {{\sbl{1}}\XS}
\def\deuxb   {{\sbv{2}}\XS}        \def\deuxbb   {{\sbl{2}}\XS}
\def\troisb  {{\sbv{3}}\XS}        \def\troisbb  {{\sbl{3}}\XS}
\def\quatreb {{\sbv{4}}\XS}        \def\quatrebb {{\sbl{4}}\XS}
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% Author: J.I.          Date : 07/96    Derniere modif le 10/07/96
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snoussi1.tex
% Article: Penalized maximum likelihood for multivariate Gaussian mixture
% Maxent 2001
\documentclass[nomathfonts]{aipproc}
\layoutstyle{6s}
\usepackage{xspace,epic,eepic,graphicx}
\graphicspath{{Eps/}}
\title[Bayesian source separation]
      {Penalized maximum likelihood for multivariate 
		Gaussian mixture}  
\author{Hichem Snoussi}
  {
  address = {Laboratoire des Signaux et Syst\`emes (L2S), \linebreak   
             Sup\'elec, Plateau de Moulon,  
				 91192 Gif-sur-Yvette Cedex, France}
  ,email   = snoussi@lss.supelec.fr
  }
\author{Ali Mohammad-Djafari}
  {
  address = {Laboratoire des Signaux et Syst\`emes (L2S), \linebreak  
             Sup\'elec, Plateau de Moulon, 
				 91192 Gif-sur-Yvette Cedex, France}
  ,email   = djafari@lss.supelec.fr
  }

%----------- MACROS et Definitions Speciales pour CE document ----------%%
\input{alphabet}       % Definitions de macros pour les alphabets
\input{abrege}         % Abbreviations latines, anglaises et fran�aises
\input{abrmath}        % Abbreviations math�matiques        
\input{beginend}       % Abbreviations des \begin{} et des \end{}

%----------- MACROS et Definitions Speciales pour CE document ----------%%
\def\pxta{p\left( \xb(1),\ldots,\xb(T) | \Ab \right)}
\def\pxas{p\left( \xb_{1..T} | \Ab, \sb_{1..T} \right)}
\def\pasx{p\left( \Ab,\sb_{1..T} | \xb_{1..T} \right)}
\def\pax{p\left( \Ab | \xb_{1..T} \right)}
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\def\Sbh{\wh{\Sb}}
\def\Abh{\wh{\Ab}}
\def\Bbh{\wh{\Bb}}
\def\zbh{\wh{\zb}}
\def\xbh{\wh{\xb}}
\def\Reps{\Rb_{\epsilon}}
\def\Rz{\Rb_{z}}
\def\muz{\mub_{z}}
\def\pz{p_{z}}
\def\thetaz{\thetab_{z}}
\def\Gz{\Gammab_{z}}
\def\Tr#1{\mbox{Tr}\left( #1 \right)}

\bdoc


\begin{abstract}
In this paper, we first consider the parameter estimation of a multivariate random process distribution using multivariate Gaussian mixture law. The labels of the mixture are allowed to have a general probability law  which gives the possibility to  modelize a temporal structure of the process under study. We generalize the case of univariate Gaussian mixture in  \cite{ridolfi99} to show that the likelihood is unbounded and goes to infinity when one of the covariance matrices approaches the boundary of singularity of the non negative definite matrices set. We characterize the parameter set of these singularities. As a solution to this degeneracy problem, we show that the penalization of the likelihood by an Inverse Wishart prior on covariance matrices results to a penalized  or maximum \apost criterion which is  bounded.  Then, the existence of positive definite matrices optimizing this criterion can be guaranteed. We also show that with a modified EM procedure or with a Bayesian sampling scheme, we  can constrain covariance matrices to belong to a particular  subclass of covariance matrices. Finally, we study degeneracies in the source separation problem where the characterization of parameter singularity set is more complex. We show, however, that Inverse Wishart prior on covariance matrices eliminates the degeneracies in this case too.  
\end{abstract}

