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</div>		<h1 class="firstHeading">Bayes' theorem</h1>
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			<p><b>Bayes' theorem</b> (also known as <b>Bayes' rule</b> or <b>Bayes' law</b>) is a result in <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">probability theory</a>, which relates the <a href="http://en.wikipedia.org/wiki/Conditional_probability" title="Conditional probability">conditional</a> and <a href="http://en.wikipedia.org/wiki/Conditional_probability" title="Conditional probability">marginal</a> <a href="http://en.wikipedia.org/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> of <a href="http://en.wikipedia.org/wiki/Random_variable" title="Random variable">random variables</a>. In some interpretations of <a href="http://en.wikipedia.org/wiki/Probability" title="Probability">probability</a>, Bayes' theorem tells how to update or revise beliefs in light of new evidence <i><a href="http://en.wikipedia.org/wiki/A_posteriori" title="A posteriori">a posteriori</a></i>.</p>
<p>The probability of an <a href="http://en.wikipedia.org/wiki/Event_%28probability_theory%29" title="Event (probability theory)">event</a> <i>A</i> conditional on another event <i>B</i> is generally different from the probability of <i>B</i> conditional on <i>A</i>. However, there is a definite relationship between the two, and Bayes' theorem is the statement of that relationship.</p>
<p>As a formal <a href="http://en.wikipedia.org/wiki/Theorem" title="Theorem">theorem</a>, Bayes' theorem is valid in all interpretations of probability. However, <a href="http://en.wikipedia.org/wiki/Frequentist" title="Frequentist">frequentist</a> and <a href="http://en.wikipedia.org/wiki/Bayesian_probability" title="Bayesian probability">Bayesian</a>
interpretations disagree about the kinds of things to which
probabilities should be assigned in applications: frequentists assign
probabilities to random events according to their frequencies of
occurrence or to subsets of populations as proportions of the whole;
Bayesians assign probabilities to propositions that are uncertain. A
consequence is that Bayesians have more frequent occasion to use Bayes'
theorem. The articles on <a href="http://en.wikipedia.org/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> and <a href="http://en.wikipedia.org/wiki/Frequency_probability" title="Frequency probability">frequentist probability</a> discuss these debates at greater length.</p>
<table id="toc" class="toc" summary="Contents">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="javascript:toggleToc()" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1"><a href="#Statement_of_Bayes.27_theorem"><span class="tocnumber">1</span> <span class="toctext">Statement of Bayes' theorem</span></a></li>
<li class="toclevel-1"><a href="#Derivation_from_conditional_probabilities"><span class="tocnumber">2</span> <span class="toctext">Derivation from conditional probabilities</span></a></li>
<li class="toclevel-1"><a href="#Alternative_forms_of_Bayes.27_theorem"><span class="tocnumber">3</span> <span class="toctext">Alternative forms of Bayes' theorem</span></a>
<ul>
<li class="toclevel-2"><a href="#Bayes.27_theorem_in_terms_of_odds_and_likelihood_ratio"><span class="tocnumber">3.1</span> <span class="toctext">Bayes' theorem in terms of odds and likelihood ratio</span></a></li>
<li class="toclevel-2"><a href="#Bayes.27_theorem_for_probability_densities"><span class="tocnumber">3.2</span> <span class="toctext">Bayes' theorem for probability densities</span></a></li>
<li class="toclevel-2"><a href="#Abstract_Bayes.27_theorem"><span class="tocnumber">3.3</span> <span class="toctext">Abstract Bayes' theorem</span></a></li>
<li class="toclevel-2"><a href="#Extensions_of_Bayes.27_theorem"><span class="tocnumber">3.4</span> <span class="toctext">Extensions of Bayes' theorem</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Examples"><span class="tocnumber">4</span> <span class="toctext">Examples</span></a>
