Maximum Likelihood from Incomplete Data via the EM Algorithm

• p. 7 Section 2, unnumbered display after 'In the special case of exponential families'

  E(b(x)|y, φ(p))

• p. 5 Section 2, paragraph after (2.3)

  log f(x|φ) = −log a(φ) + log b(x) + φt(x)T

Only the authors can say what is right here.

Judge's note: Viewed page 7 (printed p. 6): the display reads E(b(x)|y,φ(p)). By (2.17), Q is the conditional expectation of log f(x|φ), and page 5 itself writes log f(x|φ) = −log a(φ) + log b(x) + φt(x)T. Taking the expectation gives E(log b(x)|y,φ(p)), so the display is not an identity as printed. The term does not involve φ, so the M-step conclusion that follows is unaffected. Evident fix: insert 'log'.

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• p. 8 Theorem 2 conclusion

  Then the sequence φ⁽ᵖ⁾ converges to some φ* in the closure of Ω.

• p. 9 Proof of Theorem 2, passage from (3.13) to (3.14)

  whence

• p. 9 sentence after (3.14)

  as required to prove convergence of

Only the authors can say what is right here.

Judge's note: I read Theorem 2 on PDF p. 8 and (3.11)–(3.14) on p. 9. (3.13) bounds the sum of the squared step lengths, Σ||d_j||². (3.14) needs ||Σd_j||², which can be as large as rΣ||d_j||², with r unbounded. Scalar check: with λ = 1, d1 = d2 = 1 and ε = 3, (3.13) holds (2 < 3) but (3.14) fails (4 > 3). I also checked the reviewer's counterexample against the theorem's stated assumptions. Take Ω = (1,∞), L(φ) = −1/φ (bounded on Ω), Q(φ'|φ) = −1/φ' − log 2 and M(φ) = φ + φ^−2. Then ΔQ = 1/(φ^4 + φ) and d² = φ^−4, so ΔQ/d² = φ³/(φ³ + 1) > 1/2 and assumption (2) holds with λ = 1/2. M increases Q, so this is a GEM. Yet φ_{n+1}³ ≥ φ_n³ + 3, so the iterates go to infinity. Both assumptions hold and the stated convergence fails. The same gap carries over to the second part of Theorem 3, which relies on Theorem 2. The files contain no erratum.

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• p. 10 Proof of Theorem 3, displayed expansion

  +(φ − φ(p+1)) D20 Q(φ0(p+1)|φ(p)) (φ − φ(p+1))T

• p. 9 Theorem 3, equation (3.17)

  = −(φ(p+1) − φ(p)) D20 Q(φ0(p+1)|φ(p)) (φ(p+1) − φ(p))T

• p. 9 Theorem 3, equation (3.17)

  −(φ⁽ᵖ⁺¹⁾−φ⁽ᵖ⁾)D²⁰Q

• p. 10 Proof of Theorem 3, opening Taylor expansion

  +(φ−φ⁽ᵖ⁺¹⁾)D²⁰Q

Only the authors can say what is right here.

Judge's note: Viewed (3.17) on page 9 and the expansion on page 10: neither carries a 1/2. A second-order expansion with the Hessian at an intermediate point has ½ on the quadratic form. Checked with constant Hessian: Q(φ'|φ)=C−φ'²/2, φ(p)=1, φ(p+1)=0, D20Q=−1 everywhere. The actual improvement is 1/2, while the printed right side is −(−1)(−1)(−1)=1 for any intermediate point. D20 is the plain second derivative (see (3.24)), so the factor is not absorbed by notation. The qualitative convergence argument is unaffected because the constant can go into λ, but (3.17) is off by a factor of 2 as printed.

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• p. 10 Proof of Theorem 3, first lines

  Expand Q(φ|φp) about φ(p+1)

• p. 10 Proof of Theorem 3, Taylor expansion of Q(φ|φ^(p)) about φ^(p+1)

  Expand

• p. 10 Theorem 3, equation (3.17)

  (3.17)

Only the authors can say what is right here.

Judge's note: I read (3.17) on PDF p. 9 and the proof on p. 10. D^20 is the plain matrix of second derivatives (see (3.16) and (3.24)). A Lagrange-remainder second-order expansion therefore carries a 1/2, and printing it without the 1/2 makes (3.17) false in general. Check: take Q = −φ², φ^(p+1) = 0 (where D^10 = 0) and φ^(p) = 1. The left side is 1. The printed right side is 2. Because D^20 is constant here, no choice of intermediate point can absorb the factor. The qualitative conclusion (condition (2) of Theorem 2 with λ equal to half the eigenvalue bound) is unaffected, but the displayed identity is wrong as printed. The missing superscript parentheses are confirmed on p. 10 as a minor typographical point.