\maketitle
\thispagestyle{empty}

\section{Introduction}
We consider a double stochastic process:
\bit
\item A discrete process $(z_t)_{t=1..T}$, with $z_t$ taking its values in the discrete set $\Zc=\{1..K\}$.
\item A continuous process $(\sb_t)_{t=1..T}$ which is white conditionally to the first process $(z_t)_{t=1..T}$ and following a distribution:
\[
p(\sb | z)=f(\sb ; \zeta_z)
\]  
In the following, without loss of generality of the considered model, we restrict the function $f(.)$ to be a Gaussian: $f(.|z)=\Nc(\muz, \Rz)$.
\eit
This double process is called in literature "Mixture model". When the hidden process $z_{1..T}$ is white, we have an i.i.d mixture model: $p(\sb)=\sum_{z}\, p_z \Nc(\muz,\,\Rz)$ and when  $z_{1..T}$ is Markovian, the model is called HMM 
(Hidden Markov Model). For application of these two models see \cite{snoussi00a} and \cite{snoussi01c}.
Mixture models present an interesting alternative to non parametric modeling. By increasing the number of mixture components, we are able to approximate any probability density and the time dependence structure of the hidden process $z_{1..T}$ allows to take into account a correlation structure of the resulting process.   In the following, for clarity of demonstrations, we assume an i.i.d. mixture model. 
\section{Characterization of likelihood degeneracy}
We consider $T$ observations $(\sb_t)_{t=1..T}$ of a random $n$-vector  following a multivariate Gaussian mixture law:
\[
p(\sb_t)=\sum_{z=1}^{K}\, p_z \Nc(\sb_t;\muz,\,\Rz)
\]
Where $p_z=P(Z=z)$ is the probability that the random hidden label $Z$ takes the value $z \in \Zc=\{1..K\}$,  $\muz$ is the $n$-vector mean of the Gaussian component $z$ and $\Rz$ its $n \times n$ covariance matrix. We intend to estimate the parameters $\thetab_z=(p_z,\,\muz,\,\Rz)_{z \in {1..K}}$ by maximizing its likelihood given the observations $\sb_{1..T}=[\sb_t]_{t=1..T}$:
\[
{\wh \thetab}=\argmax_{\thetab \in \Thetab}{p(\sb_{1..T}\,|\,\thetab)}
\]
Where
\[
p(\sb_{1..T}\,|\,\thetab)=\displaystyle{\prod_{t=1}^{T}}\,\displaystyle{\sum_{z}}\,p_z\,|2\pi \Rb_z|^{(-1/2)}\Exp{-\frac{1}{2}(\sb_t-\mub_z)^{T}\Rb_z^{-1}(\sb_t-\mub_z)}
\]
and 
\beq\barr{lll}\label{ensemble}
\Thetab=\left\{ \thetab_z=(p_z,\,\muz,\,\Rz)\, | \,p_z \in  \RD_+,\, \sum_{z=1}^{K}p_z=1; 
                  \Rz \in \Rc;\, \muz \in \RD^n \right\}
\earr
\eeq
$\Rc$ is a closed subset of covariance matrices. Some examples of $\Rc$ are considered later in section $4$ and in \cite{Burg82}.

\vspace*{0.5cm}

{\bf Proposition $1$} [Likelihood function is unbounded]: $\forall$ $\sb_{1..T}$ $\in$ $(\RD^{n})^{T}$, $\exists$ a singularity point $\thetab_{s}$ in the parameter space $\Thetab$ such that: $\displaystyle{\lim_{\thetab \rightarrow \thetab_{s}}}\,\,p(\sb_{1..T}\,|\,\thetab)= \infty$. These points are the $\thetab=(p_z,\,\mub_z,\,\Rb_z)_{\zb \in \Zc}$ such that,  at least one of  the $\Rb_z$ (but not all of them together) is a singular non negative matrix and  the correspondent mean  $\mub_z$ lies in the  intersection of $n-rank(\Rb_z)$ hyperplans of $\RD^{n}$. 
\vspace*{0.5cm}
  

