# Q4 (3 marks)

Subject: Natural Language Processing

Exam: M2

Reader: https://deep-learning-study-room.vercel.app/nlp/#m2/mocks/mock-c/q4

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/m2/mocks/mock-c--q4/

M2 · confirmed 2 + 2 + any two of 3 + 3 + 3 · worked practice

### Q4

Training sentences with tags are: “the/D cat/N sleeps/V” and “a/D dog/N runs/V”. Estimate start/end-inclusive transition probabilities and the joint probability of “the cat sleeps” with path D,N,V. (3 marks)

**Given**

Two paths start→D→N→V→end. Four distinct words the,a,cat,dog and two verbs sleeps,runs. Use unsmoothed estimates and lowercase tokens.

**To find**

Transitions, required emissions and full path score.

### Formula and why it applies



$$
P(t_j\mid t_i)=\frac{C(t_i\to t_j)}{\sum_k C(t_i\to t_k)}
$$





$$
P(w\mid t)=\frac{C(w,t)}{C(t)}
$$



Transition denominators count outgoing transitions from the previous state. Emission denominators count all tokens tagged with the emitting state.

### Solution

#### Step 1. Count transitions

start→D, D→N, N→V and V→end each occur twice. Each state has just one observed next state.

#### Step 2. Divide to get probabilities



$$
P(D\mid start)=P(N\mid D)=P(V\mid N)=P(end\mid V)=\frac22=1
$$



#### Step 3. Estimate emissions



$$
P(\text{the}\mid D)=\frac12
$$





$$
P(\text{cat}\mid N)=\frac12
$$





$$
P(\text{sleeps}\mid V)=\frac12
$$



#### Step 4. Multiply in word order



$$
P=1\times\frac12\times1\times\frac12\times1\times\frac12\times1=\frac18=0.125
$$



Joint probability=0.125. Each of the four observed transition probabilities is 1.