<story><title>Modes, Medians and Means: A Unifying Perspective</title><url>http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medians-and-means-an-unifying-perspective/</url></story><parent_chain></parent_chain><comment><author>bowaggoner</author><text>There is a much wider generalization here which is studied under the name of &amp;quot;property elicitation&amp;quot; in computer science, machine learning, and statistics.&lt;p&gt;The generic question is: Given a loss function, what &amp;quot;property&amp;quot; of the distribution minimizes average loss; and given a &amp;quot;property&amp;quot;, characterize all such loss functions. For example, Bregman divergences are (essentially) all losses that &amp;quot;elicit&amp;quot; the mean of a distribution. If you have any monotone continuous function g(), then |g(x) - g(s)| actually also elicits the median, and these are essentially all that do.&lt;p&gt;Apologies for self-promotion, but you can read more at references on this page (disclaimer: I&amp;#x27;m one of the researchers who posted it): &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;sites.google.com&amp;#x2F;site&amp;#x2F;informationelicitation&amp;#x2F;&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;sites.google.com&amp;#x2F;site&amp;#x2F;informationelicitation&amp;#x2F;&lt;/a&gt;&lt;p&gt;or tutorials on this subject at my blog: &lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;bowaggoner.com&amp;#x2F;blog&amp;#x2F;series.html#convexity-elicitation&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;bowaggoner.com&amp;#x2F;blog&amp;#x2F;series.html#convexity-elicitation&lt;/a&gt;</text></comment>
<story><title>Modes, Medians and Means: A Unifying Perspective</title><url>http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medians-and-means-an-unifying-perspective/</url></story><parent_chain></parent_chain><comment><author>Scene_Cast2</author><text>I should note that square-error loss is not the only one that gives the average. Log-loss is another one, for example (you can prove it by taking the derivative of your minimization, setting it to zero and solving)</text></comment>