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<H3><CENTER>Universal Alignment Probability (Blind Picking)<BR>
P(|<b>G</b>«<b>S</b>|>=<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> -> 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>
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<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>
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