1. Page 1 of 19 Confidence measure using word posteriors Sridhar Raghavan Confidence Measure using Word Graphs 3/6 2/6 4/6 2/6 1/6 4/6 1/6 4/6 1/6 4/6 5/6 Sil This is a test sentence Sil this is the is a the guest a quest sentence sense 1/6 Sil

2. Page 2 of 19 Confidence measure using word posteriors Abstract Confidence measure using word posterior: There is a strong need for determining the confidence of a word hypothesis in a LVCSR system because conventional viterbi decoding just generates the overall one best sequence, but the performance of a speech recognition system is based on Word error rate and not sentence error rate. Word posterior probability in a hypothesis is a good estimate of the confidence. The word posteriors can be computed from a word graph where the links correspond to the words. A forward-backward type algorithm is used to compute the link posteriors.

3. Page 3 of 19 Confidence measure using word posteriors What is a word posterior? A word posterior is a probability that is computed by considering a words acoustic score, language model score and its presence is a particular path through the word graph. An example of a word graph is given below, note that the nodes are the start- stop times and the links are the words. The goal is to determine the link posterior probabilities. Every link holds an acoustic score and a language model probability. 3/6 2/6 4/6 2/6 1/6 4/6 1/6 4/6 1/6 4/6 5/6 Sil This is a test sentence Sil this is the is a the guest a quest sentence sense 1/6 Sil

4. Page 4 of 19 Confidence measure using word posteriors Example Let us consider an example as shown below: 3/6 2/6 4/6 2/6 1/6 4/6 1/6 4/6 1/6 4/6 5/6 Sil This is a test sentence Sil this is the is a the guest a quest sentence sense 1/6 Sil The values on the links are the likelihoods.

5. Page 5 of 19 Confidence measure using word posteriors Forward-backward algorithm Using forward-backward algorithm for determining the link probability. The equations used to compute the alphas and betas are as follows: Computing alphas: Step 1: Initialization: In a conventional HMM forward-backward algorithm we would perform the following – We need to use a slightly modified version of the above equation for processing a word graph. The emission probability will be the language model probability and the initial probability in this case has been taken as 0.01 (assuming we have 100 words in a loop grammar and hence all the words are equally probable with probability 1/100).

6. Page 6 of 19 Confidence measure using word posteriors Forward-backward algorithm continue… The α for the first node in the word graph is computed as follows: Step 2: Induction This step is the main reason we use forward-backward algorithm for computing such probabilities. The alpha values computed in the previous step is used to compute the alphas for the succeeding nodes. Note: Unlike in HMMs where we move from left to right at fixed intervals of time, over here we move from one start time of a word to the next closest words start time.

7. Page 7 of 19 Confidence measure using word posteriors Forward-backward algorithm continue… Let us see the computation of the alphas from node 2, the alpha for node 1 was computed in the previous step during initialization. Node 2: Node 3: Node 4: The alpha calculation continues in this manner for all the remaining nodes

8. Page 8 of 19 Confidence measure using word posteriors Forward-backward algorithm continue… Once we compute the alphas using the forward algorithm we begin the beta computation using the backward algorithm. The backward algorithm is similar to the forward algorithm, but we start from the last node and proceed from right to left. Step 1 : Initialization Step 2: Induction

9. Page 9 of 19 Confidence measure using word posteriors Forward-backward algorithm continue… Let us see the computation of the beta values from node 14 and backwards. Node 14: Node 13: Node 12:

10. Page 10 of 19 Confidence measure using word posteriors Forward-backward algorithm continue… Node 11: In a similar manner we obtain the beta values for all the nodes till node 1. We can compute the probabilities on the links (between two nodes) as follows: Let us call this link probability as Γ. Therefore Γ(t-1,t) is computed as the product of α(t-1)*ß(t). These values give the un-normalized posterior probabilities of the word on the link considering all possible paths through the link.

11. Page 11 of 19 Confidence measure using word posteriors Word graph showing the computed alphas and betas 1 3 4 5 6 7 10 11 12 13 9 15 2 3/6 2/6 4/6 2/6 1/6 4/6 1/6 4/6 1/6 4/6 5/6 Sil This is a test sentence Sil this is the is a the guest quest 14 sentence sense 1/6 Sil α =1E-04 β=2.8843E-16 8 α =5E-07 β=2.87E-16 α =5.025E-07 β=5.740E-14 α=1.117E-11 β=2.514E-9 α=1.675E-09 β=1.5422E-13 α=3.35E-9 β=8.534E-12 α=1.675E-11 β=4.626E-11 α=2.79E-14 β=2.776E-8 α=1.861E-14 β=2.776E-8 α=7.446E-14 β=3.703E-7 α=7.75E-17 β=1.666E-5 α=4.964E-16 β=5.555E-5 α=3.438E-18 β=8.33E-3 α=1.2923E-19 β=1.667E-3 α=2.886E-20 β=1 Assumption here is that the probability of occurrence of any word is 0.01. i.e. we have 100 words in a loop grammar This is the word graph with every node with its corresponding alpha and beta value.

