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BayesTheorem.slides.html
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<h2 id="Bayes-Theorem">Bayes Theorem<a class="anchor-link" href="#Bayes-Theorem">¶</a></h2><h3 id="Rohan-L.-Fernando">Rohan L. Fernando<a class="anchor-link" href="#Rohan-L.-Fernando">¶</a></h3><h3 id="November-2015">November 2015<a class="anchor-link" href="#November-2015">¶</a></h3>
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<h2 id="Motivation">Motivation<a class="anchor-link" href="#Motivation">¶</a></h2><ul>
<li>In whole-genome analyses, the number $k$ of marker covariates typically exceeds the number $n$ of observations </li>
<li>Cannot use least squares methods to simultaneously estimate the effects of all the $k$ covariates</li>
<li>Bayesian inference is widely used to overcome this problem by combining prior information with the observed data</li>
<li>In the Bayesian approach, inferences are based on
conditional probabilities, and the Bayes theorem is a statement on
conditional probability</li>
</ul>
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<h2 id="Conditional-Probability-of-$X$-Given-$Y$">Conditional Probability of $X$ Given $Y$<a class="anchor-link" href="#Conditional-Probability-of-$X$-Given-$Y$">¶</a></h2><p>Suppose $X$ and $Y$are two random variables with joint probability
distribution $\Pr(X,Y)$. Then, the conditional probability of $X$ given
$Y$ is given by Bayes theorem as</p>
$$\Pr(X|Y) = \frac{\Pr(X,Y)}{\Pr(Y)} \tag{1}$$<p>where $\Pr(Y)$ is the probability distribution of $Y$.</p>
<p>Similarly, thethe conditional probability of $Y$ given $X$ is</p>
$$\Pr(Y|X)=\frac{\Pr(X,Y)}{\Pr(X)},$$<p></p>
<p>which upon rearranging gives</p>
$$\Pr(X,Y)=\Pr(Y|X)\Pr(X). \tag{2}$$<p>Then, substituting (2) in (1) gives</p>
$$\begin{eqnarray}
\Pr(X|Y) &= &\frac{\Pr(X,Y)}{\Pr(Y)}\\
&= &\frac{\Pr(Y|X)\Pr(X)}{\Pr(Y)},
\end{eqnarray}$$<p>which is the form of the formula that is used for inference of $X$ given
$Y.$</p>
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<h2 id="Bayes-Theorem-by-Example">Bayes Theorem by Example<a class="anchor-link" href="#Bayes-Theorem-by-Example">¶</a></h2><p>Here we consider a simple example to justify the formula
(1). The following table gives the joint distribution of
smoking and lung cancer in a hypothetical population of 1,000,000
individuals.</p>
$$
\begin{array}{c|lc|r}
& \text{Smoking} \\
\hline
\text{Cancer} & \text{Yes} & \text{No} & \text{} \\
\hline
\text{Yes} & 42,500 & 7,500 & 50,000 \\
\text{No} & 207,500 & 742,500 & 950,000 \\
\hline
& 250,000 & 750,000 & 1,000,000
\end{array}
$$<ul>
<li>Given these numbers, consider how you would compute the relative
frequency of lung cancer among smokers</li>
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<ul>
<li>There are a total of 250,000 smokers in this population, and among these 250,000 individuals, 42,500
have lung cancer</li>
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<ul>
<li>So, relative frequency of lung cancer among smokers is
$\frac{42,500}{250,000}$</li>
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<ul>
<li>As we reason next, this relative frequency is also the conditional probability of lung cancer given the
individual is a smoker</li>
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<ul>
<li>The frequentist definition of probability of an event is the
limiting value of its relative frequency in a “large” number of
trials.</li>
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<li><p>Suppose we sample with replacement individuals from the 250,000
smokers and compute the relative frequency of the incidence of lung
cancer.</p>
</li>
<li><p>It can be shown that as the sample size goes to infinity, this
relative frequency will approach $\frac{42,500}{250,000}=0.17$.</p>
</li>
</ul>
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<ul>
<li><p>This ratio can also be written as
$$\frac{42,500/1,000,000}{250,000/1,000,000}=0.17.$$</p>
</li>
<li><p>The ratio in the numerator is the joint probability of smoking and
lung cancer, and the ratio in the denominator is the marginal
probability of smoking.</p>
</li>
<li><p>So, $$\Pr(X|Y) = \frac{\Pr(X,Y)}{\Pr(Y)} $$</p>
</li>
</ul>
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<div class="prompt input_prompt">In [12]:</div>
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<div class=" highlight hl-julia"><pre><span class="p">;</span><span class="n">ipython</span> <span class="n">nbconvert</span> <span class="o">--</span><span class="n">to</span> <span class="n">slides</span> <span class="n">BayesTheorem</span><span class="o">.</span><span class="n">ipynb</span>
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<pre>[NbConvertApp] Converting notebook BayesTheorem.ipynb to slides
[NbConvertApp] Writing 197689 bytes to BayesTheorem.slides.html
</pre>
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