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<h1 class="title">Note de lecture de <em>A tutorial on approximate Bayesian computation</em> de Brandon M. Turner, Trisha Van Zandt.</h1>
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<div class="quarto-category">literature note</div>
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<div class="callout callout-style-default callout-note callout-titled" title="Synthèse">
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Synthèse
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<p><strong>Contribution</strong>:</p>
<p><strong>Related</strong>:</p>
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<p><strong>FirstAuthor</strong>: Turner, Brandon M.<br>
<strong>Author</strong>: Van Zandt, Trisha</p>
<p><strong>Title</strong>: A tutorial on approximate Bayesian computation<br>
<strong>Year</strong>: 2012<br>
<strong>Citekey</strong>: turnerTutorialApproximateBayesian2012<br>
<strong>itemType</strong>: journalArticle<br>
<strong>Journal</strong>: <em>Journal of Mathematical Psychology</em><br>
<strong>Volume</strong>: 56<br>
<strong>Issue</strong>: 2<br>
<strong>Pages</strong>: 69-85<br>
<strong>DOI</strong>: 10.1016/j.jmp.2012.02.005</p>
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Pièces-jointes
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<blockquote class="blockquote">
<p><a href="file:///home/louis/snap/zotero-snap/common/Zotero/storage/N4E59NBM/Turner%20et%20Van%20Zandt%20-%202012%20-%20A%20tutorial%20on%20approximate%20Bayesian%20computation.pdf">PDF</a>.</p>
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<div class="callout callout-style-default callout-note callout-titled" title="Abstract">
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Abstract
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<p>This tutorial explains the foundation of approximate Bayesian computation (ABC), an approach to Bayesian inference that does not require the specification of a likelihood function, and hence that can be used to estimate posterior distributions of parameters for simulation-based models. We discuss briefly the philosophy of Bayesian inference and then present several algorithms for ABC. We then apply these algorithms in a number of examples. For most of these examples, the posterior distributions are known, and so we can compare the estimated posteriors derived from ABC to the true posteriors and verify that the algorithms recover the true posteriors accurately. We also consider a popular simulation-based model of recognition memory (REM) for which the true posteriors are unknown. We conclude with a number of recommendations for applying ABC methods to solve real-world problems..</p>
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<section id="prise-de-notes" class="level1">
<h1>Prise de notes</h1>
<p>%% begin user_notes %%</p>
<section id="introduction" class="level2">
<h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
<p>La première partie est une explication du point de vue bayésien avec un regard sur les modèles en psychologie cognitive</p>
<p>Les auteurs différencie les modèles “mathématiques” : ont une vraisemblance explicite, des modèles de “simulation” : pas de vraisemblance explicite et génèrent des prédictions en répliquant/modélisant le mécanisme de génération de données</p>
</section>
<section id="approximate-bayesian-computation" class="level2">
<h2 class="anchored" data-anchor-id="approximate-bayesian-computation">Approximate Bayesian computation</h2>
<p><span class="math display">\[
\newcommand{\ts}{\theta^{\star}}
\newcommand{\tss}{\theta^{\star\star}}
\]</span></p>
<p><strong>Idée principale</strong>: Remplacer un calcul de vraisemblance coûteux (ou impossible?) par des simulations qui produisent un jeu de données artificiel <span class="math inline">\(X\)</span>.</p>
<p>Donc on cherche à simuler des jeux de données <span class="math inline">\(X\)</span> qui approchent bien <span class="math inline">\(Y\)</span> notre vrai jeu de données en utilisant diverses valeurs de <span class="math inline">\(\theta^{\star}\)</span> puis en retenant celle qui donnent <span class="math inline">\(\rho(X,Y) \leq \epsilon_{0}\)</span> avec lidée dapprocher la posterior <span class="math inline">\(\pi(\theta\mid Y)\)</span> par <span class="math inline">\(\pi(\theta\mid \rho(X,Y) \leq \epsilon_{0})\)</span></p>
<p><strong>Procédure</strong>:</p>
<ol type="1">
<li>Échantillonner <span class="math inline">\(\theta^{\star}\)</span> de la prior <span class="math inline">\(\pi(\theta)\)</span></li>
