A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition
Proceedings of the IEEE, 1989 · paper · Computer science
A tutorial reviewing the theory of hidden Markov models, a statistical model for signals, and showing how they have been applied to selected problems in machine speech recognition.
Checked: PDF from web.ece.ucsb.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
14 findings
Substance Sec. IV-F prints "s = (p + q − 2pq)/(1 − 2r)", which leaves out −r in the numerator and does not give the stated "s = 13/30 ≈ 0.433".
- p. 15 Section IV-F, equivalence condition and solution for s
s = (p + q − 2pq)/(1 − 2r)
- p. 15 Section IV-F numeric example
we get s = 13/30 ≈ 0.433
- p. 15 Section IV-F, below the equivalence condition
s = p + q - 2pq / 1 - 2r
Only the authors can say what is right here.
Judge's note: Page 15 image, right column. Reworked: LHS pq + (1−p)(1−q) = 1 − p − q + 2pq and RHS = 1 − r − s + 2rs, so s(1 − 2r) = p + q − 2pq − r, giving s = (p + q − 2pq − r)/(1 − 2r). With p = 0.6, q = 0.7, r = 0.2, the corrected formula gives 0.26/0.6 = 13/30 (matching the printed value), while the printed formula gives 0.46/0.6 ≈ 0.767. The printed numerator shows only p + q − 2pq.
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Substance Eq. (88) is printed as D(λ1, λ2) = (1/T)[log P(O^(2)|λ1) − log P(O^(2)|λ2)], with O^(2) generated by λ2. In expectation this is ≤ 0, yet the text calls it a distance (dissimilarity) measure and interprets it as divergence or discrimination information.
- p. 15 Eq. (88)
D(λ1, λ2) = (1/T)[log P(O^(2)|λ1) − log P(O^(2)|λ2)]
Only the authors can say what is right here.
Judge's note: I checked (88) on the p.15 image and it reads as quoted. O^(2) is generated by λ2, so E[log P(O2|λ2) − log P(O2|λ1)] = T·KL(λ2||λ1) ≥ 0. The printed difference therefore has nonpositive expectation. That contradicts calling it a dissimilarity and interpreting it as divergence, both of which are nonnegative. The two log terms look reversed, but only the authors can say which form they intended.
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Substance The denominator of Eq. (90) sums t = 1 to T ("Σ_{t=1}^{T} Σ_{j=1}^{N} α_t(i) a_ij b_j(O_{t+1}) β_{t+1}(j)"), while its numerator, (40b), and the scaled forms (95) and (98) sum to T−1.
- p. 16 Eq. (90) denominator
Σ_{t=1}^{T} Σ_{j=1}^{N} α_t(i) a_ij b_j(O_{t+1}) β_{t+1}(j)
- p. 16 Eq. (95) denominator
Σ_{t=1}^{T−1} Σ_{j=1}^{N}
Judge's note: The p.16 image shows the upper limit T in the (90) denominator and T−1 in the (95) and (98) denominators. A t = T term would need O_{T+1} and β_{T+1}, which are undefined. (39a) and (40b) count transitions over t = 1..T−1.
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Substance The scaled forward recursion (92a), (92b) and (93b) is printed as α_t(i) = Σ_{j=1}^{N} α̂_{t−1}(j) a_ij b_j(O_t). By the induction (20), the transition is j→i, so the terms should be a_ji b_i(O_t).
- p. 16 Eq. (92a)
α_t(i) = Σ_{j=1}^{N} α̂_{t−1}(j) a_ij b_j(O_t)
- p. 16 Eq. (92b), (93b)
α̂_{t−1}(j) a_ij b_j(O_t)
Judge's note: The p.16 image shows a_ij b_j(O_t) inside the sums over j in (92a), (92b) and (93b). By (20), α_{t+1}(j) = [Σ_i α_t(i) a_ij] b_j(O_{t+1}). Renaming the indices, state i at time t needs Σ_j α_{t−1}(j) a_ji · b_i(O_t). As printed, the result is not the forward variable for state i.
