3 ms·

If I'm not wrong you can't obtain a lognormal as a mixture of Gaussian distributions. The processes that generate these distributions are completely different.

If I'm not wrong you can't obtain a lognormal as a mixture of Gaussian distributions. The processes that generate these distributions are completely different. Gaussian distribution arise when the factors are additive, and lognormal when factors are multiplicative, so these are two completely different distributions. The same way I don't think you can get a normal distribution just by conditioning on an exponential distribution.

jefftk 3y ago
Say you get a lognormal by combining five gaussian-distributed factors (as in the past). Then you condition on four of those factors. The remaining factor, and hence your overall result, will be gaussian. Now, maybe in real life there are far too many factors and we're only holding a few constant so the math doesn't apply, but I do think the math is valid.
- alexmolas 3y ago
  Good point, I was wrong. However, it can also happen that the factors are not gaussian, and still the multiplication would follow a lognormal distribution by the CLT. So saying that "if you condition on enough factors the distribution will be normal" requires the assumption that the factors are normally distributed.
salty_biscuits 3y ago
A gaussian mixture is a universal density approximator, so you can definitely approximate a lognormal with such a model.
leto_ii 3y ago
I was thinking about this and how you might get a lognormal from normals. Peer comments I think have good suggestions. The thing is a while back I took a look at the times achieved by competitive athletes at the 100m dash (from worldathletics.org), and they looked normally distributed. Intuitively that made sense to me - many other human attributes are normally distributed, so why not this?
- eszed 3y ago
  Just a guess: all of those competitive athletes are in the far, far upper-right quadrant of the performance graph, where the curve flattens out. Their distribution will approximate to a normal curve, whereas zooming out to include everyone reveals a gaussian distribution.

jefftk 3y ago

Say you get a lognormal by combining five gaussian-distributed factors (as in the past). Then you condition on four of those factors. The remaining factor, and hence your overall result, will be gaussian. Now, maybe in real life there are far too many factors and we're only holding a few constant so the math doesn't apply, but I do think the math is valid.

alexmolas 3y ago
Good point, I was wrong. However, it can also happen that the factors are not gaussian, and still the multiplication would follow a lognormal distribution by the CLT. So saying that "if you condition on enough factors the distribution will be normal" requires the assumption that the factors are normally distributed.

salty_biscuits 3y ago

A gaussian mixture is a universal density approximator, so you can definitely approximate a lognormal with such a model.

leto_ii 3y ago

I was thinking about this and how you might get a lognormal from normals. Peer comments I think have good suggestions. The thing is a while back I took a look at the times achieved by competitive athletes at the 100m dash (from worldathletics.org), and they looked normally distributed. Intuitively that made sense to me - many other human attributes are normally distributed, so why not this?

eszed 3y ago
Just a guess: all of those competitive athletes are in the far, far upper-right quadrant of the performance graph, where the curve flattens out. Their distribution will approximate to a normal curve, whereas zooming out to include everyone reveals a gaussian distribution.