JaynesProbabilityTheory/chapter14.html at master · MaksimIM/JaynesProbabilityTheory · GitHub

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<p>#Simple applicatons of decision theory.</p>
<p><span class="math inline">\(\leftarrow\)</span> <a href="./index.html">Back to Chapters</a></p>
<h3 id="comments-on-14.1-and-14.2">Comments on 14.1 and 14.2</h3>
<p>On p. 428 Jaynes seems to be describing a simple probabilistic graphical model (and is making some analogy between belief propagation in a PGM and Huygens principle).</p>
<p><span class="math inline">\(Y\)</span> — <span class="math inline">\(V\)</span> — <span class="math inline">\(D\)</span></p>
<p>Then <span class="math inline">\(V\)</span> screens <span class="math inline">\(D\)</span> from <span class="math inline">\(Y\)</span> and also screens <span class="math inline">\(Y\)</span> from <span class="math inline">\(D\)</span>.</p>
<p>See also <a href="http://ksvanhorn.com/bayes/jaynes/node16.html">this</a> on Kevin Van Horn’s page.</p>
<h3 id="comments-on-14.4">Comments on 14.4</h3>
<p>Small nitpick: There is a discrepancy between the caption of Fig 14.1 and the text (<span class="math inline">\(L_a=2, L_r=1\)</span> gives <span class="math inline">\(L_a=2L_r\)</span> and not <span class="math inline">\(L_a=(3/2)L_r\)</span>).</p>
<p>A key point seems to be that (in the example at hand) both the minimax and Neyman-Pearson decision rules are reproduced by a Bayesian decision rule with appropriate prior. It would be interesting to know how general this phenomenon is. I guess by Wald’s theorem one only needs to show that minimax/Neyman-Pearson is an dmissible decision rule, as Jaynes says: “Wald showed in great generality what we have just illustrated by one simple example”.</p>
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