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Substance Eq. (111) keeps the explicit weight 1/P_k on sums of scaled products α̂_t^k(i)…β̂_{t+1}^k(j). By (96)–(101), those products already equal (1/P_k)·α a b β, so (111) as printed does not reduce to (109).
- p. 17 Eq. (111) and following text
In this manner, for each sequence O^(k), the same scale factors will appear in each term of the sum over t as appears in the P_k term
Only the authors can say what is right here.
Judge's note: My working from the p.16–17 images: α̂_t(i)β̂_{t+1}(j) = C_t D_{t+1} α_t(i)β_{t+1}(j) by (96)–(97). By (99), C_t D_{t+1} = C_T = Π_{t=1}^{T} c_t. By (101), this equals 1/P(O^(k)|λ) = 1/P_k, with P_k defined in (108). So each numerator term of (111) is (1/P_k)·(1/P_k)·α a b β, whereas (109) has (1/P_k)·α a b β. The paper's sentence that the scale factors "cancel exactly" with the P_k term would hold only if P_k meant a scaled quantity, and the paper does not define one. Only the authors can say which form they meant.
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Reference or label Sec. III-C cites "proven by Baum and his colleagues [6], [3]" twice, but [6] is "J. K. Baker, 'The dragon system—An overview'", not a Baum paper.
- p. 9 Section III-C, left column after (40)
proven by Baum and his colleagues [6], [3] that either 1)
- p. 9 Section III-C, right column after (41)
It has been proven by Baum and his colleagues [6], [3]
- p. 28 Reference [6]
J. K. Baker, "The dragon system—An overview,"
- p. 9 Sec. III-C
proven by Baum and his colleagues [6], [3]
Only the authors can say what is right here.
Judge's note: Page 9 image: the citation '[6], [3]' appears both after (40c) (left column) and after (41) (right column). In the page 28 reference list, [6] is Baker's Dragon system paper and [1]-[5] are Baum's papers. The text layer (line 106) also has 'Baum and his colleagues [1]-[5]', and [6] is cited for Baker at CMU. Probably [4] was meant, but only the author can say which reference.
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Reference or label Sec. V-A says "If we write the reestimation formula (41) directly in terms of the forward and backward variables", but (41) is Baum's auxiliary function Q(λ, λ̄). The a_ij reestimation formula is (40b).
- p. 16 Section V-A Scaling, paragraph before (90)
If we write the reestimation formula (41) directly in terms of the forward and backward variables
- p. 16 Sec. V-A, before Eq. (90)
we write the reestimation formula (41) directly
- p. 16 Sec. V-A
If we write the reestimation formula (41) directly in terms
- p. 9 Eq. (41)
Baum's auxiliary function
Judge's note: The p.9 image shows (41) is Q(λ,λ̄) = Σ_Q P(Q|O,λ) log[P(O,Q|λ̄)] and (40b) is the ā_ij formula. The p.16 image shows the text cites (41) just before (90), which is ā_ij.
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Reference or label Section VI-G opens "In Section IV-C we discussed the theoretically correct method of incorporating state duration information", but explicit state duration is treated in Section IV-D.
- p. 24 Section VI-G, first sentence
In Section IV-C we discussed the theoretically correct method of incorporating state duration information
- p. 24 Sec. VI-G, first sentence
In Section IV-C we discussed the theoretically correct
- p. 13 heading of Section IV-D
D. Inclusion of Explicit State Duration Density in HMMs
Judge's note: The page 13 image shows the heading "D. Inclusion of Explicit State Duration Density in HMMs [29], [30]", and its content (D², the D-fold storage cost) matches what VI-G describes. Section IV-C is "Variants on HMM Structures—Null Transitions and Tied States" (from the text layer, line 1271). The page 24 image confirms the pointer reads "IV-C".
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Reference or label Sec. IV-E cites "The proposed alternative design criterion [31] is the maximum mutual information (MMI) criterion", but [31] is Lowerre and Reddy, "The HARPY speech understanding system"; the MMI paper is [32] (Bahl et al.).
