Protein Modeling

• Using Bayesian models
• Task : Estimate Probability mass function(because discrete) for a finite, discrete distribution → given a histogram from a sample
• Large number of categories and small number of observations
• 
  • Estimate Probability distrib of amino acids in each column in a protein class. 20 dim PMF (one for each site)
  • Can be aligned
  • High chances of class not being present in data
    • MLE will assign 0 Probability to X
    • Wrong decision made for a lot of them that were not in the training set
    • Cannot use
• 20 dim PMF for amnio acid distrib : $\theta =({\theta }_1,\dots ,{\theta }_{20}{)}^{\prime }=(P(X=A),\dots ,P(X=Y){)}^{\prime }$
  • count vectors of amino acids found in a given site in training data D
  • Distributed according to Multinomial Distribution with l = 20

Using Prior

• 0 probabilities should not occur. $\mathcal{H}=({\theta }_1,\dots ,{\theta }_{20}{)}^{\prime }\in {\mathbb{R}}^{20}|{\theta }_j\in (0,1)$ and ${\Sigma }_j{\theta }_j=1$
  • 19 dim hypervolume
  • Continuous space and so can use PDF
  • Dirichlet Distribution is used to represent it because parameterized with l = 20
• 
• $\alpha$s fixed beforehand