{\bf Proof}: Let $z_0 \in \Zc$ and $\Rb_{z_0}$  be a singular NND matrix  of rank $p < n$. $\Rb_{z_0}$ can be diagonalized in the orthogonal group:
\[\barr{cc}
\Rb_{z_0}=\Ub^{T}\Lambdab\Ub, \hspace*{10mm} & \Lambdab=\left[\barr{llllll}0\\~&\ddots\\~&~&0\\~&~&~&\lambda_{n-p+1}\\~&~&~&~&\ddots\\
~&~&~&~&~&\lambda_{n}\earr\right]
\earr
\]
Consider now  a sequence of positive definite matrices $\left( \Rb_{z_0}^{(n)} \right)_{n \in \ND}$ defined by:
\[
\Rb_{z_0}^{(n)}=\Ub^{T}\left[\barr{llllll}\lambda_{1}^{(n)}\\~&\ddots\\~&~&\lambda_{n-p}^{(n)}\\~&~&~&\lambda_{n-p+1}\\~&~&~&~&\ddots\\
~&~&~&~&~&\lambda_{n}\earr\right]\Ub
\]
With the $(n-p)$ strictly positive numeric sequences  $\left( \lambda_{i}^{(n)} \right)_{i=1..(n-p)}$ which  tend to $0$. Thus the sequence of $\left( \Rb_{z_0}^{(n)} \right)_{n \in \ND}$ converges  to $\Rb_{z_0}$.
Likelihood function evaluated at $ \Rb_{z_0}^{(n)}$ is:
\[\barr{lll}
 p_n(\sb_{1..T}\,|\,\thetab)=\displaystyle{\prod_{t=1}^{T}}\left(\,p_{z_0}\,|2\pi \Rb_{z_0}^{(n)}|^{(-1/2)}\Exp{-\frac{1}{2}(\sb_t-\mub_{z_0})^{T}{\Rb_{z_0}^{(n)}}^{-1}(\sb_t-\mub_{z_0})} \right.\\
\hspace{2cm} \left. +\displaystyle{\sum_{z}} p_z\,\Nc(\mub_z,\,\Rb_z)\right)
\earr
\] 
Expending the exponent of the component $z_0$ in canonical form :
\[
(\sb_t-\mub_z)^{T}{\Rb_{z_0}^{(n)}}^{-1}(\sb_t-\mub_z)=\sum_{i}\frac{[\Ub(\sb_t-\mub_z)]_{i}^{2}}{\lambda_{i}^{(n)}},
\]
We can see that when the eigenvalues $(\lambda_{i}^{(n)})_{i=1..(n-p)}$ tend to zero, or equivalently, when the covariance $\Rb_{z_0}^{(n)}$ tends to $\Rb_{z_0}$ and when  $\mub_{z_0}$ lies in the intersection of the  hyperplans $\left( \Hc_{i}=\left\{ \mub \,|\, [\Ub(\sb_t-\mub)]_{i}=0 \right\} \right)_{i=1..(n-p)}$, the likelihood function goes to infinity.
So we have proved that any singular NND matrix is a point of degeneracy provided that the means lie in specific hyperplans.    In one dimensional  case, this corresponds  to the fact that $\sigma$ goes to zero and the correspondent mean coincides with one observation.

 Figure $1$ shows an example of this degeneracy. In this example, we take an original distribution of a $2$-D random vector which is a mixture of $10$ Gaussians. The Gaussians have their means  located on a cercle and have the same covariance. Figure $1$-a shows the graph of this distribution from which we generated $100$ samples in order to estimate its parameters. Figure $1$-b shows the estimated distribution.  We can note the failure of the maximum likelihood estimator and its tendency to converge to sharp Gaussians. 

Here, we highlight the effect of growing the dimension $n$ which increases the occurrence of degeneracy. We have, for $n > 1$ an infinite number of singularities. Moreover, even if we fix the means of the mixture components,  the unboundedness of likelihood might occur if some covariances go to particular singular matrices . But, we think that this second kind of degeneracy is less likely to happen particularly if the number of samples grows. We note that the occurrence of degeneracy increases when the dimension grows and decreases when the number of samples grows.
\bcc
\btabu{ccc}
\includegraphics[width=50mm,height=50mm]{distri_ori}&~&
\includegraphics[width=50mm,height=50mm]{distri_est}\\
Original distribution&~& ML Estimated distribution\\
~&~&with $100$ samples
\etabu
\ecc
Fig-$1$. Failure of the ML estimation of the parameters of a $10$ component Gaussian  mixture distribution.



\section{Bayesian solution to degeneracy}
This degeneracy was noted by many authors (Day, 1969 \cite{day69}) and in (Hathaway 1986 \cite{hathaway86}), a constraint formulation of the EM algorithm has been proposed to eliminate this degeneracy. In (Ormoneit 1998 \cite{ormoneit98}), a penalization by an Inverse Wishart prior was employed to eliminate it. Our contribution leads to the same penalization but in different manner. In (Ormoneit 1998), the Inverse Wishart prior was chosen because it is  a conjugate prior. In the one dimensional case \cite{ridolfi99}, the  penalization by an Inverse Gamma prior on variances was used to eliminate degeneracy. 

