a. [10 pts] Write down the formula for Dirichlet Prior Smoothing. Then, mathematically prove the following two lemmas: o Show, in the limit where document length tends to infinity, that a unigram language model smoothed with a Dirichlet prior becomes equivalent to one estimated using the maximum likelihood estimate. o Show, in the limit where the parameter u tends to infinity, that a unigram language model smoothed with a Dirichlet prior becomes equivalent to the background language model used in the smoothing. b. [5 pts] Point out one advantage of Jelinek-Mercer smoothing over Katz-Backoff smoothing. Explain why. a. [10 pts] Write down the formula for Dirichlet Prior Smoothing. Then, mathematically prove the following two lemmas: o Show, in the limit where document length tends to infinity, that a unigram language model smoothed with a Dirichlet prior becomes equivalent to one estimated using the maximum likelihood estimate. o Show, in the limit where the parameter u tends to infinity, that a unigram language model smoothed with a Dirichlet prior becomes equivalent to the background language model used in the smoothing. b. [5 pts] Point out one advantage of Jelinek-Mercer smoothing over Katz-Backoff smoothing. Explain why