<ul>
<li class="toclevel-2"><a href="#Example_.231:__Conditional_probabilities"><span class="tocnumber">4.1</span> <span class="toctext">Example #1: Conditional probabilities</span></a>
<ul>
<li class="toclevel-3"><a href="#Tables_of_occurrences_and_relative_frequencies"><span class="tocnumber">4.1.1</span> <span class="toctext">Tables of occurrences and relative frequencies</span></a></li>
</ul>
</li>
<li class="toclevel-2"><a href="#Example_.232:__Drug_testing"><span class="tocnumber">4.2</span> <span class="toctext">Example #2: Drug testing</span></a></li>
<li class="toclevel-2"><a href="#Example_.233:__Bayesian_inference"><span class="tocnumber">4.3</span> <span class="toctext">Example #3: Bayesian inference</span></a></li>
<li class="toclevel-2"><a href="#Example_.234:_The_Monty_Hall_problem"><span class="tocnumber">4.4</span> <span class="toctext">Example #4: The Monty Hall problem</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Historical_remarks"><span class="tocnumber">5</span> <span class="toctext">Historical remarks</span></a></li>
<li class="toclevel-1"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2"><a href="#Versions_of_the_essay"><span class="tocnumber">7.1</span> <span class="toctext">Versions of the essay</span></a></li>
<li class="toclevel-2"><a href="#Commentaries"><span class="tocnumber">7.2</span> <span class="toctext">Commentaries</span></a></li>
<li class="toclevel-2"><a href="#Additional_material"><span class="tocnumber">7.3</span> <span class="toctext">Additional material</span></a></li>
</ul>
</li>
</ul>
</td>
</tr>
</tbody></table>
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<p><a name="Statement_of_Bayes.27_theorem" id="Statement_of_Bayes.27_theorem"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=1" title="Edit section: Statement of Bayes' theorem">edit</a>]</span> <span class="mw-headline">Statement of Bayes' theorem</span></h2>
<p>Bayes' theorem relates the conditional and marginal probabilities of <a href="http://en.wikipedia.org/wiki/Stochastic_process" title="Stochastic process">stochastic</a> <a href="http://en.wikipedia.org/wiki/Event_%28probability_theory%29" title="Event (probability theory)">events</a> <i>A</i> and <i>B</i>:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/246afc51d522fc2830a4e2ab6a52459d.png" alt="\begin{align}   P(A|B)  &amp;  =       \frac{P(B | A)\, P(A)}{P(B)}  \\             &amp;  \propto L(A | B)\, P(A)   \end{align}"></dd>
</dl>
<p>where <i>L</i>(<i>A</i>|<i>B</i>) is the <a href="http://en.wikipedia.org/wiki/Likelihood_function" title="Likelihood function">likelihood</a> of <i>A</i> given fixed <i>B</i>. Although in this case the relationship <span class="texhtml"><i>P</i>(<i>B</i> | <i>A</i>) = <i>L</i>(<i>A</i> | <i>B</i>)</span>, in other cases <a href="http://en.wikipedia.org/wiki/Likelihood_function" title="Likelihood function">likelihood</a> <i>L</i> can be multiplied by a constant factor, so that it is proportional to, but does not equal <a href="http://en.wikipedia.org/wiki/Probability" title="Probability">probability</a> <i>P</i>.</p>
<p>Each term in Bayes' theorem has a conventional name:</p>
<ul>
<li>P(<i>A</i>) is the <a href="http://en.wikipedia.org/wiki/Prior_probability" title="Prior probability">prior probability</a> or <a href="http://en.wikipedia.org/wiki/Marginal_probability" title="Marginal probability">marginal probability</a> of <i>A</i>. It is "prior" in the sense that it does not take into account any information about <i>B</i>.</li>
<li>P(<i>A</i>|<i>B</i>) is the <a href="http://en.wikipedia.org/wiki/Conditional_probability" title="Conditional probability">conditional probability</a> of <i>A</i>, given <i>B</i>. It is also called the <a href="http://en.wikipedia.org/wiki/Posterior_probability" title="Posterior probability">posterior probability</a> because it is derived from or depends upon the specified value of <i>B</i>.</li>