In this work, after characterizing the origin of these singularities, we extend this procedure to the multivariate case to  propose an Inverse Wishart prior on covariances $\Rb_z$ which guarantees the boundness of the likelihood:
\[
p_{\alpha,\beta,\Jb}(\Rb_z)=\frac{K}{|\Rb_z|^{\beta}}\Exp{-\alpha\,\Tr{ {\Rb_z}^{-1}\,\Jb}}
\] 
where K is a normalization constant, $\alpha$ and $\beta$ two strictly positive constants which contain \aprio information about the power level (scale parameter) and $\Jb$ is a positive definite symmetric matrix which contains \aprio information  on the covariance structure. In fact, the mode of this law is given by:
\[
\frac{\partial \Log{p(\Rb_z)}}{\partial \Rb_z}=-\beta \Rb_z^{-1}+\alpha\Rb_z^{-1}\Jb\Rb_z^{-1}=0
\]
Leading to:
\[
\Rb_z=\frac{\alpha}{\beta}\Jb
\]
\vspace*{0.5cm}
 
{\bf Proposition $2$:} $\forall$ $\sb_{1..T}$ $\in$ $(\RD^{n})^{T}$, the \apost distribution  $p(\thetab\,|\,\sb_{1..T})$ with the \aprio:
\[
p(\thetab)=\displaystyle{\prod_{\zb \in \Zc}}p_{\alpha_z,\beta_z,\Jb_z}(\Rb_z)
\]
is bounded and goes to $0$ when one of the covariance matrices $\Rb_z$ approaches a singular matrix.
\vspace*{0.5cm}

{\bf Proof:} The penalized likelihood is:
\[
p(\sb_{1..T}\,|\,\thetab)\,p(\thetab)=\displaystyle{\prod_{t=1}^{T}}\left( (\displaystyle{\prod_{\zb \in \Zc}}p(\Rb_z))^{1/T}\displaystyle{\sum_{z}}\,p_z\,\Nc(\mub_z,\Rb_z)\right)
\]
For each label $z$, we have the following inequality: 
\[
(\displaystyle{\prod_{\zb \in \Zc}}p(\Rb_z))^{1/T} \Nc(\mub_z,\Rb_z) \leq \frac{A}{|\Rb_z|^{1/2}}\displaystyle{\prod_{\zb \in \Zc}}\frac{K_z}{|\Rb_z|^{\beta_z}}\Exp{-\alpha_z\,\Tr{ {\Rb_z}^{-1}\,\Jb_z}}
\]

\noindent Thus, to prove the proposition, we need to show that $\forall$ $a > 0$,  $b > 0$ and $\Rb_s$ a singular matrix, we have:
\[
\displaystyle{\lim_{\Rb_z \longrightarrow \Rb_s}} \frac{1}{|\Rb_z|^{b}}\Exp{-a\,\Tr{  \Rb_z^{-1}\,\Jb}}=0
\]
Using the inequality 
\[
(det \,\Ab)^{1/n} \leq \frac{1}{n}\Tr{\Ab}
\]
valid for any real symmetric $n \times n$ matrix $\Ab$,
We have:
\[
\frac{1}{|\Rb_z|^{b}}\Exp{-a\,\Tr{ \Rb_z^{-1}\,\Jb}} \leq \frac{1}{|\Rb_z|^{b}}\Exp{-a\,n\,\frac{|\Jb|^{1/n}}{|\Rb_z|^{1/n}}}
\]
In the above inequality, the right hand side term goes to zero when $\Rz$ approaches the boundary of singularity. Therefore, the penalized likelihood is bounded and is null on the boundary of singularity. 

At this point, we can also follow the arguments in \cite{Burg82} to prove the existence of positive definite matrices corrresponding to the modes of the penalized likelihood. Figure $2$ illustrates the regularization effect of this penalization. Here we used the same samples generated for the figure $1$ and estimated the parameters of the mixture by optimizing the penalized likelihood criterion. The probability of degeneracy is zero. 
    