- p. 15 Section IV-E, after (84)
The proposed alternative design criterion [31] is the maximum mutual information (MMI) criterion
- p. 29 Reference [31]
B. Lowerre and R. Reddy, "The HARPY speech understanding system,"
- p. 15 Sec. IV-E, para 2
The proposed alternative design criterion [31] is the max-
- p. 29 Reference list
[31] B. Lowerre and R. Reddy, "The HARPY speech understanding
Judge's note: p.15 image shows [31] in this sentence. Reference list (p.29 image): [31] is HARPY and [32] is Bahl, Brown, de Souza, Mercer, "Maximum mutual information estimation...". The section heading ("[32], [33]"), "The first alternative [32]" and "steepest descent methods [32]" all cite [32] for MMI.
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Reference or label Sec. V-D says "Using the model of Fig. 9, the value of ε can be estimated from the training data", but the deleted-interpolation model is Fig. 10. The preceding sentence cites Fig. 10 for it.
- p. 18 Section V-D, paragraph on estimating ε
Using the model of Fig. 9, the value of ε can be estimated from the training data
- p. 18 Sec. V-D, right column
Using the model of Fig. 9, the value of
Judge's note: The p.18 image shows the text cites Fig. 10 for the expanded HMM and then "Fig. 9". Fig. 10 on p.18 is the deleted-interpolation state diagram (s̃, λ, λ', ε, 1−ε). Fig. 9 shows state-duration connections, per the reviewers' report of its caption on p.13.
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Reference or label Reference [2] gives the venue as "Bull. Amer. Meteorol. Soc., vol. 73, pp. 360-363, 1967." for the Baum–Eagon inequality paper.
- p. 28 Reference [2]
Bull. Amer. Mete- orol. Soc., vol. 73, pp. 360–363, 1967.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the Baum–Eagon paper appeared in Bull. Amer. Math. Soc., vol. 73 (1967). The page 28 image clearly prints "Bull. Amer. Mete- orol. Soc."
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Reference or label Reference [42] is dated "AT&T Tech. J., vol. 64, no. 6, pp. 1211-1222, July-Aug. 1986", while [25] (pp. 1235-1249) and [35] (pp. 1251-1270) in the same vol. 64, no. 6 are dated 1985.
- p. 29 Reference [42]
AT&T Tech. J., vol. 64, no. 6, pp. 1211–1222, July–Aug. 1986.
- p. 29 Reference [25]
Tech. J., vol. 64, no. 6, pp. 1235–1249, July–Aug. 1985.
- p. 29 Reference [35]
AT&T Tech. J., vol. 64, no. 6, pp. 1251–1270, July– Aug. 1985.
- p. 29 Reference [42]
vol. 64, no. 6, pp. 1211-1222, July-Aug. 1986.
Judge's note: The page 29 image confirms all three entries. The page ranges are consecutive within the same volume, issue and months, so a single issue carries two years. Internal evidence (two of the three entries) points to 1985.
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Wording EM is expanded as "the EM (expectation-modification) method [23]" (p. 264) and the M step is called "the M (modification) step is the maximization over λ̄" (p. 265); it should be expectation-maximization / maximization.
- p. 8 Section III-C, first paragraph
the EM (expectation-modification) method [23]
- p. 9 Notes on the Reestimation Procedure
the M (modification) step is the maximization over λ̄
Judge's note: Page 8 image, right column: 'EM (expectation-modification) method [23]'. Page 9 image, right column: 'the M (modification) step is the maximization over λ̄'. The paper's own sentence describes the M step as a maximization, and the title of [23] (p. 28) begins 'Maximum like-'. The wrong expansion is internally inconsistent.
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Wording Eq. (110) for b̄_j(ℓ) sums "Σ_{t=1}^{T_k−1} α_t^k(i) β_t^k(i)". It runs to T_k−1, whereas the single-sequence formula (40c) sums γ_t(j) to T, and it uses index i in a formula for state j.
- p. 17 Eq. (110)
Σ_{t=1}^{T_k−1} α_t^k(i) β_t^k(i)
- p. 9 Eq. (40c)
Σ_{t=1}^{T} γ_t(j)
Only the authors can say what is right here.
Judge's note: The p.17 image shows upper limits T_k−1 and α_t^k(i)β_t^k(i) in both the numerator and denominator of (110). The p.9 image shows (40c) summing γ_t(j) over t = 1..T. For emission counts the last frame should be included, and the state index should be j. These look like carry-overs from (109).
