Question

Deriving MLE for a General Multinomial Model: Likelihood 1 point possible (graded) In the following problems, we will derive the maximum likelihood estimates for a multinomial model with more than 2 parameters. We will employ the method of lagrange multipliers for the optimization problem. Let the document $ D $ be a sequence of words $ w_{1}, \ldots, w_{n} $ from a collection $ W $ consisting of $ N $ words. For simplicity, we assume that $ w_{i} $ 's are independent, and that the probability of a word $ w $ is given by the parameter $ \theta_{w} $, and denote by $ \theta=\left\{\theta_{w}\right\}_{w \in W} $. Let $ P(D \mid \theta) $ be the probability of $ D $ being generated by the simple model described above. Find $ P(D \mid \theta) $. $ P(D \mid \theta)=\sum_{w \in W} \theta_{w}^{\operatorname{count}(w)} $ $ P(D \mid \theta)=\prod_{w \in W} \theta_{w}^{\operatorname{count}(w)} $ $ P(D \mid \theta)=\prod_{w \in W} \operatorname{count}(w)^{\theta_{w}} $ $ P(D \mid \theta)=\prod_{w \in W} \theta_{w}+\operatorname{count}(w) $

          Deriving MLE for a General Multinomial Model: Likelihood
1 point possible (graded)
In the following problems, we will derive the maximum likelihood estimates for a multinomial model with more than 2 parameters. We will employ the method of lagrange multipliers for the optimization problem.

Let the document \( D \) be a sequence of words \( w_{1}, \ldots, w_{n} \) from a collection \( W \) consisting of \( N \) words. For simplicity, we assume that \( w_{i} \) 's are independent, and that the probability of a word \( w \) is given by the parameter \( \theta_{w} \), and denote by \( \theta=\left\{\theta_{w}\right\}_{w \in W} \).

Let \( P(D \mid \theta) \) be the probability of \( D \) being generated by the simple model described above.
Find \( P(D \mid \theta) \).
\( P(D \mid \theta)=\sum_{w \in W} \theta_{w}^{\operatorname{count}(w)} \)
\( P(D \mid \theta)=\prod_{w \in W} \theta_{w}^{\operatorname{count}(w)} \)
\( P(D \mid \theta)=\prod_{w \in W} \operatorname{count}(w)^{\theta_{w}} \)
\( P(D \mid \theta)=\prod_{w \in W} \theta_{w}+\operatorname{count}(w) \)

$Deriving MLE for a General Multinomial Model: Likelihood 1 point possible (graded) In the following problems, we will derive the maximum likelihood estimates for a multinomial model with more than 2 parameters. We will employ the method of lagrange multipliers for the optimization problem. Let the document D be a sequence of words w1, …, wn from a collection W consisting of N words. For simplicity, we assume that wi 's are independent, and that the probability of a word w is given by the parameter θw, and denote by θ={θw}w ∈ W. Let P(D |θ) be the probability of D being generated by the simple model described above. Find P(D |θ). P(D |θ)=∑w ∈ Wθw^count(w) P(D |θ)=∏w ∈ Wθw^count(w) P(D |θ)=∏w ∈ Wcount(w)^θw P(D |θ)=∏w ∈ Wθw+count(w)$

Added by Blanca H.

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