A complete unseen-bigram question from corpus to final answer

Question. Consider the corpus “I love to code in Python. I love to learn new things. I code every day.” Obtain the bigram probabilities needed for “I love Python”. Explain the zero-probability issue, then use Laplace smoothing to score it. Treat each sentence separately, preserve case, remove punctuation, and include an end token among predicted outcomes.

Formula and why it applies

LaTeX: P(w\mid h)=\frac{C(h,w)}{C(h)},\qquad P_L(w\mid h)=\frac{C(h,w)+1}{C(h)+12}

The test contains an unseen transition love→Python. Smoothing assigns probability to that transition and renormalizes every required history row.

Solution

Step 3. Compute the unsmoothed product

LaTeX: P=\frac33\times\frac23\times\frac02\times\frac11=0

Both words love and Python were seen, but their adjacent pair was not. This is an unseen bigram, not an unknown-word case.

Step 4. Smooth the first event

LaTeX: P_L(I\mid\langle s\rangle)=\frac{3+1}{3+12}=\frac4{15}

Step 5. Smooth the second event

LaTeX: P_L(love\mid I)=\frac{2+1}{3+12}=\frac3{15}

Step 6. Smooth the unseen event

LaTeX: P_L(Python\mid love)=\frac{0+1}{2+12}=\frac1{14}

Step 7. Smooth the end event

LaTeX: P_L(\langle/s\rangle\mid Python)=\frac{1+1}{1+12}=\frac2{13}

Step 8. Multiply the smoothed factors

LaTeX: P_L=\frac4{15}\times\frac3{15}\times\frac1{14}\times\frac2{13}

LaTeX: P_L=\frac{24}{40950}=\frac4{6825}\approx0.000586081