Due: Monday 25 Feb 2002

Estimating distributions of amino acids

This exercise is to test your understanding of estimating discrete probability distributions given a small sample. Apply each method to the following seven count vectors:

1. 1 isoleucine (rest zero)
2. 1 isoleucine, 1 valine (rest zero)
3. 1 isoleucine, 1 phenylalanine (rest zero)
4. 1 isoleucine, 1 histidine (rest zero)
5. 1 isoleucine, 1 aspartic acid (rest zero)
6. 2 isoleucines (rest zero)
7. 3 isoleucines (rest zero)

1. Maximum-likelihood method. Normalize each count vector to probabilities and compute the entropy. You many want to write a short program to compute the entropy, since the summation is tedious to do on a hand calculator.
2. Pseudocounts (Dirichlet prior). Use the following three pseudocount vectors and compute the mean posterior estimate and the entropy for each. You may want to write a short program.
   1. all pseudocounts 1 (the uninformed prior)
   2. all pseudocounts 0.05 (a prior that favors conserved residues)
   3. Order = A C D E F G H I K L M N P Q R S T V W Y
      0.0943893 0.0193798 0.0643931 0.0721827 0.0430317 0.0759875 0.0257852 0.0654119 0.0713842 0.0900674 0.0263642 0.0551132 0.0458893 0.047656 0.0544527 0.0767514 0.0693522 0.0798881 0.0136764 0.0385562
      (This is available in /projects/compbio/lib/rev4.1comp)
3. Dirichlet mixture. Write a program (in any programming language that provides lgamma) to compute the mean posterior estimate from a count vector and a Dirichlet mixture. Use the Dirichlet mixture recode3.20comp (also available in /projects/compbio/lib/recode3.20comp). Note that the "DirichletReg" format provides 21 numbers in the the "Alpha" vector. The first number is the sum of the 20 pseudocounts, and can be ignored.

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