<li>P(<i>B</i>|<i>A</i>) is the conditional probability of <i>B</i> given <i>A</i>.</li>
<li>P(<i>B</i>) is the prior or marginal probability of <i>B</i>, and acts as a <a href="http://en.wikipedia.org/wiki/Normalizing_constant" title="Normalizing constant">normalizing constant</a>.</li>
</ul>
<p>With this terminology, the theorem may be paraphrased as</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/3e294d8052aa226062a1a1158be30079.png" alt="\mbox{posterior} = \frac{\mbox{likelihood} \times \mbox{prior}} {\mbox{normalizing constant}}"></dd>
</dl>
<p>In words: the posterior probability is proportional to the product of the prior probability and the likelihood.</p>
<p>In addition, the ratio P(<i>B</i>|<i>A</i>)/P(<i>B</i>) is sometimes called the standardised likelihood, so the theorem may also be paraphrased as</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/9c1f50e099b3ae7e1ece4cb35ca21356.png" alt="\mbox{posterior} = {\mbox{standardised likelihood} \times \mbox{prior} }.\,"></dd>
</dl>
<p><a name="Derivation_from_conditional_probabilities" id="Derivation_from_conditional_probabilities"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=2" title="Edit section: Derivation from conditional probabilities">edit</a>]</span> <span class="mw-headline">Derivation from conditional probabilities</span></h2>
<p>To derive the theorem, we start from the definition of <a href="http://en.wikipedia.org/wiki/Conditional_probability" title="Conditional probability">conditional probability</a>. The probability of event <i>A</i> given event <i>B</i> is</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/a6df7814a46fc4c9670cc510b72bb318.png" alt="P(A|B)=\frac{P(A \cap B)}{P(B)}."></dd>
</dl>
<p>Likewise, the probability of event <i>B</i> given event <i>A</i> is</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/8b6c81124815aad54c91c42b3261165d.png" alt="P(B|A) = \frac{P(A \cap B)}{P(A)}. \!"></dd>
</dl>
<p>Rearranging and combining these two equations, we find</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/efaf8fda8a92eeb2d8cf70468c20ed5a.png" alt="P(A|B)\, P(B) = P(A \cap B) = P(B|A)\, P(A). \!"></dd>
</dl>
<p>This <a href="http://en.wikipedia.org/wiki/Lemma_%28mathematics%29" title="Lemma (mathematics)">lemma</a> is sometimes called the product rule for probabilities. Dividing both sides by Pr(<i>B</i>), providing that it is non-zero, we obtain Bayes' theorem:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/f9fdf7c75c7f4ddae1140dd2a9e73714.png" alt="P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}. \!"></dd>
</dl>
<p><a name="Alternative_forms_of_Bayes.27_theorem" id="Alternative_forms_of_Bayes.27_theorem"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=3" title="Edit section: Alternative forms of Bayes' theorem">edit</a>]</span> <span class="mw-headline">Alternative forms of Bayes' theorem</span></h2>
<p>Bayes' theorem is often embellished by noting that</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/07cf5446f48e526e1abcdb471e60089a.png" alt="P(B) = P(A\cap B) + P(A^C\cap B) = P(B|A) P(A) + P(B|A^C) P(A^C)\,"></dd>
</dl>
<p>where <i>A</i><sup><i>C</i></sup> is the <a href="http://en.wikipedia.org/wiki/Complement_%28set_theory%29#Absolute_complement" title="Complement (set theory)">complementary</a> event of <i>A</i> (often called "not A"). So the theorem can be restated as</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/b337da08b983c9e9c6f741d856b4b72c.png" alt="P(A|B) = \frac{P(B | A)\, P(A)}{P(B|A) P(A) + P(B|A^C) P(A^C)}.  \!"></dd>
</dl>
<p>More generally, where {<i>A</i><sub><i>i</i></sub>} forms a <a href="http://en.wikipedia.org/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of the event space,</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/daa4b03e8ea857a6957e4a1d0b8ef6b1.png" alt="P(A_i|B) = \frac{P(B | A_i)\, P(A_i)}{\sum_j P(B|A_j)\,P(A_j)} , \!"></dd>
</dl>