\bcc
\btabu{ccc}
\includegraphics[width=50mm,height=50mm]{distri_ori}&~&
\includegraphics[width=50mm,height=50mm]{distri_est_penal}\\
Original distribution&~& Penalized EM Estimated \\
~&~&distribution with $100$ samples  
\etabu
\ecc
Fig-$2$. Regularization effect of the penalized EM algorithm.


 
\section{Estimation of structured covariance matrices}
In this paragraph, we generalize the work in \cite{Burg82} to estimate covariance matrices of specified structure in the mixture case.  The constraints are summarized in the closed subset $\Rc$ introduced in the definition of the parameter set $\Thetab$ (\ref{ensemble}).

\subsection{Unconstrained case:}
The unconstrained case was treated in many works. In  \cite{ormoneit98}, three methods were proposed: Averaging, maximum penalized likelihood and Bayesian sampling. We briefly recall the EM algorithm and the Bayesian sampling which both can be seen as data augmentation algorithms:
\bit
\item EM algorithm:   It consists of two steps: 

{\bf (i)} E (Expectation)-step: Consider observations $\sb_{1..T}$ as incomplete data and $(\sb_{1..T}, z_{1..T})$ as complete data and compute the functional $\Qc(\thetab\,|\,\thetab^{(k)})=E\{\log{p(\sb_{1..T}, z_{1..T}\,|\,\thetab)}+\log{p(\thetab)}\,|\, \sb_{1..T}, \thetab^{(k)}\}$;

{\bf (ii)} M (Maximization)-step: Update $\thetab^{(k+1)}=\argmax_{\thetab}{\Qc(\thetab\,|\,\thetab^{(k)})}$.

\vspace{0.5cm}
\item Bayesian sampling: It consists of two steps:

{\bf (i)} Generate $z^{(k+1)}_{1..T} \sim p(z_{1..T}\,|\,\sb_{1..T}, \thetab^{(k)})$;  

{\bf (ii)} Generate $\thetab^{k+1} \sim p(\thetab\,|\,\sb_{1..T}, z^{(k+1)}_{1..T})$.
\eit
In the unconstrained case, one obtains, in both first steps of the above algorithms, functionals which have only one maximum obtained by canceling the  gradient to zero. 
\subsection{Constrained case:}
In both EM algorithm and Bayesian sampling methods presented above, the second step which consists in updating $\thetab$ was unconstrained.  We see in the following how we are able to combine the data augmentation algorithms with the iterative gradient algorithm proposed in \cite{Burg82} to constrain the covariance matrix $\Rz$ to be in the closed set $\Rc$.
\subsubsection{Strutured EM }
The functional $\Qc(\thetab\,|\,\thetab^{(k)})$ can be decomposed as follows:
\[
\Qc(\thetab\,|\,\thetab^{(k)})=\sum_{z=1}^{K}g(\Rz, \Sb_z)\,+\,f(\pb, \mub\,|\,\thetab^{(k)}) 
\]
with:
\[\left\{\barr{lll}
g(\Rz, \Sb_z)=-(1+\frac{\beta}{N_z})\log{|\Rz |}-\Tr{\Rz^{-1}\,(\Sb_z+\frac{\alpha\,\Jb}{N_z})}\\
N_z=\sum_{t=1}^{T}p(z(t)=z\,|\,\sb(t),\,\thetab^{(k)})
\earr\right.
\]
and $\Sb_z$ the weighted sample covariance matrix:
\[
\Sb_z=\frac{\sum_{t=1}^{T}(\sb(t)-\muz^{(k+1)})\,(\sb(t)-\muz^{(k+1)})^{*}\,p(z(t)=z\,|\,\sb(t),\,\thetab^{(k)})}{\sum_{t=1}^{T}p(z(t)=z\,|\,\sb(t),\,\thetab^{(k)})}
\]

