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I hope someone could help me with the following problem: If I create a linear model that predicts the height of people (y) with the following parameters:

y = a + b*(xi-xavg)

with "a" the normally distributed (178,20) the prior distribution of heights and "b" a coefficient for the effect of the weight on the height prediction. The question is if I have to choose the normal distribution or the lognormal distribution for coefficient "b". I understand that the normal distribution has the maximum entropy for a given variance (sigma = Uniform(0,50) here) so do I just pick the normal distribution here?

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  • $\begingroup$ FWIW the lognormal is the maximum entropy distribution with a specified geometric mean and variance en.wikipedia.org/wiki/… , so that principle doesn't help you decide ... $\endgroup$
    – Ben Bolker
    Commented Sep 16, 2021 at 2:22

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As a rule of thumb:

Normal models an additive product, LogNormal is a multiplicative product of many 'weak' influences.

Normal can be negative, LogNormal is strictly non-negative.

The first point doesn't really matter in this case. As for the second, if you used LogNormal without any additional processing, you're assuming the weight can only influence the height positively, which seems like an ungrounded and unnecessary assumption.

For example. you could argue that people with a lithe build tend to be taller, and that the impact of the added height is overshadowed by the impact of the body type.

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