{"id":1893,"date":"2019-03-28T12:00:00","date_gmt":"2019-03-28T11:00:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/conditional-probability-introduction-to-bayes-rule\/"},"modified":"2026-07-05T03:22:24","modified_gmt":"2026-07-05T01:22:24","slug":"conditional-probability-introduction-to-bayes-rule","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/conditional-probability-introduction-to-bayes-rule\/","title":{"rendered":"Conditional Probability \u2013 Introduction to Bayes\u2019 Rule"},"content":{"rendered":"<p>Bayes Rule is based on Conditional Probability which we have been discussing in the previous 3 lessons.<\/p>\n<ul>\n<li><strong><a class=\"row-title\" href=\"https:\/\/kindsonthegenius.com\/tempsite\/basics-of-conditional-probability\/\" label=\"\u201cConditional Probability \u2013 Basics\u201d (Edit)\">Conditional Probability \u2013 Basics<\/a><\/strong><\/li>\n<li><strong><a class=\"row-title\" href=\"https:\/\/kindsonthegenius.com\/tempsite\/conditional-probability-multiplication-law\/\" label=\"\u201cConditional Probability \u2013 Multiplication Law\u201d (Edit)\">Conditional Probability \u2013 Multiplication Law<\/a><\/strong><\/li>\n<li><strong><a class=\"row-title\" href=\"https:\/\/kindsonthegenius.com\/tempsite\/conditional-probability-law-of-total-probability\/\" label=\"\u201cConditional Probability \u2013 Law of Total Probability\u201d (Edit)\">Conditional Probability \u2013 Law of Total Probability<\/a><\/strong><\/li>\n<\/ul>\n<p>So I recommend you start from the basics. This\u00a0 is because you need these basics to understand Bayes&#8217; Rule.<\/p>\n<p>Now, we would state Bayes&#8217; rule and you will see it&#8217;s the same as what we already derived.<\/p>\n<p>Let A and B1, . . . , Bn be events such that Bi are disjoint,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-604\" src=\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Bayes-Rule-pre.jpg\" alt=\"\" width=\"118\" height=\"75\" \/><\/p>\n<p>and P(Bi) &gt; 0 for all i, then:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-605\" src=\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Bayes-Rule-300x63.jpg\" alt=\"\" width=\"300\" height=\"63\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Let&#8217;s take some examples to illustrate how it works.<\/p>\n<hr \/>\n<p><strong>Example 1<\/strong><\/p>\n<p>Bayes&#8217; rule was applied in diagnosing coronary artery disease. This is through a procedure called cardiac fluoroscopy. This procedure determines if there is calcification of the coronary arteries and hence diagnose coronary artery disease.<\/p>\n<p>From the test carried out, number of arteries calcified could be determined. If it is 0, 1, 2 or 3.<\/p>\n<p>Let T0, T1, T2, T3 denote these events.<\/p>\n<p>Also let D+ denote that the disease is present<\/p>\n<p>And D- denotes that the disease is absent<\/p>\n<p>The results of the studies is presented in the table below<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-610\" src=\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Coronary-Artery-Disease-1-300x186.jpg\" alt=\"\" width=\"300\" height=\"186\" \/><\/p>\n<p>Now we need to calculate:<\/p>\n<ol>\n<li>chances that the patient have the disease given that no artery is calcified &#8211; P(D+ | T0)<\/li>\n<\/ol>\n<p>Let&#8217;s assume that the P(D+) is 0.05<\/p>\n<hr \/>\n<p>Applying Bayes&#8217; rule to the above table, we have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-609 \" src=\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Coronary-Artery-Disease-Bayes-Rule.jpg\" alt=\"\" width=\"508\" height=\"62\" \/><\/p>\n<p>Now, we have P(D+) = 0.5<\/p>\n<p>Therefore P(D-) = 1 &#8211; P(D+) = 0.95<\/p>\n<p>So :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-611 \" src=\"https:\/\/www.kindsonthegenius.com\/wp-content\/uploads\/2020\/09\/Bayes-1.jpg\" alt=\"\" width=\"451\" height=\"218\" \/><\/p>\n<hr \/>\n<p><b>Exercises<\/b><\/p>\n<ol>\n<li>Find\u00a0 the\u00a0<span style=\"font-size: 1rem;\">chances that patient have the disease given that only on artery is calcified &#8211; P(D+ | T1)<\/span><\/li>\n<li>Calculate the same values in Example 1, this time using P(D+) = 0.92<\/li>\n<\/ol>\n<hr \/>\n<p>In the next lesson, we would further examine Bayes&#8217; Rule using other examples.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bayes Rule is based on Conditional Probability which we have been discussing in the previous 3 lessons. Conditional Probability \u2013 Basics Conditional Probability \u2013 Multiplication &hellip; <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[414],"tags":[],"class_list":["post-1893","post","type-post","status-publish","format-standard","hentry","category-programming"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1893","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=1893"}],"version-history":[{"count":1,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1893\/revisions"}],"predecessor-version":[{"id":2061,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1893\/revisions\/2061"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=1893"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=1893"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=1893"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}