{"categories":["Probability Theory"],"contentHtml":"<p>A probability measure \\(\\mathbb P\\) assigns a number in \\([0,1]\\) to every event in \\(\\mathcal F\\). The notes begin with \\(\\mathbb P(\\Omega)=1\\) and countable additivity for disjoint events:</p>\n<p>$$\\mathbb P\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)=\\sum_{n=1}^{\\infty}\\mathbb P(A_n).$$</p>\n<p>From those axioms follow the familiar rules for complements, inclusion, and finite unions. Conditioning then changes the measure after new information arrives. For \\(\\mathbb P(B)&gt;0\\),</p>\n<p>$$\\mathbb P(A\\mid B)=\\frac{\\mathbb P(A\\cap B)}{\\mathbb P(B)}.$$</p>\n<p>The total-probability and Bayes formulas update a partition of the sample space when an observation occurs. The notes emphasize that Bayes' rule is a bookkeeping identity for the same joint event, not a new probability model. Independence is the special case in which \\(\\mathbb P(A\\cap B)=\\mathbb P(A)\\mathbb P(B)\\).</p>","contentMarkdown":"A probability measure \\(\\mathbb P\\) assigns a number in \\([0,1]\\) to every event in \\(\\mathcal F\\). The notes begin with \\(\\mathbb P(\\Omega)=1\\) and countable additivity for disjoint events:\n\n$$\\mathbb P\\left(\\bigcup_{n=1}^{\\infty}A_n\\right)=\\sum_{n=1}^{\\infty}\\mathbb P(A_n).$$\n\nFrom those axioms follow the familiar rules for complements, inclusion, and finite unions. Conditioning then changes the measure after new information arrives. For \\(\\mathbb P(B)>0\\),\n\n$$\\mathbb P(A\\mid B)=\\frac{\\mathbb P(A\\cap B)}{\\mathbb P(B)}.$$\n\nThe total-probability and Bayes formulas update a partition of the sample space when an observation occurs. The notes emphasize that Bayes' rule is a bookkeeping identity for the same joint event, not a new probability model. Independence is the special case in which \\(\\mathbb P(A\\cap B)=\\mathbb P(A)\\mathbb P(B)\\).","dataUrl":"https://sharifhsn.dev/api/posts/probability-measures-and-bayes-rule.json","date":"2024-09-09","datePublished":"2024-09-09","description":"A probability measure \\(\\mathbb P\\) assigns a number in \\([0,1]\\) to every event in \\(\\mathcal F\\). The notes begin with \\(\\mathbb P(\\Omega)=1\\) and countable additivity for disjoi…","site":"https://sharifhsn.dev","slug":"probability-measures-and-bayes-rule","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Conditional Probability","Bayes Rule"],"title":"Probability Measures and Bayes' Rule","url":"https://sharifhsn.dev/blog/probability-measures-and-bayes-rule/","version":"1","wordCount":110}