\noindent Thus, the maximization of $\Qc$ with respect to $\Rz$ is equivalent to the maximization of $g(\Rz, \Sb_z)$ with respect to $\Rz$.  The necessary gradient equations are:
\beq\label{gradient}
\delta g(\Rz, \Sb_z)=\Tr{(\Rz^{-1}(\Sb_z+\frac{\alpha\,\Jb}{N_z})\Rz^{-1}-(1+\frac{\beta}{N_z})\Rz^{-1})\delta \Rz}=0
\eeq
In the unconstrained case, the solution of (\ref{gradient}) is $\Rz=\frac{\Sb_z+\frac{\alpha\,\Jb}{N_z}}{1+\frac{\beta}{N_z}}$. Constraint maximization of $g$ with $\Rz \in \Rc$ for any $\Rc$ is not easy. However, if $\Rc$ is such that $\Rb \in \,\Rc \Rightarrow \delta\Rb \in \,\Rc$ (for example the set of Toeplitz matrices) then we replace the second step of the EM algorithm by the following: 
\begin{enumerate}
\item Find $\Db_z^{(k+1)}$ belonging to $\Rc$ so that $g(\Rz^{(k)}, \Sb_z-\Db_z^{(k+1)})$ satisfies the necessary gradient conditions.
\item Put $\Rz^{(k+1)}=\Rz^{(k)}+\Db_z^{(k+1)}$
\end{enumerate}
This modification preserves the monotonicity of the EM algorithm and makes the problem linear in $\Db_z$ and so it is easier to impose constraints with the condition that the variation of $\Rz$ still belongs to $\Rc$, which is true for a wide range of constraints such in the Toeplitz case.

\subsubsection{Structured Bayesian sampling }
We propose the following Bayesian sampling scheme:
\begin{enumerate}
\item Generate $z^{*}_{1..T} \sim p(z_{1..T}\,|\,\sb_{1..T}, \thetab^{(k)})$;
\item Generate $\Db_z^{(k+1)}$ belonging to $\Rc$ according to the \apost distribution $p(\Db_z\,|\,\sb_{1..T},\,z^{*}_{1..T})  \sim \Exp{g(\Rz^{(k)}, \Sb_z-\Db_z^{(k+1)})} $.
\item Update $\Rz^{(k+1)}=\Rz^{(k)}+\Db_z^{(k+1)}$
\end{enumerate}

$\Sb_z$ is the sample covariance depending on the partition defined by $z^{*}_{1..T}$:
\[\left\{\barr{lll}
\Sb_z=\frac{\sum_{t \in \Tc_z}\sb(t)\sb(t)^{*}}{Card(\Tc_z)} \\
\Tc_z=\{t\,|\,z(t)=z\}
\earr \right.
\]

To be sure that the sampling keeps $\Db_z$ in the closed set $\Rc$, we define a basis $(\Qb_l)_{l=1..L}$ of $\Rc$ and we sample the projection of $\Db_z$ on $\Rc$: $\xb_{1..L} \sim p(\xb_{1..L}\,|\, \sb_{1..T},\,z^{*}_{1..T})$, where the vector $\xb_{1..L}$ is defined as:
\[
\Db_z=\sum_{l=1}^{L}\,x_l\,\Qb_l
\] 





\section{Mixed sources}
We consider now the case where sources are not directly observed, but mixed with an unknown mixing matrix $\Ab$ and we want to take into account measurement errors so that observations are modeled by the following equation:
\[
\xb(t)=\Ab\,\sb(t)+\nb(t)
\]
In this section, we show that when we are interested in estimating jointly the mixing matrix $\Ab$, noise covariance matrix $\Reps$ and  the parameters of the mixture, by maximizing the  likelihood $p ( \xb_{1..T}\,|\,\Ab,\,\Reps,\,\thetab_z )$, we encounter the same problems of degeneracy mentioned above. 
Likelihood function has the following expression:
\[
p ( \xb_{1..T}\,|\,\Ab,\,\Reps,\,\thetab_z )=\prod_{t=1}^{T}\sum_{z=1}^{K}\pz(t)\Nc(\Ab\muz,\,\Ab\Rz\Ab^{*}+\Reps)
\]
with $\thetab_z=(\muz,\,\Rz,\,\pz)$.

The expression $\pz(t)=\displaystyle{\sum_{z_{1..T},z(t)=z}}p(z_{1..T})$ represents the marginal law of $z(t)$. Indeed, the hidden variables do not need  necessarily to be white and so the mixture to be  i.i.d. We can rewrite the expression of the likelihood in a more general form in which  the marginalization is not performed :
\[
p ( \xb_{1..T}\,|\,\Ab,\,\Reps,\,\thetab_z )=\sum_{z_{1..T}}p(z_{1..T})\prod_{t=1}^{T}\Nc(\Ab\muz,\,\Ab\Rz\Ab^{*}+\Reps)
\] 
It is obvious, under this form, that degeneracy happens when one of the terms constituting the sum tends to infinity  and this is  independently of the  law  $p(z_{1..T})$.