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26 minor (typos and the like; most checked by the judge only)
Reference or label The Section VIII heading cites "[6]–[13], [31], [37], [38], [51], [64]–[66]"; [51] is Atal & Hanauer on linear prediction, while [50] (TANGORA 20,000-word recognizer) is the large-vocabulary entry that fits.
- p. 27 Section VIII heading
[6]–[13], [31], [37], [38], [51], [64]–[66]
- p. 29 Reference [51]
B. S. Atal and S. L. Hanauer, "Speech analysis and synthesis by linear prediction of the speech wave,"
Only the authors can say what is right here.
Judge's note: The page 27 heading includes [51]. The page 29 reference list gives [51] as Atal & Hanauer, "Speech analysis and synthesis by linear prediction of the speech wave", and [50] as Averbuch et al., TANGORA. [51] is already cited under "C. LPC Feature Analysis [51]-[54]", and [50] is cited for the IBM TANGORA system on p. 20. An off-by-one slip is likely, but only the author can confirm which reference was intended.
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Wording Sec. II-A: when only one coin is tossed, using the 3-coin model (9 parameters) is called "an underspecified system", but the argument describes a model with more parameters than the process needs.
- p. 4 Section II-A, coin toss models paragraph
we would be using an underspecified system
- p. 4 left column, end of first paragraph
we would be using an underspecified system.
Only the authors can say what is right here.
Judge's note: Page 4 image, left column: 'it might just be the case that only a single coin is being tossed. Then using the 3-coin model of Fig. 2(c) would be inappropriate, since the actual physical event would not correspond to the model being used—i.e., we would be using an underspecified system.' The same paragraph says the 3-coin model has 9 unknown parameters and the 1-coin model has 1, so the model has too many parameters, i.e. it is overspecified. A strained reading (the physical system under-represents the model) is possible, so this is left as a query to the author.
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Wording Sec. VI-G recalls "a D²-fold increase in computation and a D-fold increase in storage", but Sec. IV-D states "D times the storage and D²/2 times the computation".
- p. 14 Section IV-D, drawbacks paragraph
D times the storage and D²/2 times the computation is required
- p. 24 Section VI-G
namely a D²-fold increase in computation and a D-fold increase in storage
Judge's note: The page 14 image (right column) says D²/2 and gives a factor of 300 for D = 25. The page 24 image (Sec. VI-G) says 'We also showed ... a D²-fold increase in computation'. This may be an order-of-magnitude paraphrase, but it is presented as the earlier result. Separately, VI-G points to 'Section IV-C', but the duration material is in Sec. IV-D (page 13).
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Wording In the scaling derivation, α_t(i) has two meanings. In (91) and (92a) it is the intermediate computed from the scaled α̂_{t−1}. In (93a), (93b) and (96) ("α̂_t(i) = [Π c_s] α_t(i) = C_t α_t(i)") it is the unscaled forward variable of (18)/(20).
- p. 16 Eq. (91), (92a)
c_t = 1 / Σ_{i=1}^{N} α_t(i)
- p. 16 Eq. (93a), (96)
α̂_t(i) = [Π_{s=1}^{t} c_s] α_t(i) = C_t α_t(i)
Only the authors can say what is right here.
Judge's note: On the p.16 image, (92a) builds α_t(i) from α̂_{t−1}(j), and c_t in (91) normalises that quantity. (93a) and (96) relate α̂ to the unscaled α through products of c. The scheme only works if c_t normalises the intermediate quantity. If it normalised the true α_t, (96) and (100)–(101) would not follow. The same symbol is used for both quantities and the text does not separate them. This is an ambiguity in notation and the stated results stand.
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Wording The scaled backward variable is defined as "β̂_t(i) = c_t β_t(i)" in (94) but as "β̂_{t+1}(j) = [Π_{s=t+1}^{T} c_s] β_{t+1}(j) = D_{t+1}β_{t+1}(j)" in (97). The symbol β is used both for the intermediate value and for the unscaled backward variable.
