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<p><b>21 September 2010</b></p>
<h3 id="a001353" class="Bigger">What is the likelihood function?</h3>
<p>An interesting 1992 paper by Bayarri and DeGroot entitled &#8220;Difficulties and Ambiguities in the Definition of a Likelihood Function&#8221; (<a href="http://www.springerlink.com/content/4322774u233m3gq0/" title="Gated Version">gated version</a>) grapples with the problem of defining the likelihood when auxiliary variables are at hand. Here is the abstract:</p>
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<p>The likelihood function plays a very important role in the development of both the theory and practice of statistics. It is somewhat surprising to realize that no general rigorous definition of a likelihood function seem to ever have been given. Through a series of examples it is argued that no such definition is possible, illustrating the difficulties and ambiguities encountered specially in situations involving &#8220;random variables&#8221; and &#8220;parameters&#8221; which are not of primary interest. The fundamental role of such auxiliary quantities (unfairly called &#8220;nuisance&#8221;) is highlighted and a very simple function is argued to convey all the information provided by the observations.</p>
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<p>The example that resonates with me in on pages 4-6, where they describe the ambiguity of using defining the likelihood function when there is an observation <em>y</em> which is a measurement of <em>x</em> subject to (classical) error. There are several different ways of writing a likelihood in that case, depending on how you handle the latent, unobserved data <em>x</em>.  One can condition on it, marginalize across it, or include it in the joint distribution of the data. Each of these can lead to a different MLE. </p>
<p>Their point is that situations like this involve subjective choices (though, all modeling requires subjective choice) and the hermetic seal between the &#8220;model&#8221; and the &#8220;prior&#8221; is less airtight than we think. </p>
<p class="posted">
Posted by <a href="http://www.iq.harvard.edu/blog/sss/archives/author/matt-blackwell/">Matt Blackwell</a> at <a href="http://www.iq.harvard.edu/blog/sss/archives/2010/09/what_is_the_lik.shtml"> 4:41 PM</a>

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<p><b>14 September 2010</b></p>
<h3 id="a001352" class="Bigger">You are not so smart</h3>
<p><a href="http://youarenotsosmart.com/">You are not so smart</a> is a blog dedicated to explaining self-delusions. The most recent <a href="http://youarenotsosmart.com/2010/09/11/the-texas-sharpshooter-fallacy/#more-689">post</a> is on the <a href="http://en.wikipedia.org/wiki/Texas_sharpshooter_fallacy">Texas sharpshooter fallacy</a>:<br />
<blockquote><br />
<b>The Misconception:</b> You take randomness into account when determining cause and effect.<br />
<b>The Truth:</b> You tend to ignore random chance when the results seem meaningful or when you want a random event to have a meaningful cause.<br />
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<p class="posted">
Posted by <a href="http://www.iq.harvard.edu/blog/sss/archives/author/matt-blackwell/">Matt Blackwell</a> at <a href="http://www.iq.harvard.edu/blog/sss/archives/2010/09/you_are_not_so.shtml"> 6:33 PM</a>

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