Consider now the matrices $\Gz=\Ab\Rz\Ab^{*}+\Reps$. It's clear that degeneracy is produced when, among matrices $\Gz$, at least one is singular  and  one is regular. We  show in the following that this situation can  occur.

We recall that the matrices $\Rz$ and $\Reps$ belong to a closed subset of the set of the non negative definite  matrices. Constraining matrices to be positive definite leads to complicated solutions. The main origin  of this complication is the fact that the set of positive definite matrices is not closed. For the same reason, we don't constrain the mixing matrix $\Ab$ to be of full rank.

\vspace{1cm}
{\bf Proposition 3}: $\forall$ $\Ab$ non null, $\exists$ matrices $\{ \Gz=\Ab\Rz\Ab^{*}+\Reps$ for  $z=1..K$\} such that $\{z \;|\; \Gz \; is \; singular\} \neq \emptyset$ and $\{z \;|\; \Gz \; is \; regular\} \neq \emptyset$. \\
$\Reps$ is necessarily a singular NND matrix and $Card\left(\{z \;|\; \Rz \; is \; regular\}\right) < K$.
\vspace{1cm}

{\bf Proof}: Without affecting the generality of the problem, we show how to   construct a singular matrix $\Gammab_{1}$ and the others matrices $\Gz$ regular.
We consider NND matrices. Therefore, the kernel of the correspondent linear mapping coincides with its  isotropic cone. Thus, we have:
\[
Ker(\Gz)=Ker(\Ab\Rz\Ab^{*}) \cap Ker(\Reps)
\] 
It is sufficient to prove the existence of $\Reps$ and $(\Rz)_{z=1..K}$ that  verify the following condition:
\beq\left\{ \barr{lll}\label{condition}
Ker(\Ab\Rb_{1}\Ab^{*}) \cap Ker(\Reps) & \neq & \{0\}\\
Ker(\Ab\Rz\Ab^{*}) \cap Ker(\Reps) &=& \{0\}\,,\,\zb=2..K
\earr \right.
\eeq
If the matrix $\Reps$ is regular, there is no degeneracy: According to  the mini-max principle applied to the  characterization of the eigenvalues  of the sum of two hermitian matrices, the eigenvalues of $\Gz$ are greater than those of  $\Reps$ and then strictly positive which imply that all of  the matrices $\Gz$ are regular. 

We have:
\beq\label{inclusion}
Ker(\Ab^{*}) \subseteq Ker(\Ab\Rz\Ab^{*}),\,\,z=1..K
\eeq
Equality holds if $\Rz$ is regular or if $Ker(\Rz) \cap Im(\Ab^{*})=\{0\}$. Note that if all the matrices  $\Rz$ are regular, there is no degeneracy.

Suppose then that the matrices $\Rz$, except the first matrix $\Rb_1$, are regular. We will try now to  construct the matrices $\Rb_1$ and $\Reps$ making the  condition (\ref{condition}) verified. Let  a non null vector $\xb_s$  belong to $[Ker(\Ab^{*})]^{\perp}$. There  exist  NND matrices  $\Rb_1$ and $\Reps$ such that  $\xb_s \in Ker(\Ab\Rb_{1}\Ab^{*}) \cap Ker(\Reps)$. In fact, consider the family of vectors $(\xb_{j})_{j \in J}$ belonging to $Ker(\Ab^{*})$ such that the family $\{\xb_s\} \cup (\xb_{j})_{j \in J}$ is orthogonal (this is insured  by  the principle of the incomplete basis). The  matrices $\Rb_1=\sum_{j \in J}\alpha_j\,\xb_{j}\xb_{j}^{*}$ ($\alpha_j \geq 0$) and $\Reps=\sum_{j \in J}\beta_j\,\xb_{j}\xb_{j}^{*}$ ($\beta_j \geq 0$) are such that $\xb_s \in Ker(\Ab\Rb_{1}\Ab^{*}) \cap Ker(\Reps)$ by construction and $Ker(\Ab\Rz\Ab^{*}) \cap Ker(\Reps)=\{0\}$. We have then constructed  matrices which verify the  degeneracy  condition.


