Example
The following is taken from Sornette's book.
Consider a random variable, that takes on values with probability . Furthmore, lets assume for another parameter
for some fixed . We then want to minimize
subject to the constraint
Using Lagrange multipliers, this gives
giving us a power law. The global optimization of minimizing the energy along with the power law dependence between and gives us a power law distribution in probability.
Read more about this topic: Highly Optimized Tolerance
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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