12. Page 12 of 19 Confidence measure using word posteriors Link probabilities calculated from alphas and betas Γ=4.649E-19 1 3 4 5 15 2 3/6 2/6 4/6 2/6 1/6 4/6 1/6 4/6 1/6 4/6 5/6 Sil This is a test sentence Sil this is the is a the guest quest 14 sentence sense 1/6 Sil Γ=5.74E-18 Γ=2.87E-20 Γ=4.288E-18 Γ=7.749E-20 Γ=1.549E-19 Γ=8.421E-18 Γ=4.649E-19 Γ=3.1E-19 Γ=4.136E1-18 Γ=3.1E-19 Γ=4.136E-18 Γ=6.46E-19 Γ=1.292E-19 Γ=3.438E-18 The following word graph shows the links with their corresponding link posterior probabilities (not yet normalized). 6 7 8 9 10 11 12 13 Γ=2.87E-20 By choosing the links with the maximum posterior probability we can be certain that we have included most probable words in the final sequence.

13. Page 13 of 19 Confidence measure using word posteriors Using it on a real application Using the algorithm on real application: * Need to perform word spotting without using a language model i.e. we can only use a loop grammar. * In order to spot the word of interest we will construct a loop grammar with just this one word. * Now the final one best hypothesis will consist of a sequence of the same word repeated N times. So, the challenge here is to determine which of these N words actually corresponds to the word of interest. * This is achieved by computing the link posterior probability and selecting the one with the maximum value.

14. Page 14 of 19 Confidence measure using word posteriors 1-best output from the word spotter The recognizer puts out the following output :- 0000 0023 !SENT_START -1433.434204 0023 0081 BIG -4029.476440 0081 0176 BIG -6402.677246 0176 0237 BIG -4080.437500 0237 0266 !SENT_END -1861.777344 We have to determine which of the three instances of the word actually exists.

15. Page 15 of 19 Confidence measure using word posteriors 01 2 3 4 567 sent_start sent_end -1433 -1095 -1888 -2875 -4029 -912 -1070 -1232 -6402 -1861 8 -4056 Lattice from one of the utterances For this example we have to spot the word BIG in an utterance that consists of three words (BIG TIED GOD). All the links in the output lattice contains the word BIG. The values on the links are the acoustic likelihoods in log domain. Hence a forward backward computation just involves addition of these numbers in a systematic manner.

16. Page 16 of 19 Confidence measure using word posteriors Alphas and betas for the lattice 0 1 2 3 4 567 sent_start sent_end -1433 -1095 -1888 -2875 -4029 -912 -1070 -1232 -6402 -1861 8 -4056 α =0 β=-67344 α =-1433 β=-65911 α =-2528 β=-15533 α =-6761 β=-14621 α =-12139 β=-13551 α =-18833 β=-12319 α =-25235 β=-5917 α =-29291 β=-1861 α =-31152 β=0 Let the initial probability at both the nodes in this case be 1. So, its logarithmic value is 0. The initial value can be any constant as it will not change the net result. The language model probability of the word is also 1 since it is the only word in the loop grammar.

17. Page 17 of 19 Confidence measure using word posteriors Link posterior calculation 0 1 2 3 4 567 sent_start sent_end 8 Γ=-67344 Γ=-18061 Γ=-17942 Γ=-17859 Γ=-17781 Γ=-21382 Γ=-25690 Γ=-31152 It is observed that we can obtain a greater discrimination in confidence levels if we also multiply the final probability with the likelihood of the link other than the corresponding alphas and betas. In this example we add the likelihood since it is in log domain.

18. Page 18 of 19 Confidence measure using word posteriors Inference from the link posteriors Link 1 to 5 corresponds to the first word time instance while 5 to 6 and 6 to 7 correspond to the second and third word instances respectively. It is very clear from the link posterior values that the first instance of the word BIG has a much higher probability than the other two. Note: The part that is missing in this presentation is the normalization of these probabilities, this is needed to make comparison between various link posteriors.

19. Page 19 of 19 Confidence measure using word posteriors References: F. Wessel, R. Schlüter, K. Macherey, H. Ney. "Confidence Measures for Large Vocabulary Continuous Speech Recognition". IEEE Trans. on Speech and Audio Processing. Vol. 9, No. 3, pp. 288-298, March 2001 Wessel, Macherey, and Schauter, "Using Word Probabilities as Confidence Measures, ICASSP'97 G. Evermann and P.C. Woodland, Large Vocabulary Decoding and Confidence Estimation using Word Posterior Probabilities in Proc. ICASSP 2000, pp. 2366-2369, Istanbul.