<p>for any <i>A</i><sub><i>i</i></sub> in the partition.</p>
<p>See also the <a href="http://en.wikipedia.org/wiki/Law_of_total_probability" title="Law of total probability">law of total probability</a>.</p>
<p><a name="Bayes.27_theorem_in_terms_of_odds_and_likelihood_ratio" id="Bayes.27_theorem_in_terms_of_odds_and_likelihood_ratio"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=4" title="Edit section: Bayes' theorem in terms of odds and likelihood ratio">edit</a>]</span> <span class="mw-headline">Bayes' theorem in terms of odds and likelihood ratio</span></h3>
<p>Bayes' theorem can also be written neatly in terms of a <a href="http://en.wikipedia.org/wiki/Likelihood_function" title="Likelihood function">likelihood</a> ratio Λ and <a href="http://en.wikipedia.org/wiki/Odds" title="Odds">odds</a> <i>O</i> as</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/6aa65315e6ae2da1b417789e43b4eb72.png" alt="O(A|B)=O(A) \cdot \Lambda (A|B)"></dd>
</dl>
<p>where <img class="tex" src="Bayes%27_theorem%20Files/81be09df6b27e9b3d3ee7c8e4e6ced85.png" alt="O(A|B)=\frac{P(A|B)}{P(A^C|B)} \!"> are the <i>odds</i> of <i>A</i> given <i>B</i>,</p>
<p>and <img class="tex" src="Bayes%27_theorem%20Files/c4451387126a4c6a51739382cacb9e99.png" alt="O(A)=\frac{P(A)}{P(A^C)} \!"> are the odds of <i>A</i> by itself,</p>
<p>while <img class="tex" src="Bayes%27_theorem%20Files/32146e2a0949cb320bfd837305e6c133.png" alt="\Lambda (A|B) = \frac{L(A|B)}{L(A^C|B)} = \frac{P(B|A)}{P(B|A^C)} \!"> is the likelihood ratio.</p>
<p><a name="Bayes.27_theorem_for_probability_densities" id="Bayes.27_theorem_for_probability_densities"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=5" title="Edit section: Bayes' theorem for probability densities">edit</a>]</span> <span class="mw-headline">Bayes' theorem for probability densities</span></h3>
<p>There is also a version of Bayes' theorem for continuous
distributions. It is somewhat harder to derive, since probability
densities, strictly speaking, are not probabilities, so Bayes' theorem
has to be established by a limit process; see Papoulis (citation
below), Section 7.3 for an elementary derivation. Bayes's theorem for
probability densities is formally similar to the theorem for
probabilities:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/402214a6b5fb1babe545afc206a96d92.png" alt="f(x|y) = \frac{f(x,y)}{f(y)} = \frac{f(y|x)\,f(x)}{f(y)} \!"></dd>
</dl>
<p>and there is an analogous statement of the law of total probability:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/908da7e26f756ae79f5ac0bfce87c5ce.png" alt="f(x|y) = \frac{f(y|x)\,f(x)}{\int_{-\infty}^{\infty} f(y|x)\,f(x)\,dx}. \!"></dd>
</dl>
<p>As in the discrete case, the terms have standard names. <i>f</i>(<i>x</i>, <i>y</i>) is the joint distribution of <i>X</i> and <i>Y</i>, <i>f</i>(<i>x</i>|<i>y</i>) is the posterior distribution of <i>X</i> given <i>Y</i>=<i>y</i>, <i>f</i>(<i>y</i>|<i>x</i>) = <i>L</i>(<i>x</i>|<i>y</i>) is (as a function of <i>x</i>) the likelihood function of <i>X</i> given <i>Y</i>=<i>y</i>, and <i>f</i>(<i>x</i>) and <i>f</i>(<i>y</i>) are the marginal distributions of <i>X</i> and <i>Y</i> respectively, with <i>f</i>(<i>x</i>) being the prior distribution of <i>X</i>.</p>
<p>Here we have indulged in a conventional <a href="http://en.wikipedia.org/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>, using <i>f</i>
for each one of these terms, although each one is really a different
function; the functions are distinguished by the names of their
arguments.</p>
<p><a name="Abstract_Bayes.27_theorem" id="Abstract_Bayes.27_theorem"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=6" title="Edit section: Abstract Bayes' theorem">edit</a>]</span> <span class="mw-headline">Abstract Bayes' theorem</span></h3>
