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In language modeling, L(lambda1, lambda2, lambda3) is defined as:

Sum(count of trigram(u,v,w) x q(w|u,v)) 

where u, v, w are words in the corpus and perplexity is defined as:

2^-l 

where

l = (1/M)Sum(log(q(w|u,v)).

where M is the total no. of words in the corpus.

Also,

q(w|u,v) = lambda1*q(w|u,v) + lambda2*q(w|v) + lamdba3*q(w)

The q-values on the right hand side are maximum likelihood estimates. Some of the materials for Natural Language Processing state that maximizing L is same as minimizing perplexity. I don't see how that is true or can be mathematically proved.

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