Note that the fact that the matrices $\Rb_1$ and $\Reps$ are singular is a necessary condition but not sufficient; the matrix $\Rb_1$ can be  singular with $Ker(\Ab\Rb_1\Ab^{*})=Ker(\Ab^{*})$ and so there is no degeneracy, or as well, $\Reps$ is singular but $Ker(\Ab\Rz\Ab^{*}) \cap Ker(\Reps) \neq \{0\}$, $\forall z \in \{ 1..K\}$, which implies that all matrices $\Gz$ are singular and  so no degeneracy occurs.

\section{degeneracy elimination in the mixed case}

In the light of what we presented in the two first paragraphs,   one possible way to eliminate this degeneracy  consists in penalizing the likelihood by an  Inverse Wishart \aprio for covariance matrices. In fact, we know that the origin of degeneracy is that the covariance matrices  $\Rz$ and $\Reps$ approach the boundary of singularity (in a non arbitrary way). Thus, if we  penalize the likelihood such that  when one of the covariance matrices approaches the boundary, the \apost distribution goes to zero, eliminating  the infinity value at the boundary and even forcing it to zero.

\vspace{1cm}
{\bf Proposition $5$:} $\forall$ $\xb_{1..T}$ $\in$ $(\RD^{m})^{T}$, the likelihood  $p(\xb_{1..T}\,|\,\thetaz,\,\Reps,\,\Ab)$ penalized by an \aprio Inverse Wishart for the noise covariance matrix $\Reps$ or by  an \aprio Inverse Wishart for the matrices $\Rz$ is bounded and goes to  $0$ when one of the covariance matrices approaches the boundary of singularity.
\vspace*{1cm} 

  
{\bf Proof 5:} The  proof is based upon the proof of the proposition $4$, except the fact that here the \aprio is not directly related  to the  matrices $\Gz=\Ab\Rz\Ab^{*}+\Reps$, but to covariance matrices $\Rz$ or $\Reps$. Then, we have the following alternative:  
\bit
\item If one penalizes by an \aprio Inverse Wishart on the matrix $\Reps$, we have the following inequality:
\[
(p(\Reps))^{1/T} \Nc(\Ab\muz,\Gz) \leq \frac{A}{|\Gz|^{1/2}}\frac{K}{|\Reps|^{\beta}}\Exp{-\alpha\,\Tr{ {\Reps}^{-1}\,\Jb}}
\]
Now according to the mini-max principle applied to the characterization of eigenvalues, we have:
\[
| \Gz |=|\Ab\Rz\Ab^{*}+\Reps| \geq |\Reps|
\]
which yields the following inequality:
\[
(p(\Reps))^{1/T} \Nc(\Ab\muz,\Gz) \leq \frac{1}{|\Reps|^{b}}\Exp{-a\,\Tr{ \Reps^{-1}\,\Jb}}
\]
This insures the convergence to $0$ of the penalized likelihood when  $\Reps$ goes to a singular matrix and insures, as well,  the elimination of degeneracy which one the necessary conditions  is the singularity of the covariance  $\Reps$.
\item If we  penalize only by an Inverse Wishart prior  on the   matrices $\Rz$  with an uniform  \aprio on the matrix $\Reps$, we have a similar inequality:
\[
(p(\Rz))^{1/T} \Nc(\Ab\muz,\Gz) \leq \frac{1}{|\Ab\Rz\Ab^{*}|^{b_z}}\Exp{-a_z\,trace \left( \Rz^{-1}\,\Jb_z\right)}
\]
Here, the only query is that the determinant $|\Ab|$ goes to zero faster than the exponential of $|\Rz|$ but, in  this situation, the degeneracy condition (\ref{condition}) is not verified because of the inclusion relation (\ref{inclusion}).     
\eit


\section{Conclusion}
The set of parameter singularities which characterizes the likelihood degeneracy of a multivariate Gaussian mixture is identified. A Bayesian solution to this degeneracy is proposed. We proposed a modified version of the data augmentation algorithms which allows to  account for some constraints on the structure of the covariance matrices of the Gaussian mixture distribution. It consists essentially in the introduction of an inverse iteration to make the problem linear with respect to the matrix estimate.  The case of source separation with Gaussian mixture model sources is also considered and discussed.
 
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