<p>Given two <a href="http://en.wikipedia.org/wiki/Absolutely_continuous" title="Absolutely continuous">absolutely continuous</a> probability measures <span class="texhtml"><i>P</i>˜<i>Q</i></span> on the probability space <img class="tex" src="Bayes%27_theorem%20Files/b9474d45d82accf03434710c10871795.png" alt="(\Omega, \mathcal{F})"> and a sigma-algebra <img class="tex" src="Bayes%27_theorem%20Files/6956beed6709f29c47056603dd448e37.png" alt="\mathcal{G} \subset \mathcal{F}">, the abstract Bayes theorem for a <img class="tex" src="Bayes%27_theorem%20Files/26afd73f8c17f310707120691ccc4a35.png" alt="\mathcal{F}">-measurable random variable <span class="texhtml"><i>X</i></span> becomes</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/bcb58d4f262347072d35dea4a65977dd.png" alt="E_P[X|\mathcal{G}] = \frac{E_Q[\frac{dP}{dQ} X |\mathcal{G}]}{E_Q[\frac{dP}{dQ}|\mathcal{G}]}">.</dd>
</dl>
<p>This formulation is used in <a href="http://en.wikipedia.org/wiki/Kalman_filtering" title="Kalman filtering">Kalman filtering</a> to find <a href="http://en.wikipedia.org/wiki/Zakai_equation" title="Zakai equation">Zakai equations</a>. It is also used in <a href="http://en.wikipedia.org/wiki/Financial_mathematics" title="Financial mathematics">financial mathematics</a> for change of <a href="http://en.wikipedia.org/wiki/Numeraire" title="Numeraire">numeraire</a> techniques.</p>
<p><a name="Extensions_of_Bayes.27_theorem" id="Extensions_of_Bayes.27_theorem"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=7" title="Edit section: Extensions of Bayes' theorem">edit</a>]</span> <span class="mw-headline">Extensions of Bayes' theorem</span></h3>
<p>Theorems analogous to Bayes' theorem hold in problems with more than two variables. For example:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/01b89e228d286117e3f007b0bc3b67c0.png" alt="P(A|B,C) = \frac{P(A) \, P(B|A) \, P(C|A,B)}{P(B) \, P(C|B)}"></dd>
</dl>
<p>This can be derived in several steps from Bayes' theorem and the definition of conditional probability:</p>
<dl>
<dd><img class="tex" src="Bayes%27_theorem%20Files/0000c57867374dbd1b7837882c5f7e5c.png" alt="P(A|B,C) = \frac{P(A,B,C)}{P(B,C)} = \frac{P(A,B,C)}{P(B) \, P(C|B)} ="></dd>
<dd><img class="tex" src="Bayes%27_theorem%20Files/2203d39e737f922023985141bee96496.png" alt="= \frac{P(C|A,B) \, P(A,B)}{P(B) \, P(C|B)} = \frac{P(A) \, P(B|A) \, P(C|A,B)}{P(B) \, P(C|B)} ."></dd>
</dl>
<p>A general strategy is to work with a decomposition of the <a href="http://en.wikipedia.org/wiki/Joint_probability" title="Joint probability">joint probability</a>,
and to marginalize (integrate) over the variables that are not of
interest. Depending on the form of the decomposition, it may be
possible to prove that some integrals must be 1, and thus they fall out
of the decomposition; exploiting this property can reduce the
computations very substantially. A <a href="http://en.wikipedia.org/wiki/Bayesian_network" title="Bayesian network">Bayesian network</a>, for example, specifies a factorization of a <a href="http://en.wikipedia.org/wiki/Joint_distribution" title="Joint distribution">joint distribution</a>
of several variables in which the conditional probability of any one
variable given the remaining ones takes a particularly simple form (see
<a href="http://en.wikipedia.org/wiki/Markov_blanket" title="Markov blanket">Markov blanket</a>).</p>
<p><a name="Examples" id="Examples"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=8" title="Edit section: Examples">edit</a>]</span> <span class="mw-headline">Stochastic Hoody-hoo</span></h2>
<p><a name="Example_.231:__Conditional_probabilities" id="Example_.231:__Conditional_probabilities"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=9" title="Edit section: Example #1:  Conditional probabilities">edit</a>]</span> <span class="mw-headline">Blahbitty Bloohbitty Blee</span></h3>
<p>Suppose there are blah blah blitty blah. Yada yada, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. Yada yada, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. Yada yada, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee.</p>

