1. Assume that we are in an alien world and their languages are different and only contains the following vocabulary [delta, gamma, alpha, beta, sigma, derivative, summation]. Their parts-of-speech tags are given as [A,B,C,D].

You are given a task to assign tags using Hidden Markov Model for the given test sentence.

Given the following sentences as training examples.

Sentence1: delta gamma sigma summation Tags:[A,B,C,A]

Sentence 2: alpha, sigma, beta derivative Tags:[ A,C,D,A]

Sentence 3: derivative gamma delta beta Tags:[A,B,B,D ]

Sentence 4: sigma summation beta alpha Tags:[ C,B,C,D ]

Sentence 5: [alpha beta sigma derivative] Tags:[ ABCA]

Test Sentence: [alpha gamma beta sigma]

1. Please calculate transition probabilities
2. Calculate emission probabilities
3. How many possible tag sequences (paths) can be generated for the given test sentence?
4. Assign possible tags for the given test sentence following maximum likelihood calculation. Please show details likelihood calculation for all the optimized tag sequences.