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Understanding the Probabilistic Spell Checker Based on Noisy Channel Model

This lecture note outlines the principles of a probabilistic spell checker utilizing the Noisy Channel Model. It explores the formulation of the problem, where given a potentially misspelled word (t), the goal is to find the most probable correct word (w) that maximizes P(w|t). Key concepts include the application of Bayes' rule, the construction of a confusion matrix for capturing error patterns, and the computation of probabilities for insertion, deletion, substitution, and transposition errors. Understanding these elements is crucial for developing robust spelling correction systems.

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Understanding the Probabilistic Spell Checker Based on Noisy Channel Model

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  1. CS621/CS449Artificial IntelligenceLecture Notes Set 8 : 27/10/2004 CS-621/CS-449 Lecture Notes

  2. Outline • Probabilistic Spell Checker (continued from Noisy Channel Model) • Confusion Matrix CS-621/CS-449 Lecture Notes

  3. Probabilistic Spell Checker Noisy Channel Model • The problem formulation for spell checker is based on the Noisy Channel Model w t (wn, wn-1, … , w1) (tm, tm-1, … , t1) • Given t, find the most probable w : Find that ŵ for which P(w|t) is maximum, where t, w and ŵ are strings: Noisy Channel ŵ Guess at the correct word Correct word Wrongly spelt word CS-621/CS-449 Lecture Notes

  4. Probabilistic Spell checker • Applying Bayes rule, • Why apply Bayes rule? • Finding p(w|t) Vs p(t|w) ? • P(w|t) or P(t|w) have to be computed by counting c(w,t) or c(t,w) and then normalizing them • Assumptions : • t is obtained from w by a single error of the above type. • The words consist of only alphabets ŵ CS-621/CS-449 Lecture Notes

  5. Confusion Matrix Confusion Matrix: 26x26 • Data structure to store c(a,b) • Different matrices for insertion, deletion, substitution and transposition • Substitution • The number of instances in which a is wrongly substituted by b in the training corpus (denoted sub(x,y) ) CS-621/CS-449 Lecture Notes

  6. Confusion Matrix • Insertion • The number of times a letter y is inserted after x wrongly( denoted ins(x,y) ) • Transposition • The number of times xy is wrongly transposed to yx ( denoted trans(x,y) ) • Deletion • The number of times y is deleted wrongly after x ( denoted del(x,y) ) CS-621/CS-449 Lecture Notes

  7. Confusion Matrix • If x and y are alphabets, • sub(x,y) = # times y is written for x (substitution) • ins(x,y) = # times x is written as xy • del(x,y) = # times xy is written as x • trans(x,y) = # times xy is written as yx CS-621/CS-449 Lecture Notes

  8. Probabilities • P(t|w) = P(t|w)S + P(t|w)I + P(t|w)D + P(t|w)X • Where P(t|w)S = sub(x,y) / count of x P(t|w)I = ins(x,y) / count of x P(t|w)D = del(x,y) / count of x P(t|w)X = trans(x,y) / count of x • These are considered to be mutually exclusive events CS-621/CS-449 Lecture Notes

  9. Example • Correct document has ws • Wrong document has ts • P(maple|aple) = # (maple was wanted instead of aple) / # (aple) • P(apple|aple) and P(applet|aple) calculated similarly • Leads to problems due to data sparsity. • Hence, use Bayes rule. CS-621/CS-449 Lecture Notes

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