<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=9" title="Edit section: Example #2:  Conditional probabilities">edit</a>]</span> <span class="mw-headline">You are getting very sleepy...</span></h3>

<p>Then you can yada yada bloopy blah. Yada yada, yada yada bleepity Blee, yada blee blah yaditty yada ya. Blahbloopy yada yada blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, yada yada yaddity yada ya, yada yada blahbloopy yada yada bloohbloopy blee. Yada yada, yada yada bleepity Blee, yada blee blah yaditty yada ya. Blahbloopy yada yada blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, yada yada yaddity yada ya, yada yada blahbloopy yada yada bloohbloopy blee. Yada yada, yada yada bleepity Blee, yada blee blah yaditty yada ya. Blahbloopy yada yada blech blah, blechitty blah yada yada ying yang yong. Foo bar baz, yada yada yaddity yada ya, yada yada blahbloopy yada yada bloohbloopy blee.</p>


<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=9" title="Edit section: Example #2:  Conditional probabilities">edit</a>]</span> <span class="mw-headline">Verrrrry Sleeeeeepy.......</span></h3>

<p>We find that blah blah blitty blah. blah blah, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. blah blah, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. blah blah, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. blah blah, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee. blah blah, blah blah bleepity Blee, yada blee blah yaditty yada ya. Blahbitty blah blah blech blah, blechitty blah blah blah ying yang yong. Foo bar baz, blah blah yaddity yada ya, blah blah blahbitty blah blah bloohbitty blee.</p>