- p. 16 Eq. (94)
β̂_t(i) = c_t β_t(i)
- p. 16 Eq. (97)
β̂_{t+1}(j) = [Π_{s=t+1}^{T} c_s] β_{t+1}(j) = D_{t+1}β_{t+1}(j)
Only the authors can say what is right here.
Judge's note: The p.16 image confirms both definitions. They agree only if β_t in (94) is the intermediate computed from β̂_{t+1}, and the paper does not say so. In (97), β is the true backward variable of (23). This is a notation ambiguity of the same kind as C022 but for β.
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Typo Introduction, right column, para 2 reads "These are several possible choices for what type of signal model is used"; 'These' should be 'There'.
- p. 1 Section I, right column
These are several possible choices for what type of signal model
- p. 1 Introduction, right column, para 2
These are several possible choices for what type of signal model is used
Judge's note: Page 1 image, right column, start of the second paragraph: the text reads 'These are several possible choices'. This is a wording slip, not an OCR error.
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Typo Sec. III-A reads "until we make the list transition (at time T)"; 'list' should be 'last'.
- p. 6 Section III-A, interpretation of (17)
until we make the list transition (at time T)
Judge's note: Page 6 image, left column, interpretation of (17): the text reads 'until we make the list transition (at time T)'.
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Typo Sec. III-B misspells probability as "have zero probabilty".
- p. 8 Section III-B, first paragraph
have zero probabilty
Judge's note: Page 8 image, left column, first paragraph: 'which have zero probabilty (aij = 0 for some i and j)'.
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Typo After (40c) the reestimated model is defined as "λ̄ = Ā, B̄, π̄)", with the opening parenthesis missing.
- p. 9 Paragraph after (40c)
λ̄ = Ā, B̄, π̄)
Judge's note: Page 9 image, left column below (40c): 'define the reestimated model as λ̄ = Ā, B̄, π̄)'. The earlier form is λ = (A, B, π).
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Typo Sec. IV-B reads "Gaussian autoregressive (or order p)"; 'or' should be 'of'.
- p. 11 Section IV-B
Gaussian autoregressive (or order p)
Judge's note: Page 11 image, right column, Sec. IV-B: 'is Gaussian autoregressive (or order p)'.
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Typo End of Sec. IV-B reads "The new autocorrection vectors of the autoregression coefficients"; it should be 'autocorrelation'.
- p. 12 Top of right column, end of Section IV-B
The new autocorrection vectors of the autoregression coefficients
Judge's note: Page 12 image, top of right column: 'The new autocorrection vectors'. Page 12, left column, says r_a(i) of (57c) is 'the autocorrelation of the autoregressive coefficients'.
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Typo Sec. IV-C misspells deletions as "(i.e., symbol delections)".
- p. 12 Section IV-C, Fig. 8(b) discussion
(i.e., symbol delections)
Judge's note: Page 12 image, right column, discussion of Fig. 8(b): 'alternate word pronunciations (i.e., symbol delections)'.
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Typo Sec. IV-D step 3 reads "Generally we assume independent of observations"; it should be 'independence'.
- p. 13 Section IV-D, step 3
Generally we assume independent of observations
Judge's note: Page 13 image, right column, item 3): 'Generally we assume independent of observations so that...'.
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Typo The numerator of (80) has the summation condition "s.t. O_t = k", while the denominator and the text use "s.t. O_t = v_k".
- p. 14 Eq. (80) numerator
s.t. O_t = k
Judge's note: Page 14 image, left column, Eq. (80): the numerator reads 's.t. O_t = k' and the denominator reads 's.t. O_t = v_k'. The text in the right column says 'observation O_t = v_k'. This is a notational slip.
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Typo In Eq. (89) the numerator is printed "D(λ1λ2) + D(λ2, λ1)". The first pair has no comma.
- p. 15 Eq. (89)
D(λ1λ2) + D(λ2, λ1)
Judge's note: The p.15 image shows "D(λ1λ2)" with no comma, unlike every other use of D(·,·).
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Typo Sec. VI-A item 4 reads "only a single word from a finite set of equiprobable is required to be recognized". A noun (e.g., "words") is missing after "equiprobable".
