<BODY bgcolor="#aaeebb"> <H3><CENTER>Universal Alignment Probability (Blind Picking)<BR> P(|<b>G</b>&#171;<b>S</b>|&gt;=<i>k</i>; sigma<sup>2</sup>, <i>g</i>=<i>s</i>, <i>N</i>, OPC class) </CENTER></H3> <BR> For sigma<sup>2</sup> -&gt; Infinity, i.e., blind pick <BR> <P> <CENTER> <i>P</i> = <i><sub>g</sub>C<sub>i</sub></i> * <i><sub>N-g</sub>C<sub>g-i</sub></i> / <i><sub>N</sub>C<sub>g</sub></i> </CENTER> <br> <P>For sigma<sup>2</sup> being finite, <i>P</i> will be problem dependent.<BR> <BR> <BR> <tt> <A HREF="Slide04.html" target="_parent">To Next Slide</A> / <A HREF="Slide02.html" target="_parent">To Previous Slide</A> / <A HREF="../OOTOC.html" target="_parent">Table of Content</A> </tt> <BR> <hr> <hr> <P> Let us give a simple example. Suppose we use the crudest selection method, namely we blindly pick |<B>S</B>| members of the <i>N</i> samples and ask the probability that among these |<B>S</B>| selected designs there are at least <i>k</i> designs that will rank in the top-<i>g</i> of the performance order (recall the engineering design example in the earlier transparency). Blind Pick (BP) is equivalent to a selection method where the error variance of selection is infinity. It turns out this probability can be calculated in closed form. There are <sub>g</sub>C<sub>i</sub> ways of picking <i>i</i> elements out of the top-<i>g</i> designs and <sub>N-g</sub>C<sub>g-i</sub> ways picking the remaining (<i>g</i>-<i>i</i>) designs out the <i>N</i>-</i>g</i> samples; this is divided by the total number of ways of selecting <i>g</i> designs out of <i>N</i> samples <sub>N</sub>C<sub>g</sub> to arrive at the alignment probability. The curves shown are for <i>N</i> = 1000. As we can see for <i>k</i>=1, <i>P</i> ~ 1 for size of |<B>G</B>|=<i>g</i>=50, i.e., without any knowledge we can effect a 20:1 reduction and still be sure that at least one of the top-50 design is in the blindly picked subset.<BR> <P> For sigma<sup>2</sup>, we no longer have a closed form answer. And we need a model to calculate the alignment probability. For example, let us assume that <BR> <P><CENTER>Performance<sub>observed</sub> = Performance<sub>actual</sub> + error .................... (*)</CENTER> <P> which is known as the <b>Thurstone</b> Model. We shall take the observed top-|<b>S</b>| performances as the selected subset <b>S</b> and compute experimentally via simple simulation using Eq.(*) the alignment probability. However, it is also clear that this probability will be problem-dependent. We need to introduce some other concepts before we can characterizing this probability.<BR> </BODY> </HTML>