<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=9" title="Edit section: Example #2:  Conditional probabilities">edit</a>]</span> <span class="mw-headline">Your eyelids are sooooo heavy.......</span></h3>

<p>Blah blah blah</p>

<p>Yada yada yada</p>

<p>Blah blah</p>

<p>Yada yada</p>

<p>Blah</p>

<p>Yada</p>

<p>blah</p>

<p>yada</p>

<p>blah</p>

<p>blah</p>

<p>blah</p>

<p>blah</p>

<p>blah</p>

<p>blah</p>

<p>Zzzzz.............</p>

<p><a name="See_also" id="See_also"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=15" title="Edit section: See also">edit</a>]</span> <span class="mw-headline">See also</span></h2>
<div class="references-small" style="-moz-column-count: 2;">
<ul>
<li><a href="http://en.wikipedia.org/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bayesian_spam_filtering" title="Bayesian spam filtering">Bayesian spam filtering</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bogofilter" title="Bogofilter">Bogofilter</a></li>
<li><a href="http://en.wikipedia.org/wiki/Conjugate_prior" title="Conjugate prior">Conjugate prior</a></li>
<li><a href="http://en.wikipedia.org/wiki/Empirical_Bayes_method" title="Empirical Bayes method">Empirical Bayes method</a></li>
<li><a href="http://en.wikipedia.org/wiki/Monty_Hall_problem" title="Monty Hall problem">Monty Hall problem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Occam%27s_razor" title="Occam's razor">Occam's razor</a></li>
<li><a href="http://en.wikipedia.org/wiki/Prosecutor%27s_fallacy" title="Prosecutor's fallacy">Prosecutor's fallacy</a></li>
<li><a href="http://en.wikipedia.org/wiki/Raven_paradox" title="Raven paradox">Raven paradox</a></li>
<li><a href="http://en.wikipedia.org/wiki/Recursive_Bayesian_estimation" title="Recursive Bayesian estimation">Recursive Bayesian estimation</a></li>
<li><a href="http://en.wikipedia.org/wiki/Revising_opinions_in_statistics" title="Revising opinions in statistics">Revising opinions in statistics</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sequential_bayesian_filtering" title="Sequential bayesian filtering">Sequential bayesian filtering</a></li>
</ul>
</div>
<p><a name="References" id="References"></a></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=16" title="Edit section: References">edit</a>]</span> <span class="mw-headline">References</span></h2>
<ol class="references"></ol>
<p><a name="Versions_of_the_essay" id="Versions_of_the_essay"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=17" title="Edit section: Versions of the essay">edit</a>]</span> <span class="mw-headline">Versions of the essay</span></h3>
<ul>
<li>Thomas Bayes (1763), "An Essay towards solving a Problem in the
Doctrine of Chances. By the late Rev. Mr. Bayes, F. R. S. communicated
by Mr. Price, in a letter to John Canton, A. M. F. R. S.", <i><a href="http://en.wikipedia.org/wiki/Philosophical_Transactions" title="Philosophical Transactions">Philosophical Transactions</a>, Giving Some Account of the Present Undertakings, Studies and Labours of the Ingenious in Many Considerable Parts of the World</i> 53:370–418.</li>
<li>Thomas Bayes (1763/1958) "Studies in the History of Probability and
Statistics: IX. Thomas Bayes' Essay Towards Solving a Problem in the
Doctrine of Chances", <i><a href="http://en.wikipedia.org/wiki/Biometrika" title="Biometrika">Biometrika</a></i> 45:296–315. <i>(Bayes' essay in modernized notation)</i></li>
<li>Thomas Bayes <a href="http://www.stat.ucla.edu/history/essay.pdf" class="external text" title="http://www.stat.ucla.edu/history/essay.pdf" rel="nofollow">"An essay towards solving a Problem in the Doctrine of Chances"</a>. <i>(Bayes' essay in the original notation)</i></li>
</ul>
<p><a name="Commentaries" id="Commentaries"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=18" title="Edit section: Commentaries">edit</a>]</span> <span class="mw-headline">Commentaries</span></h3>
<ul>
<li><a href="http://en.wikipedia.org/wiki/George_Alfred_Barnard" title="George Alfred Barnard">G. A. Barnard</a>
(1958) "Studies in the History of Probability and Statistics: IX.
Thomas Bayes' Essay Towards Solving a Problem in the Doctrine of