<li>Générer <span class="math inline">\(X\)</span> en utilisant <span class="math inline">\(\theta^{\star}\)</span></li>
<li>Comparer <span class="math inline">\(\rho(X,Y)\)</span> à <span class="math inline">\(\epsilon_{0}\)</span>
<ol type="a">
<li>Si <span class="math inline">\(\rho(X,Y) \leq \epsilon_{0}\)</span> alors <span class="math inline">\(\theta^{\star}\)</span> est conservé comme valeur probable de la posterior</li>
<li>Sinon on fait quelque chose avec <span class="math inline">\(\theta^{\star}\)</span>, dans lacceptation-rejet : on rejette <span class="math inline">\(\theta^{\star}\)</span></li>
</ol></li>
</ol>
<p>Pour faciliter les calculs on remplace <span class="math inline">\(X, Y\)</span> par des statistiques. Si elles sont suffisantes (limitation à la famille exponentielle) très bon remplacement sans perte dinformations.</p>
<p><em>Remarque sur Acceptation-Rejet (AR-ABC) :</em> pour <span class="math inline">\(\epsilon_{0}\)</span> très petit le taux de rejet peut être trop élevé et lalgo très inefficace.</p>
<section id="abc-mcmc" class="level3">
<h3 class="anchored" data-anchor-id="abc-mcmc">ABC MCMC</h3>
<p>Au lieu de calculer la proba dacceptation par la vraisemblance, on utilise <span class="math inline">\(\theta^{\star}\)</span> pour générer <span class="math inline">\(X\)</span> et alors la proba dacceptation est :</p>
<p><span class="math display">\[
\alpha = \begin{cases} \min(1,\frac{\pi(\theta^{\star})q(\theta_{i}\mid \theta^{\star})}{\pi(\theta_{i})q(\theta^{\star}\mid \theta_{i})}) &amp; \text{si } \rho(X,Y)\leq \epsilon_{0} \\
0 &amp; \text{sinon}
\end{cases}
\]</span></p>
<p><span class="math inline">\(q\)</span> est la loi de proposition.</p>
<p>Si elle est symétrique, les deux termes sannulent et <span class="math inline">\(\alpha\)</span> ne dépend que du prior, de la distance aux données réelles et de <span class="math inline">\(\epsilon_{0}\)</span>.</p>
<p><strong>Limites importantes</strong>:</p>
<ul>
<li>Très probable de se retrouver bloquée dans les zones de faibles probas où les <span class="math inline">\(\theta^{\star}\)</span> génèreront des <span class="math inline">\(X\)</span> improbable et donc dy rester bloqué. Et ainsi <em>fort taux de rejet</em>.</li>
<li>Pas possible de paralléliser une chaîne car dépendance.</li>
</ul>
</section>
<section id="particle-filtering-monte-carlo-séquentiel" class="level3">
<h3 class="anchored" data-anchor-id="particle-filtering-monte-carlo-séquentiel"><em>Particle filtering</em>, Monte-Carlo séquentiel</h3>
<p><strong>Principe de lalgorithme</strong>:</p>
<ol type="1">
<li>Générer une <em>pool</em> de valeurs <span class="math inline">\(\{\theta_{1}, \dots, \theta_{i}, \dots, \theta_{N}\}\)</span> selon <span class="math inline">\(\pi(\theta)\)</span></li>
<li>Initialiser les poids de sélection <span class="math inline">\(w_{i} = \frac{1}{N}\)</span></li>
<li>Itérer pour <span class="math inline">\(T\)</span> étapes :
<ol type="a">
<li>Tirer <span class="math inline">\(\theta^{*}\)</span> de la <em>pool</em> avec les poids <span class="math inline">\(\pmb{w}\)</span></li>
<li>Utiliser le noyau de transition <em>forward</em>[^1] <span class="math inline">\(q(\theta^{\star\star}|\theta^{\star})\)</span> pour proposer la nouvelle particule <span class="math inline">\(\tss\)</span></li>
<li>Mettre à jour <span class="math inline">\(\pmb{w}\)</span></li>
</ol></li>
</ol>
<p>À la fin la <em>pool</em> est (on espère) distribuée selon <span class="math inline">\(\pi(\theta\mid Y)\)</span>.</p>
<section id="partial-rejection-control-abc-prc" class="level4">
<h4 class="anchored" data-anchor-id="partial-rejection-control-abc-prc"><em>Partial Rejection control</em> : ABC PRC</h4>
<p><span class="math display">\[
\newcommand{\qf}[1]{q_{f}(#1|\ts)}
\newcommand{\qb}[1]{q_{b}(#1|\tss)}
\newcommand{\thresh}{\rho(X,Y) \leq \epsilon_{0}}
\]</span></p>
<p>Il y a besoin de deux noyaux de transition <span class="math inline">\(\qf{.},\qb{.}\)</span> forward et backward.</p>
<p>On spécifie ici létape :</p>
<ol start="2" type="a">
<li><span class="math inline">\(\tss \sim \qf{\theta_{i}}\)</span>. Si le <span class="math inline">\(X\)</span> réalisé par <span class="math inline">\(\tss\)</span> est tel que <span class="math inline">\(\rho(X,Y) \leq \epsilon_{0}\)</span> alors <span class="math inline">\(\theta_{i} = \tss\)</span>. Sinon on rejette <span class="math inline">\(\tss\)</span> et on itère jusquà passer le critère.</li>
