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RE: distribution assumption of Eta in NONMEM

From: Eleveld, DJ <d.j.eleveld>
Date: Sun, 30 May 2010 01:20:09 +0200

I'd like to interject a slightly different point of view to the distributional assumption question here.

When I hear people speak in terms of the “distribution assumptions of some estimation method” I think its easy for people to jump to the conclusion that the normal distribution assumption is just one of many possible, equally justifiable distributional assumptions that could potentially be made. And that if the normal distribution is the “wrong” one then the results from such an estimation method would be “wrong”. This is what I used to think, but now I believe this is wrong and I'd like to help others from wasting as much time thinking along this path, as I have.

From information theory, information is gained when entropy decreases. So if you have data from some unknown distribution and if you must make some distribution assumption in order to analyze the data, you should choose the highest entropy distribution you can. This insures that your initial assumptions, the ones you do before you actually consider your data, are the most uninformative you can make. This is the principle of Maximum Entropy which is related to Principle of Indifference and the Principle of Insufficient Reason.

A normal distribution has the highest entropy of all real-valued distributions that share the same mean and standard deviation. So if you assume your data has some true SD, then the best distribution to assume would be normal distribution. So we should not think of the normal distribution assumption as one of many equally justifiable choices, it is really the “least-bad” assumption we can make when we do not know the true distribution. Even if normal is the “wrong” distribution, it still remains the “best”, by virtue of being the “least-bad”, because it is the most uninformative assumption that can be made (assuming a some finite true variance).

In the real-word we never know the true distribution and so it makes sense to always assume a normal distribution unless we have some scientifically justifiable reason to believe that some other distribution assumption would be advantageous.

The Cauchy distribution is a different animal though since its has an infinite variance, and is therefore an even weaker assumption than the finite true SD of a normal distribution. It would possibly be even better than a normal distribution because its entropy is even higher (comparing the standard Cauchy and standard normal). It would be very interesting if Cauchy distributions could be used in NONMEM. Actually, the ratio of two N(0,1) random variables is Cauchy distributed. Maybe this property could be used trick NONMEM into making a Cauchy (or nearly-Cauchy) distributed random variable?
Douglas Eleveld

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Received on Sat May 29 2010 - 19:20:09 EDT

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