Chances", <i>Biometrika</i> 45:293–295. <i>(biographical remarks)</i></li>
<li>Daniel Covarrubias. <a href="http://www.stat.rice.edu/%7Eblairc/seminar/Files/danTalk.pdf" class="external text" title="http://www.stat.rice.edu/~blairc/seminar/Files/danTalk.pdf" rel="nofollow">"An Essay Towards Solving a Problem in the Doctrine of Chances"</a>. <i>(an outline and exposition of Bayes' essay)</i></li>
<li>Stephen M. Stigler (1982). "Thomas Bayes' Bayesian Inference," <i>Journal of the Royal Statistical Society</i>, Series A, 145:250–258. <i>(Stigler argues for a revised interpretation of the essay; recommended)</i></li>
<li><a href="http://en.wikipedia.org/wiki/Isaac_Todhunter" title="Isaac Todhunter">Isaac Todhunter</a> (1865). <i>A History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace</i>, Macmillan. Reprinted 1949, 1956 by Chelsea and 2001 by Thoemmes.</li>
</ul>
<p><a name="Additional_material" id="Additional_material"></a></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&amp;action=edit&amp;section=19" title="Edit section: Additional material">edit</a>]</span> <span class="mw-headline">Additional material</span></h3>
<ul>
<li>Pierre-Simon Laplace (1774). "Mémoire sur la Probabilité des Causes par les Événements", <i>Savants Étranges</i> 6:621–656; also <i>Œuvres</i> 8:27–65.</li>
<li>Pierre-Simon Laplace (1774/1986). "Memoir on the Probability of the Causes of Events", <i>Statistical Science</i> 1(3):364–378.</li>
<li>Stephen M. Stigler (1986). "Laplace's 1774 memoir on inverse probability", <i>Statistical Science</i> 1(3):359–378.</li>
<li>Stephen M. Stigler (1983). "Who Discovered Bayes' Theorem?" <i>The American Statistician</i> 37(4):290–296.</li>
<li>Jeff Miller et al. <a href="http://members.aol.com/jeff570/mathword.html" class="external text" title="http://members.aol.com/jeff570/mathword.html" rel="nofollow">Earliest Known Uses of Some of the Words of Mathematics (B)</a>. (<i>very informative; recommended</i>)</li>
<li><a href="http://en.wikipedia.org/wiki/Athanasios_Papoulis" title="Athanasios Papoulis">Athanasios Papoulis</a> (1984). <i>Probability, Random Variables, and Stochastic Processes</i>, second edition. New York: McGraw-Hill.</li>
<li>James Joyce (2003). <a href="http://plato.stanford.edu/entries/Bayes-theorem/" class="external text" title="http://plato.stanford.edu/entries/Bayes-theorem/" rel="nofollow">"Bayes' Theorem"</a>, <i><a href="http://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</li>
<li><a href="http://www.inference.phy.cam.ac.uk/mackay/itila/" class="external text" title="http://www.inference.phy.cam.ac.uk/mackay/itila/" rel="nofollow">The on-line textbook: Information Theory, Inference, and Learning Algorithms</a>, by <a href="http://en.wikipedia.org/wiki/David_J.C._MacKay" title="David J.C. MacKay">David J.C. MacKay</a> provides an up to date overview of the use of Bayes' theorem in information theory and machine learning.</li>
<li><a href="http://plato.stanford.edu/entries/bayes-theorem/" class="external text" title="http://plato.stanford.edu/entries/bayes-theorem/" rel="nofollow">Stanford Encyclopedia of Philosophy: Bayes' Theorem</a> provides a comprehensive introduction to Bayes' theorem.</li>
<li><a href="http://en.wikipedia.org/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <i><a href="http://mathworld.wolfram.com/BayesTheorem.html" class="external text" title="http://mathworld.wolfram.com/BayesTheorem.html" rel="nofollow">Bayes' Theorem</a></i> at <a href="http://en.wikipedia.org/wiki/MathWorld" title="MathWorld">MathWorld</a>.</li>
<li><i><a href="http://planetmath.org/encyclopedia/BayesTheorem.html" class="external text" title="http://planetmath.org/encyclopedia/BayesTheorem.html" rel="nofollow">Bayes' theorem</a></i> at <a href="http://en.wikipedia.org/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li>
<li>Yudkowsky, Eliezer S. (2003), "<a href="http://yudkowsky.net/bayes/bayes.html" class="external text" title="http://yudkowsky.net/bayes/bayes.html" rel="nofollow">An Intuitive Explanation of Bayesian Reasoning</a>"</li>
</ul>

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