<story><title>Modes, Medians and Means: A Unifying Perspective (2013)</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>bo1024</author><text>One can generalize this approach further to take any given loss function, not just a p-norm, and ask what statistic is achieved by minimizing loss.&lt;p&gt;A field that studies this is called property elicitation, from the idea that the loss function elicits a certain kind of prediction be it mean, mode, etc.&lt;p&gt;This field actually originated with human experts like weather forecasters - what is a loss function (or “scoring rule”) that incents them to predict accurately? Squared loss is good if by accurately we imply we want the mean.&lt;p&gt;One fact people find surprising is that squared loss is not the only one that elicits the mean. There is an entire family of losses that do, including KL divergence.</text></comment>
<story><title>Modes, Medians and Means: A Unifying Perspective (2013)</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>ogogmad</author><text>Interestingly, since the L^1 discrepancy is convex, it&amp;#x27;s possible to analytically minimize it by using the &lt;i&gt;subdifferential&lt;/i&gt;, and thereby prove that its minimum is attained at the median. The subdifferential is a set-valued generalisation of the derivative to all convex functions, including non-differentiable ones. See &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Subderivative&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;en.wikipedia.org&amp;#x2F;wiki&amp;#x2F;Subderivative&lt;/a&gt; and &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;towardsdatascience.com&amp;#x2F;beyond-the-derivative-subderivatives-1c4e5bf20679&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;towardsdatascience.com&amp;#x2F;beyond-the-derivative-subderi...&lt;/a&gt;&lt;p&gt;Whenever the subdifferential of a convex function at a point includes 0, that point is a global minimum of the function.</text></comment>