(ENGR-E 511) Homework 1 MLE for uniform distribution solution


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P1: MLE for uniform distribution [4 points]
1. You took a class taught by Prof. K and found that only five students got an A+. You asked
around and found their total scores which are as follows:
[92, 95.8, 91.3, 94.1, 90.9]. (1)
You know that the pdf of a uniform distribution can be defined as follow:
p(x) =  1
if a ≤ x ≤ b
0 otherwise (2)
2. You assume that there must be an underlying uniform distribution for the A+ student group
(which you don’t belong to unfortunately). Estimate the parameters b and a that maximize
the likelihood of observing the five data samples you collected. Explain how you get your
solution (Hint: you can solve this problem without writing up a program).
3. Now that you are obsessed with someone else’ grades, you became to wonder what would be
the actual boundary Prof. K used, because you feel that you were so close to get an A+. You
have a feeling that he must have used some reasonable numbers, such as an integer multiple
of 5 rather than a real-valued boundary. For example, the actual a and b values must have
been, say, “from 70 to 85 get an A-”, rather than “from 72.9 to 88.32 get an A-.” Based on
this prior knowledge, adjust your a and b values for the A+ students. Justify your choice.
You don’t need to formally prove this. Instead, you could use some other choices to explain
why they are worse than your answer.
P2: Central Limit Theorem [5 points]
1. Load the luddy.jpg file. It must be a 3655 × 6496 × 3 tensor, whose third dimension holds
red, green and blue channels respectively. Let’s take the red channel, which will be a matrix
of 3655 × 6496 elements.
2. Take two patches from two random locations in the image. You may want to use a fairly
large patch size, such as 100 × 100. Get an average patch out of these two—each of the pixel
intensities of the averaged patch is the average of the two pixels at the same position in the
two original patches. See the histogram of the pixel intensities (I would use matplotlib’s
hist function. It is a way to see the sample distribution of your data, which is with 10,000
samples (i.e., the number of pixels in the averaged patch) in our case. Does it look like a
Gaussian distribution?
3. Now take 100 random patches and repeat the experiment. How does it look like? More
4. How about 1000 patches?
5. You know where we’re going. We’re talking about the central limit theorem. You must have
built a feeling that some histogram is more Gaussian-looking than the others. Is there any
way to judge the level of Gaussianity in a more objective way? If you google about it, you’ll
see some terms like kurtosis. For this homework, you’ll use the log-likelihood to measure
the Gaussianity of the pixels in your three averaged patches (that are from 2, 100, and 1000
random patches, respectively).
6. The MLE solution for the Gaussian parameters, mean and variance, is simple: the sample
mean and the sample variance. Calculate them from your three averaged patches.
7. Now you have three pairs of Gaussian parameters (sample mean and variance) for the three
averaged patches. For each averaged patch, calculate the log-likelihood of observing the 10,000
pixel intensities based . Use your MLE parameters. Compare the three log-likelihoods. Which
one is with the highest log-likelihood value? What does this mean (in terms of the central
limit theorem)?
P3: Gradient Ascent for Eigendecomposition [6 points]
1. Although you learned power iteration as an optimzation tool for your eigendecomposition
problem, this time I’d ask you to implement someting slightly different based on gradient
descent (actually, gradient ascent).
2. I prepared 1,000 randomly generated samples from a 2D Gaussian distribution (X.mat)
3. Draw a scatter plot using matplotlib.pyplot.scatter to eyeball this football-shaped distribution. I ask you to find two eigenvectors from this sample distribution.
4. You know, power iteration first finds the eigenvector that’s associated with the largest eigenvalue. Let’s mimic the process. First off, since we do this using gradients, you need to
1Look for the functions that read the .mat files, e.g. scipy.io.loadmat, into a dictionary.
initialize parameters of your model, i.e., the elements of the first eigenvector that you are
looking for, w(1) ∈ R
, with random numbers. It should be a vector of two elements, as you
have two dimensions in the original data space. Use random samples from a standard normal
distribution to initialize them.
5. One condition you have to remember is that w(1) has to be a unit vector, meaning its L2-norm
should be 1. Make sure that by normalizing it after initialization.
6. Project your data samples to your randomly-initialized-and-then-normalized eigenvector (meaning it’s not quite yet an eigenvector). Do it anyway. Let the resulting row vector be
z = (w(1))
7. You know, by definition, your eigenvalue is λ
(1) = (w(1))
⊤XX⊤w(1) = zz⊤. Since for now
we are looking for the largest eigenvalue, we would like to maximize this value, which is our
optimization goal.
8. Differentiate this objective function, λ
(1), with respect to your parameters w(1). The derivative is the gradient direction. If you are not familiar with differentiating a linear algebra
notation, revisit M01-C03-S05, where I gave you an example very similar to this case.
9. Using the gradient direction, update your parameter. Your learning rate should be a small
number so that the update is gradual. Be careful with the sign of the gradient. This time,
you are MAXIMIZING your objective function, rather than MINIMIZING it. So, the update
algorithm is actually gradient ascent, not descent.
10. Another tricky part is the constraint that the eigenvector has to be a unit vector. There
must be a different (better) way to enforce it, but for now let’s be handwavy: normalize the
newly updated parameter vector w(1) to make sure its L2-norm is 1 like you did during the
initialization process.
11. Repeat above process (P3.6-10) multiple times until the absolute values of your gradient
updates are consistently too small.
12. Now let’s move on to the next eigenvector. Revisit M01-C02-S12 to remind yourself of the
process of “taking of the effect of the first eigenvector (or equivalently the first singular
vector).” Once you subtract the contribution of the first eigenvector, your X won’t have
any variation along the direction defined by the first eigenvector. Do another scatter plot to
examine your thought.
13. Repeat your gradient ascent-based eigendecomposition process on the new X matrix which
doesn’t contain any compoent from the first eigenvector. It will give you w(2), the eigenvector
associated with the second largest eigenvalue.
14. Let’s go back to the first scatter plot you drew, which showed you the football-shaped sample
distribution of the original dataset. Overlay the two eigenvectors you found as lines, so that
your AI can verify that your solution, i.e., the directions of the two eigenvectors you found,
is correct. It is going to be basically looking like two arrows starting from the origin, while
pointing to the direction defined by the eigenvectors. They are overlayed on top of the point
15. Include all the intermediate plots that I asked you to draw to your report.