Naïve Bayesian Reasoning

Bayesian Classification

  • A form of machine learning

  • Given a bunch of evidence for and against a thing, decide whether the thing

  • Formulate as a classic probability problem

     pr(H|E1=v1, E2=v2, ... En=vn)
    
  • We can use Bayes Rule to turn this around

       pr(E1=v1, E2=v2, ... En=vn|H) pr(H)
     = -----------------------------------
         pr(E1=v1, E2=v2, ... En=vn)
    
  • Now we "just" need the probabilities: perhaps we can examine a bunch of classified examples and compute them

Binary Naïve Bayes

  • Problem: There's a huge space of possible values here; likely no way we'll get enough examples to accurately estimate probabilities

  • Let's reduce the state space:

    • Binarize the hypothesis and evidence. Now the state space size is 2^(n+1) for n features

    • Naively assume that all the features are independent (they aren't). Then we can just replace the conjunction with a product

         pr(E1=v1|H) pr(E2=v2|H) ... pr(En=vn|H) pr(H)
       = ---------------------------------------------
            pr(E1=v1) pr(E2=v2) ... pr(En=vn)
      
  • Sadly, the naive assumption means that the probabilities will be too low, since interactions won't be counted: but comparable for H true and false. So

    H iff pr(H|E) > pr(not H|E)
    
  • Note that this comparison drops the denominator, which is nice

  • Games might be played to make the computation more tractable

m-Estimation

  • What we have so far:

                        pr(E1=v1|H) ... pr(En=vn|H) pr(H)
     pr(H|E1=v1, ...) = ---------------------------------
                          pr(E1=v1) ... pr(En=vn)
    
  • Consider the case where pr(Ei=vi|H)=0. If we try to classify an instance of this form, this will make the whole product 0, regardless of what the other terms tell us

  • (Worse yet, consider pr(Ei=vi)=0. Oops. We drop the denominator in the comparison, but…)

  • The training instances are sampled from a notionally large space. If we see no count in n training instances, that means either that the frequency is less than 1/n or we got unlucky

  • Assume that there is some fraction m associated with the undersampling — usually use ½

  • Now adjust the probabilities:

                  |S[Ei=vi|H]+m|
    pr(Ei=vi|H) = --------------
                     |S[H]+m|
    
  • Zeros are gone, calculation may be more accurate

Numerics

  • We are asked to compute a product of potentially many terms:

    pr(E1=v1|H) ... pr(En=vn|H) pr(H)
    
  • All terms are ≤ 1

  • When n is large, this product will be small. Maybe too small. Floating point underflow could happen

  • We are comparing

    pr(E1=v1|H) ... pr(En=vn|H) pr(H) >
    pr(E1=v1|not H) ... pr(En=vn|not H) pr(not H) ?
    
  • Take log on both sides:

    log(pr(E1=v1|H) ... pr(En=vn|H) pr(H)) >
    log(pr(E1=v1|not H) ... pr(En=vn|not H) pr(not H)) ?
    
    log(pr(E1=v1|H)) +  ... + log(pr(En=vn|H) + log(pr(H))) >
    log(pr(E1=v1|not H)) +  ... + log(pr(En=vn|not H) + log(pr(not H))) ?
    
  • These sums should not underflow

Last modified: Monday, 2 November 2020, 12:19 AM