**3.3.1.2. Training Algorithm**

As discussed in Chapter 1 with a context-free grammar, probabilities acting as parameters can initially be generated at random, and then updated every time a new sentence is analyzed and added to the set. corpus. The training algorithm proposed by [79] aims to recalculate the parameter value after processing the input sentence. Like the context-free grammar, this algorithm relies on two parameters, the inner probability and the outer probability.

The probability in PrI ( L, R, l, r ) is the probability that words from L to R can be linked together such that the connections l and r are satisfied.

The outer probability Pro ( L, R, l, r ) is the probability that words outside the range L to R can be associated with each other such that the outer join requirements l and r are satisfied.

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The inner probability is calculated recursively according to the relations:

According to the parsing algorithm in Figure 3.4, it is clear that PI ( wi , wi+1, NIL, NIL ) = 1 with 0 ≤ i ≤ n-1.

** For example** , with the linking grammar and the sentence “

*I bought a flower*” mentioned above,

PrI ( 1, 4, NIL, NcNt3 ) = Pr (3, (McN)(NcNt3),→ | 1, 4, NIL, NcNt3 ) × PrI ( 1, 3, NIL, McN ) × PrI ( 3, 4

, NIL, NIL )

with the values of the probabilities given in (3.1) :

PrI( 1, 3, NIL, McN) = Pr(2, ( )(McN), → | 1, 3, NIL, McN) × PrI (1, 2, NIL,NIL) × PrI (2, 3,

NIL , NIL)

= 0.06 × 1 × 1 = 0.06

Pr ( 3, (McN)(NcNt3),→ | 1, 4, NIL, NcNt3 ) = 0.05

so, PrI (buy, flower, NIL, NcNt3) = 0.05 × 0.06 = 0.003 (3.5)

The probabilities outside PrO are calculated recursively: initially, for each d ∈ D(W0) there is left[d] = NIL, set

The probability is added up for the 4 possible cases in the previous step (then R and L also play the role of W):

**Figure 3.22** . Describe how to calculate probability Pr0 ⊲left(L, W, l ⊳, ⊲ left[D])

According to [79], Counts are calculated in the following formulas (3.6) to (3.9) :

The value of count(L, R, l, r) is calculated in the analysis algorithm:

where δ is a function that takes the value 1 if l = NIL, 0 otherwise, match takes the value 1 if the two matches match, 0 otherwise. Notice match(c,NIL) = match(NIL,c) = 0.

The Pr(S) value stated in the above formulas is calculated according to the following formula:

The values Count(L, R, l, r), Count(W, l, r) and Count(d, l, r) are calculated directly according to the connections and selections that appear in the corpus.