- p. 19 Section VI-A, item 4 Syntactic Analysis
only a single word from a finite set of equiprobable is required to be recognized
- p. 19 Sec. VI-A item 4
from a finite set of equiprobable is required to be recognized
Judge's note: The p.19 image, right column, item 4) Syntactic Analysis, shows the text as quoted.
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Typo The first paragraph on p.20 reads "There is one additional factor that has a significant effort on the implementation". The word should be "effect".
- p. 20 First paragraph
There is one additional factor that has a significant effort
Judge's note: The p.20 image, left column, first line, shows "significant effort".
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Typo Section VIII reads "The research in this area far outweights the research" (should be "outweighs"), and Sec. VI-A reads "from the vocabularly" (should be "vocabulary").
- p. 27 Section VIII, first paragraph
The research in this area far outweights the research
- p. 27 Section VIII, para 1
The research in this area far outweights the research in any other area
- p. 27 Sec. VIII
far outweights the research
- p. 20 Sec. VI-A end
from the vocabularly.
Judge's note: Both misspellings are printed in the page images: "outweights" on page 27, right column, Sec. VIII paragraph 1; "vocabularly" on page 20, left column. Neither is an OCR artefact.
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Typo The Fig. 14 caption reads "as a function of the size of the VQ, M (shown of a log scale)". It should read "on a log scale".
- p. 22 Fig. 14 caption
(shown of a log scale)
Judge's note: The p.22 image shows the caption as quoted. The body text says "(on a log scale)".
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Typo Text after Fig. 16 reads "similarly the second, fourth, and eight cepstral parameters show the need"; the ordinal should be "eighth".
- p. 23 Text after Fig. 16
the second, fourth, and eight cepstral parameters
- p. 23 left column, after Fig. 16
and eight cepstral parameters show the need
Judge's note: The page 23 image, left column below Fig. 16, shows "eight cepstral parameters" in an ordinal list (first … second, fourth, and eight). This is a typo.
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Typo Section VII-B: "a best string of size ℓ words (1 ≤ t ≤ L)" uses the frame index t where the level index ℓ (1 ≤ ℓ ≤ L) is meant.
- p. 26 Section VII-B, after (122c)
a best string of size ℓ words (1 ≤ t ≤ L)
Judge's note: In the page 26 image (right column, after (122c)) the range reads "(1 ≤ t ≤ L)". In the same section t is the frame index (1 ≤ t ≤ T), and L bounds the level ℓ. At this resolution the glyph looks like t rather than ℓ. This is a notational slip.
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Typo Reference [17] reads "Proc. Inst. of Acoustics, Autum Conf." (should be "Autumn").
- p. 28 Reference [17]
of Acoustics, Autum Conf., pp. 1–8, Nov. 1984.
- p. 28 Reference [17]
Autum Conf.
Judge's note: The page 28 image, reference [17], prints "Autum Conf."
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Typo Reference [12] gives the editor as "P. R. Krishnaiad, Ed." (presumably Krishnaiah).
- p. 28 Reference [12]
P. R. Krishnaiad, Ed.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the editor of Handbook of Statistics vol. II is P. R. Krishnaiah. The page 28 image prints "Krishnaiad".
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Typo Reference [2] gives the co-author as "L. E. Baum and J. A. Egon" (presumably Eagon).
- p. 28 Reference [2]
L. E. Baum and J. A. Egon,
- p. 28 Reference [2]
J. A. Egon
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the co-author's name is J. A. Eagon (the Baum–Eagon inequality). The page 28 image prints "Egon".
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Typo Reference [36] gives the editors as "E. S. Gelesma and L. N. Kanal, Eds." (presumably Gelsema).
- p. 29 Reference [36]
E. S. Gelesma and L. N. Kanal, Eds.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the co-editor of Pattern Recognition in Practice is E. S. Gelsema. The page 29 image prints "Gelesma".
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Typo Reference [55] gives the author as "Y. Tokhura" (presumably Tohkura).
- p. 29 Reference [55]
Y. Tokhura, "A weighted cepstral distance measure for speech recognition,"
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the author of the weighted cepstral distance paper is Y. Tohkura. The page 29 image prints "Tokhura".
Second check: none (judge only)