</ol>
<p>Alors le poids de la particule est mis à jour :</p>
<p><span class="math display">\[
w = \frac{\pi(\tss)\qb{\ts}}{\pi(\ts)\qf{\tss}}
\]</span></p>
<p>Et on répète jusquà avoir remplacé nos <span class="math inline">\(N\)</span> particules afin quelles satisfassent toutes <span class="math inline">\(\thresh\)</span>.</p>
<p>Puis on ré-itère <span class="math inline">\(T\)</span> fois lopération.</p>
<p><strong>Avantages</strong>:</p>
<ul>
<li>Permet de sortir des régions de faibles probabilités en enlevant les particules. <strong>Limites</strong>:</li>
<li>Estimation biaisée de la posterior</li>
</ul>
<p><a href="zotero://select/items/1_TR63KKUN">Turner. 4/2012. <i>A tutorial on approximate Bayesian computation</i></a></p>
<p>Je rajoute dautres notes nouvelles</p>
<p>%% end user_notes %%# Annotations importées%% begin annotations %%</p>
</section>
</section>
<section id="imported-2026-05-13-205-pm" class="level3">
<h3 class="anchored" data-anchor-id="imported-2026-05-13-205-pm">Imported: 2026-05-13 2:05 pm</h3>
<p><mark style="background-color: #5fb236">Quote</mark></p>
<blockquote class="blockquote">
<p>The simplest of these is the ABC rejection sampling algorithm (see Algorithm 1; Beaumont, Zhang, &amp; Balding, 2002; Pritchard et al., 1999). The ABC rejection sampler simply discards the candidate value θ if it does not meet the criterion ρ(X , Y ) ≤ ε0, as we described above.</p>
</blockquote>
</section>
<section id="imported-2026-05-13-532-pm" class="level3">
<h3 class="anchored" data-anchor-id="imported-2026-05-13-532-pm">Imported: 2026-05-13 5:32 pm</h3>
<p><mark style="background-color: #5fb236">Quote</mark></p>
<blockquote class="blockquote">
<p>The simplest of these is the ABC rejection sampling algorithm (see Algorithm 1; Beaumont, Zhang, &amp; Balding, 2002; Pritchard et al., 1999). The ABC rejection sampler simply discards the candidate value θ if it does not meet the criterion ρ(X , Y ) ≤ ε0, as we described above.</p>
</blockquote>
</section>
<section id="imported-2026-05-13-535-pm" class="level3">
<h3 class="anchored" data-anchor-id="imported-2026-05-13-535-pm">Imported: 2026-05-13 5:35 pm</h3>
<p><mark style="background-color: #5fb236">Quote</mark></p>
<blockquote class="blockquote">
<p>If the particle does not pass inspection (if ρ(X , Y ) &gt; ε0), it is discarded</p>
</blockquote>
</section>
<section id="imported-2026-05-13-542-pm" class="level3">
<h3 class="anchored" data-anchor-id="imported-2026-05-13-542-pm">Imported: 2026-05-13 5:42 pm</h3>
<p><mark style="background-color: #5fb236">Quote</mark></p>
<blockquote class="blockquote">
<p>If the particle does not pass inspection (if ρ(X , Y ) &gt; ε0), it is discarded</p>
</blockquote>
<p>%% end annotations %%</p>
<p>%% Import Date: 2026-05-13T17:44:26.337+02:00 %%</p>
</section>
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const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.getElementById(id);
if (note !== null) {
const html = processXRef(id, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
} else {
// See if we can fetch a full url (with no hash to target)
// This is a special case and we should probably do some content thinning / targeting
fetch(url)
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.querySelector('main.content');
if (note !== null) {
// This should only happen for chapter cross references
// (since there is no id in the URL)
// remove the first header
if (note.children.length > 0 && note.children[0].tagName === "HEADER") {
note.children[0].remove();
}
const html = processXRef(null, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
}, function(instance) {
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
div.style.left = 0;
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Handle positioning of the toggle
window.addEventListener(
"resize",
throttle(() => {
elRect = undefined;
if (selectedAnnoteEl) {
selectCodeLines(selectedAnnoteEl);
}
}, 10)
);
function throttle(fn, ms) {
let throttle = false;
let timer;
return (...args) => {
if(!throttle) { // first call gets through
fn.apply(this, args);
throttle = true;
} else { // all the others get throttled
if(timer) clearTimeout(timer); // cancel #2
timer = setTimeout(() => {
fn.apply(this, args);
timer = throttle = false;
}, ms);
}
};
}
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
if (citeInfo) {
tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {
citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
});
}
}
});
</script>
</div> <!-- /content -->
</body></html>