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b/_freeze/posts/factor_graphs/index/execute-results/html.json new file mode 100644 index 0000000..a9da047 --- /dev/null +++ b/_freeze/posts/factor_graphs/index/execute-results/html.json @@ -0,0 +1,12 @@ +{ + "hash": "d97a85747968d5e3e8b12a5e92ca14e8", + "result": { + "engine": "julia", + "markdown": "---\ntitle: \"Bayesian Inference and Factor Graphs\"\nauthor: \"Nico Grimm\"\ndate: \"2025-07-29\"\ntoc: true\nhtml-math-method:\n method: katex\n katex:\n use_cdn: false\ncategories: [probabilistic-ml, probability-theory]\nengine: julia\n---\n## Introduction\n\n## Bayesian Inference Model\nDerived from a fundamental result in probability theory, the *Bayesian Inference Model* is one of the basic concepts in probabilistic machine learning. It represents the elemental idea of updating prior beliefs over model parameters through data.\n$$ p(B \\: \\vert \\: A) = \\frac{p(A \\: \\vert \\: B)\\cdot p(B)}{p(A)} $$ {#eq-bayes}\nIn machine learning, we typically have a model which can be specified by some set of parameters $\\mathbf{\\Phi}$ deal with training data $\\mathcal{D}$. Usually, the parameters are directly updated by optimizing a given metric on the training data, e.g. minimizing a loss function. However, in this setting we are specifiying a so-called *prior distribution* over $\\mathbf{\\Phi}$ and update the distribution through training. Not only does this allow to incorporate assumptions over possibly important model parameters, but also captures the model's uncertainty of the weight's values after training and makes it possible to compute the uncertainty of predictions.\n\nStarting off with *Bayes' Rule* (see @eq-bayes), which relates the probability of $B$ to the probability of $B$ given $A$. By replacing $B$ with $\\mathbf{\\Phi}$ and $A$ with $\\mathcal{D}$ we end up with the basic formula of *Bayesian Inference*:\n$$p(\\mathbf{\\Phi} \\: \\vert \\: \\mathcal{D}) \\propto p(\\mathcal{D} \\: \\vert \\: \\mathbf{\\Phi}) \\cdot p(\\mathbf{\\Phi})$$\nThis is exactly the tool to achieve what was proposed earlier. From the prior belief over parameters $p(\\mathbf{\\Phi})$ we can obtain an updated distribution after seeing the data $p(\\mathbf{\\Phi} \\: \\vert \\: \\mathcal{D})$, called *posterior distribution*, by multiplying with $p(\\mathcal{D} \\: \\vert \\: \\mathbf{\\Phi})$. This term is called *likelihood* and is very important. It is the probability of the training data having been generated by the current parameter values. It basically represents the actual model in the equation, as it describes the probability of the model output being the target value given the current parameter values. \n\nNow what about $p(\\mathcal{D})$? This term is called *model evidence* and essentially captures to which extent the proposed model, including it's prior assumptions, is capable of fitting the data, because in order to calculate it, we need to integrate the product of prior and likelihood over all possible parameter values. In practice, it is often ommited, because it is\n\n* typically very hard to compute exactly\n* constant for a given model\n* unnecessary for parameter estimation of the posterior\n\n## Factor Graphs\n\n## Sum-Product-Algorithm\n\n", + "supporting": [ + "index_files" + ], + "filters": [], + "includes": {} + } +} \ No newline at end of file diff --git a/_freeze/site_libs/clipboard/clipboard.min.js b/_freeze/site_libs/clipboard/clipboard.min.js new file mode 100644 index 0000000..1103f81 --- /dev/null +++ b/_freeze/site_libs/clipboard/clipboard.min.js @@ -0,0 +1,7 @@ +/*! + * clipboard.js v2.0.11 + * https://clipboardjs.com/ + * + * Licensed MIT © Zeno Rocha + */ +!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return b}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),r=n.n(e);function c(t){try{return document.execCommand(t)}catch(t){return}}var a=function(t){t=r()(t);return c("cut"),t};function o(t,e){var n,o,t=(n=t,o="rtl"===document.documentElement.getAttribute("dir"),(t=document.createElement("textarea")).style.fontSize="12pt",t.style.border="0",t.style.padding="0",t.style.margin="0",t.style.position="absolute",t.style[o?"right":"left"]="-9999px",o=window.pageYOffset||document.documentElement.scrollTop,t.style.top="".concat(o,"px"),t.setAttribute("readonly",""),t.value=n,t);return e.container.appendChild(t),e=r()(t),c("copy"),t.remove(),e}var f=function(t){var e=10?setTimeout((function(){e(r,n,s)}),1):(t.update(),n(s))}}},"./src/filter.js":function(t){t.exports=function(t){return