Testing A joint probability

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$ \DeclareMathOperator{\argmin}{arg\,min} \DeclareMathOperator{\argmax}{arg\,max} \newcommand{\nominal}[1]{#1^{\star}} \newcommand{\psis}{\psi{^\star}} \newcommand{\phis}{\phi{^\star}} \newcommand{\hpsi}{\hat{\psi}} \newcommand{\hphi}{\hat{\phi}} \newcommand{\teps}{\varepsilon} \newcommand{\limite}[2]{\mathop{\longrightarrow}\limits_{\mathrm{#1}}^{\mathrm{#2}}} \newcommand{\DDt}[1]{\partial^2_\theta #1} \def\aref{a^\star} \def\kref{k^\star} \def\model{M} \def\hmodel{m} \def\mmodel{\mu} \def\imodel{H} \def\Imax{\text{\it Imax}} \def\id{ {\rm Id}} \def\teta{\tilde{\eta}} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\deriv}[1]{\frac{d}{dt}#1(t)} \newcommand{\pred}[1]{\tilde{#1}} \def\phis{\phi{^\star}} \def\hphi{\tilde{\phi}} \def\hw{\tilde{w}} \def\hpsi{\tilde{\psi}} \def\hatpsi{\hat{\psi}} \def\hatphi{\hat{\phi}} \def\psis{\psi{^\star}} \def\transy{u} \def\psipop{\psi_{\rm pop}} \newcommand{\psigr}[1]{\hat{\bpsi}_{#1}} \newcommand{\Vgr}[1]{\hat{V}_{#1}} \def\psig{\psi} \def\psigprime{\psig^{\prime}} \def\psigiprime{\psig_i^{\prime}} \def\psigk{\psig^{(k)}} \def\psigki{ {\psig_i^{(k)}}} \def\psigkun{\psig^{(k+1)}} \def\psigkuni{\psig_i^{(k+1)}} \def\psigi{ {\psig_i}} \def\psigil{ {\psig_{i,\ell}}} \def\phig{ {\phi}} \def\phigi{ {\phig_i}} \def\phigil{ {\phig_{i,\ell}}} \def\etagi{ {\eta_i}} \def\IIV{ {\Omega}} \def\thetag{ {\theta}} \def\thetagk{ {\theta_k}} \def\thetagkun{ {\theta_{k+1}}} \def\thetagkunm{\theta_{k-1}} \def\sgk{s_{k}} \def\sgkun{s_{k+1}} \def\yg{y} \def\xg{x} \def\neta{ {n_\eta}} \def\ncov{M} \def\npsi{n_\psig} \def\bu{\boldsymbol{u}} \def\bt{\boldsymbol{t}} \def\bT{\boldsymbol{T}} \def\by{\boldsymbol{y}} \def\bx{\boldsymbol{x}} \def\bc{\boldsymbol{c}} \def\bw{\boldsymbol{w}} \def\bz{\boldsymbol{z}} \def\bpsi{\boldsymbol{\psi}} \def\bbeta{\beta} \def\beeta{\eta} \def\logit{\rm logit} \def\transy{u} \def\so{O} \def\one{\mathbb 1} \newcommand{\prob}[1]{ \mathbb{P}\!\left(#1\right)} \newcommand{\probs}[2]{ \mathbb{P}_{#1}\!\left(#2\right)} \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\esps}[2]{\mathbb{E}_{#1}\left(#2\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \newcommand{\vars}[2]{\mbox{Var}_{#1}\left(#2\right)} \newcommand{\std}[1]{\mbox{sd}\left(#1\right)} \newcommand{\stds}[2]{\mbox{sd}_{#1}\left(#2\right)} \newcommand{\corr}[1]{\mbox{Corr}\left(#1\right)} \def\pmacro{\mathbf{p}} \def\py{\pmacro} \def\pt{\pmacro} \def\pc{\pmacro} \def\pu{\pmacro} \def\pyi{\pmacro} \def\pyj{\pmacro} \def\ppsi{\pmacro} \def\ppsii{\pmacro} \def\pcpsith{\pmacro} \def\pth{\pmacro} \def\pypsi{\pmacro} \def\pcypsi{\pmacro} \def\ppsic{\pmacro} \def\pcpsic{\pmacro} \def\pypsic{\pmacro} \def\pypsit{\pmacro} \def\pcypsit{\pmacro} \def\pypsiu{\pmacro} \def\pcypsiu{\pmacro} \def\pypsith{\pmacro} \def\pypsithcut{\pmacro} \def\pypsithc{\pmacro} \def\pcypsiut{\pmacro} \def\pcpsithc{\pmacro} \def\pcthy{\pmacro} \def\pyth{\pmacro} \def\pcpsiy{\pmacro} \def\pz{\pmacro} \def\pw{\pmacro} \def\pcwz{\pmacro} \def\pw{\pmacro} \def\pcyipsii{\pmacro} \def\pyipsii{\pmacro} \def\pypsiij{\pmacro} \def\pyipsi1{\pmacro} \def\ptypsiij{\pmacro} \def\pcyzipsii{\pmacro} \def\pczipsii{\pmacro} \def\pcyizpsii{\pmacro} \def\pcyijzpsii{\pmacro} \def\pcyi1zpsii{\pmacro} \def\pcypsiz{\pmacro} \def\pccypsiz{\pmacro} \def\pypsiz{\pmacro} \def\pcpsiz{\pmacro} \def\peps{\pmacro} \def\mlxtran{\mathbb MLXtran} $

Introduction

A model built for real-world applications can involve various types of variable, such as measurements, individual and population parameters, covariates, design, etc. The model allows us to represent relationships between these variables.

If we consider things from a probabilistic point of view, some of the variables will be random, so the model becomes a probabilistic one, representing the joint distribution of these random variables.

Defining a model therefore means defining a joint distribution. The hierarchical structure of the model will then allow it to be decomposed into submodels, i.e., the joint distribution decomposed into a product of conditional distributions.

Tasks such as estimation, model selection, simulation and optimization can then be expressed as specific ways of using this probability distribution.

- A model is a joint probability distribution.

- A submodel is a conditional distribution derived from this joint distribution.

- A task is a specific use of this distribution.


We will illustrate this approach starting with a very simple example that we will gradually make more sophisticated. Then we will see in various situations what can be defined as the model and what its inputs are.