/m17 Thanks for the information. I found dirichlet there, but Zipf distribution is not stated clearly, and there is no power law distribtution. Where can I get these info?

Edited by author 02-11-2005 09:44 PM
In problem 4, I don't get the last sentence. What did you mean by "distribution of estimate"? For example, if avg(x1,..,xn) is the MLE of \mu in Gaussian, what is the distribution of the estimate?

\m20 Just as a random variable has a distribution, so does some function of that variable. Since an estimator is just a function of the data, it will some distribution as well. In your example, if x1,...,xn are iid gaussian(\mu,\sigma^2), then avg(x1,...,xn) will follow a gaussian distribution with mean \mu and variance \frac{\sigma^2}{n}

/m19 answer: One lesson to learn from this problem is that widely used terms may not have a universally shared definition. You should make the definition you use clear.

The wikipedia is an excellent reference. Here is a paper that explicitly discusses the confusion:
http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html

For the contemporary importance of power laws see
http://www.sciencemag.org/cgi/content/full/294/5548/1849

/m18 answer: Natural parameters are explained here: http://www-cse.ucsd.edu/users/elkan/291/lect08.pdf

The issue is, when you look for an open rectangle, do you have to look in the space of natural parameters, or can you look in the space of original parameters?

/m21 answer: Yes.

/m21 /m24 Thanks. I understand what you mean. One more question. Can we assume that the number of cross-section is always n? If the organelles are distributed randomly, the number of cross-section will actually vary. So, the distribution should involve the probability distribution of n also. But since there is no description about the density of the organelles, I cannot compute the exact distribution. (Also, the problem becomes much more complicated).

/m25 answer: Yes, you may assume that the number of cross-sections n is fixed.

For problem 4,

I've figured out what the distribution looks like, and I believe I can define it mathematically, but am having trouble doing so. Basically what I would like to do is transform a uniform distribution to the desired one. Here is my logic.

If we look at a cross section of an organelle of radius r, the cross section can be thought of as cutting the organelle at a point x that is generated uniformly from -r..r. This corresponds to the front and back of the sphere relative to the cutting plane.

If we know where the sphere is cut, we can compute the observed radius, Z = sqrt(r^2 - x^2). My questions is how can I use the function for Z and the fact that x is uniform to find the pdf (likelihood function) for Z.

I know this is possible but am having difficulty actually doing it. Thanks.

/m27
There is a trick.

You can use the CDF of the uniform (P(z<Z)) to correspond to some CDF of the radii (P(r>R)). Once you have that, the PDF is the derivative of the CDF.

Thanks, I had figured that out.

Now I'm having trouble finding the MLE, because i'm having trouble breaking up the sum for the total score function.

I know that each x_i ~ p*(x_i,r) = x_i / ( r sqrt(r^2-x^2)

s(x_i,r) = -1/r - r/(r^2-x_i)^2

I need to find s(x,r) = \sum s(x_i,r)

When I do that, I can't seem to separate it into something solveable. Does anyone have any tips on manipulating the sum to give the score function in terms that will put the sum only on the x_i's

Thanks,

Stephen


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/m29 sorry that should read:

s(x_i,r) = -1/r - r/(r^2-x_i^2)

For problem 4, which variable is uniformly distributed? Is the distance of a cross-section or the radius of a sphere itself? How can we decide this? In both cases, we get similar pdf but not the same.

/m31 answer: Your question is perhaps the part of the problem that is conceptually the least clear. I think the answer is that the point at which each organelle is aliced is uniformly distributed, hence the observed radius of the disk obtained from the organelle is not uniformly distributed.

/m29 answer: Remember, you can maximize f(x) by setting its derivative to zero only when the optimal x is in the interior of the allowed range for x.