t.handlers.filterStart=t.handlers.filterStart||[],t.handlers.filterComplete=t.handlers.filterComplete||[],function(e){if(t.trigger("filterStart"),t.i=1,t.reset.filter(),void 0===e)t.filtered=!1;else{t.filtered=!0;for(var r=t.items,n=0,s=r.length;nv.page,a=new g(t[s],void 0,n),v.items.push(a),r.push(a)}return v.update(),r}m(t.slice(0),e)}},this.show=function(t,e){return this.i=t,this.page=e,v.update(),v},this.remove=function(t,e,r){for(var n=0,s=0,i=v.items.length;s-1&&r.splice(n,1),v},this.trigger=function(t){for(var e=v.handlers[t].length;e--;)v.handlers[t][e](v);return v},this.reset={filter:function(){for(var t=v.items,e=t.length;e--;)t[e].filtered=!1;return v},search:function(){for(var t=v.items,e=t.length;e--;)t[e].found=!1;return v}},this.update=function(){var t=v.items,e=t.length;v.visibleItems=[],v.matchingItems=[],v.templater.clear();for(var r=0;r=v.i&&v.visibleItems.lengthe},innerWindow:function(t,e,r){return t>=e-r&&t<=e+r},dotted:function(t,e,r,n,s,i,a){return this.dottedLeft(t,e,r,n,s,i)||this.dottedRight(t,e,r,n,s,i,a)},dottedLeft:function(t,e,r,n,s,i){return e==r+1&&!this.innerWindow(e,s,i)&&!this.right(e,n)},dottedRight:function(t,e,r,n,s,i,a){return!t.items[a-1].values().dotted&&(e==n&&!this.innerWindow(e,s,i)&&!this.right(e,n))}};return function(e){var n=new i(t.listContainer.id,{listClass:e.paginationClass||"pagination",item:e.item||"
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+ if (categoriesLoaded) { + activateCategory(category); + setCategoryHash(category); + } +}; + +window["quarto-listing-loaded"] = () => { + // Process any existing hash + const hash = getHash(); + + if (hash) { + // If there is a category, switch to that + if (hash.category) { + // category hash are URI encoded so we need to decode it before processing + // so that we can match it with the category element processed in JS + activateCategory(decodeURIComponent(hash.category)); + } + // Paginate a specific listing + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + const page = hash[getListingPageKey(listingId)]; + if (page) { + showPage(listingId, page); + } + } + } + + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + // The actual list + const list = window["quarto-listings"][listingId]; + + // Update the handlers for pagination events + refreshPaginationHandlers(listingId); + + // Render any visible items that need it + renderVisibleProgressiveImages(list); + + // Whenever the list is updated, we also need to + // attach handlers to the new pagination elements + // and refresh any newly visible items. + list.on("updated", function () { + renderVisibleProgressiveImages(list); + setTimeout(() => refreshPaginationHandlers(listingId)); + + // Show or hide the no matching message + toggleNoMatchingMessage(list); + }); + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Attach click handlers to categories + const categoryEls = window.document.querySelectorAll( + ".quarto-listing-category .category" + ); + + for (const categoryEl of categoryEls) { + // category needs to support non ASCII characters + const category = decodeURIComponent( + atob(categoryEl.getAttribute("data-category")) + ); + categoryEl.onclick = () => { + activateCategory(category); + setCategoryHash(category); + }; + } + + // Attach a click handler to the category title + // (there should be only one, but since it is a class name, handle N) + const categoryTitleEls = window.document.querySelectorAll( + ".quarto-listing-category-title" + ); + for (const categoryTitleEl of categoryTitleEls) { + categoryTitleEl.onclick = () => { + activateCategory(""); + setCategoryHash(""); + }; + } + + categoriesLoaded = true; +}); + +function toggleNoMatchingMessage(list) { + const selector = `#${list.listContainer.id} .listing-no-matching`; + const noMatchingEl = window.document.querySelector(selector); + if (noMatchingEl) { + if (list.visibleItems.length === 0) { + noMatchingEl.classList.remove("d-none"); + } else { + if (!noMatchingEl.classList.contains("d-none")) { + noMatchingEl.classList.add("d-none"); + } + } + } +} + +function setCategoryHash(category) { + setHash({ category }); +} + +function setPageHash(listingId, page) { + const currentHash = getHash() || {}; + currentHash[getListingPageKey(listingId)] = page; + setHash(currentHash); +} + +function getListingPageKey(listingId) { + return `${listingId}-page`; +} + +function refreshPaginationHandlers(listingId) { + const listingEl = window.document.getElementById(listingId); + const paginationEls = listingEl.querySelectorAll( + ".pagination li.page-item:not(.disabled) .page.page-link" + ); + for (const paginationEl of paginationEls) { + paginationEl.onclick = (sender) => { + setPageHash(listingId, sender.target.getAttribute("data-i")); + showPage(listingId, sender.target.getAttribute("data-i")); + return false; + }; + } +} + +function renderVisibleProgressiveImages(list) { + // Run through the visible items and render any progressive images + for (const item of list.visibleItems) { + const itemEl = item.elm; + if (itemEl) { + const progressiveImgs = itemEl.querySelectorAll( + `img[${kProgressiveAttr}]` + ); + for (const progressiveImg of progressiveImgs) { + const srcValue = progressiveImg.getAttribute(kProgressiveAttr); + if (srcValue) { + progressiveImg.setAttribute("src", srcValue); + } + progressiveImg.removeAttribute(kProgressiveAttr); + } + } + } +} + +function getHash() { + // Hashes are of the form + // #name:value|name1:value1|name2:value2 + const currentUrl = new URL(window.location); + const hashRaw = currentUrl.hash ? currentUrl.hash.slice(1) : undefined; + return parseHash(hashRaw); +} + +const kAnd = "&"; +const kEquals = "="; + +function parseHash(hash) { + if (!hash) { + return undefined; + } + const hasValuesStrs = hash.split(kAnd); + const hashValues = hasValuesStrs + .map((hashValueStr) => { + const vals = hashValueStr.split(kEquals); + if (vals.length === 2) { + return { name: vals[0], value: vals[1] }; + } else { + return undefined; + } + }) + .filter((value) => { + return value !== undefined; + }); + + const hashObj = {}; + hashValues.forEach((hashValue) => { + hashObj[hashValue.name] = decodeURIComponent(hashValue.value); + }); + return hashObj; +} + +function makeHash(obj) { + return Object.keys(obj) + .map((key) => { + return `${key}${kEquals}${obj[key]}`; + }) + .join(kAnd); +} + +function setHash(obj) { + const hash = makeHash(obj); + window.history.pushState(null, null, `#${hash}`); +} + +function showPage(listingId, page) { + const list = window["quarto-listings"][listingId]; + if (list) { + list.show((page - 1) * list.page + 1, list.page); + } +} + +function activateCategory(category) { + // Deactivate existing categories + const activeEls = window.document.querySelectorAll( + ".quarto-listing-category .category.active" + ); + for (const activeEl of activeEls) { + activeEl.classList.remove("active"); + } + + // Activate this category + const categoryEl = window.document.querySelector( + `.quarto-listing-category .category[data-category='${btoa( + encodeURIComponent(category) + )}']` + ); + if (categoryEl) { + categoryEl.classList.add("active"); + } + + // Filter the listings to this category + filterListingCategory(category); +} + +function filterListingCategory(category) { + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + const list = window["quarto-listings"][listingId]; + if (list) { + if (category === "") { + // resets the filter + list.filter(); + } else { + // filter to this category + list.filter(function (item) { + const itemValues = item.values(); + if (itemValues.categories !== null) { + const categories = decodeURIComponent( + atob(itemValues.categories) + ).split(","); + return categories.includes(category); + } else { + return false; + } + }); + } + } + } +} diff --git a/_quarto.yml b/_quarto.yml index 2551077..5955987 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -16,4 +16,3 @@ format: css: styles.css - diff --git a/docs/index.html b/docs/index.html index 08f7ba4..d9bac08 100644 --- a/docs/index.html +++ b/docs/index.html @@ -171,7 +171,7 @@ +
    Categories
    All (1)
    probabilistic-ml (1)
    probability-theory (1)
    @@ -200,27 +200,29 @@

    Latest Posts

    -
    -
    +
    +
     

    -Welcome To My Blog +Bayesian Inference and Factor Graphs

    -
    test
    +
    probabilistic-ml
    + +
    probability-theory
    -
    -This is a test entry! +
    +Derived from a fundamental result in probability theory, the Bayesian Inference Model is one of the basic concepts in probabilistic machine learning. It represents the…
    - +
    Jul 29, 2025
    diff --git a/docs/listings.json b/docs/listings.json index ddfdfa1..a599e10 100644 --- a/docs/listings.json +++ b/docs/listings.json @@ -2,7 +2,7 @@ { "listing": "/index.html", "items": [ - "/posts/welcome/index.html" + "/posts/factor_graphs/index.html" ] } ] \ No newline at end of file diff --git a/docs/posts/welcome/index.html b/docs/posts/factor_graphs/index.html similarity index 76% rename from docs/posts/welcome/index.html rename to docs/posts/factor_graphs/index.html index 634f1ce..f62a7b8 100644 --- a/docs/posts/welcome/index.html +++ b/docs/posts/factor_graphs/index.html @@ -9,7 +9,7 @@ -Welcome To My Blog – ncgrmm.github.io +Bayesian Inference and Factor Graphs – ncgrmm.github.io