Sridhar Raghavan and Joseph Picone URL: HUMAN AND SYSTEMS ENGINEERING: Confidence Measures Based on Word Posteriors and Word Graphs

Page 1 of 21 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 systems, this is because conventional viterbi decoding just generates the overall one best sequence while the performance of a speech recognition system is based on Word error rate and not sentence error rate. A good estimate of the confidence level is the word posterior probability. The word posteriors can be computed from a word graph. A forward-backward algorithm can be used to compute the link posteriors.

Page 2 of 21 Confidence measure using word posteriors Foundation The equation for computing the posterior of the word is as follows [Wessel.F]: The idea here is to sum up the posterior probabilities of all those word hypothesis sequences that contain the word ‘w’ with same start and end times. Time W

Page 3 of 21 Confidence measure using word posteriors Foundation: continued… We cannot compute the above posterior directly, so we decompose it into likelihood and priors using Baye’s rule. Hence the value in the numerator has to be computed using the forward backward algorithm. The denominator term is simply the sum of the numerator for all words ‘w’ occurring in the same time instant. N There are 6 different ways to reach the node N and 2 different ways to leave N, so we need to obtain the forward probability as well as the backward probability to obtain a good estimate of the probability of passing through the node N, and this is where the forward-backward algorithm comes into picture.

Page 4 of 21 Confidence measure using word posteriors What is exactly a word posterior from a word graph? A word posterior is a probability that is computed by considering a word’s 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. quest 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 sentence sense 1/6 Sil

Page 5 of 21 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.

Page 6 of 21 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 for an HMM 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 acoustic score and the initial probability is taken as 1 since we always begin with a silence.

Page 7 of 21 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 word’s start time.

Page 8 of 21 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 The forward backward calculation on word-graphs is similar to the calculations used on HMMs, but in word graphs the transition matrix is populated by the language model probabilities and the emission probability corresponds to the acoustic score /6 2/6 4/6 Sil this is α =1 α =0.5 α = α=1.675E-03

Page 9 of 21 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

Page 10 of 21 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: /6 4/6 5/6 sentence Sil 14 sentence sense 1/6 Sil β=1.66E-3 β=5.55E-3 β=0.833 β= β=1

Page 11 of 21 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)*aij. These values give the un-normalized posterior probabilities of the word on the link considering all possible paths through the link.

Page 12 of 21 Confidence measure using word posteriors Word graph showing the computed alphas and betas /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 =1 β=2.8843E-14 8 α =0.5 β=2.87E-14 α = β=5.740E-12 α=1.117E-5 β=2.512E-7 α=1.675E-03 β=1.536E-11 α=3.35E-3 β=8.527E-10 α=1.675E-5 β=4.61E-9 α=2.79E-8 β=2.766E-6 α=1.861E-8 β=2.766E-6 α=7.446E-8 β=3.7E-5 α=7.751E-11 β=1.66E-3 α=4.964E-10 β=5.55E-3 α=3.438E-12 β=0.833 α=1.2923E-13 β= α=2.88E-14 β=1 Assumption here is that the probability of occurrence of any word is 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.

Page 13 of 21 Confidence measure using word posteriors Link probabilities calculated from alphas and betas Γ=4.649E /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-12 Γ=2.87E-14 Γ=4.288E-12 Γ=7.71E-14 Γ=7.72E-14 Γ=1.549E-13 Γ=8.421E-12 Γ=4.649E-13 Γ=3.08E-13 Γ=4.136E1-12 Γ=3.08E-13 Γ=4.136E-12 Γ=6.45E-13 Γ=1.292E-13 Γ=1.292E-15 Γ=3.438E-14 The following word graph shows the links with their corresponding link posterior probabilities (not yet normalized) Γ=2.87E-14 By choosing the links with the maximum posterior probability we can be certain that we have included most probable words in the final sequence.

Page 14 of 21 Confidence measure using word posteriors Some Alternate approaches… The paper by F.Wessel (confidence Measures for Large Vocabulary Continuous Speech Recognition) describes alternate techniques to compute the posterior, because the drawback of the approach described above is that the lattice has to be very deep to accommodate sufficient links at the same time instant. To overcome the problem once can use a soft time margin instead a hard margin, and this is achieved by considering overlapping words to a certain degree. But, by doing this the author states that the normalization part will no longer work since the probabilities are not summed in the same time frame, and hence will total more than unity. Hence, the author suggests an approach where the posteriors are computed frame-by-frame so that the normalization of the posteriors is possible. In the end it was found that normalization using frame-by-frame approach did not perform significantly better than the overlapping time marks approach. The normalization of the posteriors is done by dividing the value by the sum of the posterior probabilities of all the words in the specific time instant Instead of using the probabilities as described above, once can use logarithmic approximations of the above probabilities so that the multiplications are converted to additions. Also, we can directly use the acoustic and language model scores from the ASR’s output lattice.

Page 15 of 21 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.

Page 16 of 21 Confidence measure using word posteriors 1-best output from the word spotter The recognizer puts out the following output : !SENT_START BIG BIG BIG !SENT_END We have to determine which of the three instances of the word actually exists.

Page 17 of 21 Confidence measure using word posteriors sent_start sent_end 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.

Page 18 of 21 Confidence measure using word posteriors Alphas and betas for the lattice sent_start sent_end α =0 β= α =-1433 β= α =-2528 β= α =-6761 β= α = β= α = β= α = β=-5917 α = β=-1861 α = β=0 The initial probability at both the nodes is ‘1’. So, its logarithmic value is 0. The language model probability of the word is also ‘1’ since it is the only word in the loop grammar.

Page 19 of 21 Confidence measure using word posteriors Link posterior calculation sent_start sent_end 8 Γ= Γ= Γ= Γ= Γ= Γ= Γ= Γ= 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.

Page 20 of 21 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.

Page 21 of 21 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 , 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 , Istanbul. X. Huang, A. Acero, and H.W. Hon, Spoken Language Processing - A Guide to Theory, Algorithm, and System Development, Prentice Hall, ISBN: , 2001 J. Deller, et. al., Discrete-Time Processing of Speech Signals, MacMillan Publishing Co., ISBN: , 2000