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• p. 10 Theorem 4, assumption (2)

  D¹⁰Q(φ⁽ᵖ⁺¹⁾|φ⁽ᵖ⁾) = 0

• p. 10 Equation (3.19) and passage from (3.22) to (3.23)

  DM(φ*) = D²⁰H(φ*|φ*)[D²⁰Q(φ*|φ*)]⁻¹

Only the authors can say what is right here.

Judge's note: Viewed page 10. Reworked the counterexample: Q(φ'|φ) = −||φ'||²/2 − ||φ'−φ||²/2 + const, so D10Q = φ − 2φ' and D20Q = −2I. M(a,b) = (a/2,b) gives Q(M(φ)|φ) − Q(φ|φ) = a²/4 ≥ 0, so it is a GEM. From (1,0) the sequence (2⁻ᵖ,0) converges to 0, D10Q(φ(p+1)|φ(p)) = 0 along it, and D20Q is negative definite, so assumptions (1)–(3) hold. With D²L = −I, D20H = −I, and (3.19) gives I/2, whereas the actual Jacobian is diag(1/2,1). The limit of (3.22) along a single sequence fixes DM(φ*) only in the direction of approach. The result holds for a true EM map, where the stationarity identity holds for all φ near φ*, but that is not what is stated.

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• p. 14 Section 4.2, definition of the latent observations within a group

  h(z|φ)/Pᵢ(φ)

• p. 15 Equation (4.2.6)

  nᵢ ∫

Only the authors can say what is right here.

Judge's note: Viewed pages 14 and 15. Page 14 says z_i is n_i independent draws from h(z|φ)/P_i(φ) over Z_i, so the conditional expectation is [n_i/P_i(φ(p))]∫ log h(z|φ) h(z|φ(p)) dz. The paper's own (4.2.7) confirms this: with E(m) = n(1−P)/P and density h/(1−P) on Z_t the coefficient is n/P as printed, whereas (4.2.6) as printed with n_t replaced by E(m) would give n(1−P)/P. Checked the example (h=1/2 on (0,2), group (0,1), n_i=1): the true expectation is −log 2, and the printed formula gives −(log 2)/2.

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• p. 15 Equation (4.2.6)

  = n_i ∫ log h(z|φ) h(z|φ(p)) dz. (4.2.6)

• p. 15 Equation (4.2.7)

  = [n/P(φ(p))] ∫ log h(z|φ) h(z|φ(p)) dz

• p. 14 Section 4.2, complete-data specification

  h(z|φ)/P_i(φ) over

• p. 15 Section 4.2, equation (4.2.6)

  (4.2.6)

Judge's note: On PDF p. 14 the z_i are defined as draws from "the density h(z|φ)/P_i(φ) over Z_i", with P_i(φ) = ∫_{Z_i} h. The conditional expectation therefore has to integrate against h(z|φ^(p))/P_i(φ^(p)). The printed (4.2.6) shows no divisor. The paper's own (4.2.7) confirms this. The text says (4.2.6) 'still holds' with n_t replaced by E(m) = n(1−P)/P. With truncated density h/(1−P), that substitution gives n/P, which matches the printed (4.2.7), only if (4.2.6) carries the divisor. Without it the substitution gives n(1−P)/P. The fix is evident: divide by P_i(φ^(p)).

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• p. 17 Equation (4.3.3)

  log f(x|φ)

• p. 18 Opening paragraph, Markov-chain replacement for the second term of (4.3.3)

  V*(φ) is the matrix of transition probabilities

• p. 18 Section 4.3, final paragraph

  is the matrix of transition probabilities

Judge's note: I checked (4.3.2) on PDF p. 16, where V(φ) is the vector of log v(·|φ), and (4.3.3) on p. 17, which is explicitly the complete-data log-likelihood. I also checked the sentence on p. 18. With V* as the matrix of transition probabilities, as defined, z_i^T V* z_{i−1} is a transition probability rather than its logarithm: for a 1→1 transition with probability 0.9 it gives 0.9 instead of log 0.9. Because this depends on φ, it cannot be absorbed as a constant. The same sentence also calls (4.3.3) the 'complete-data likelihood'.

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• p. 21 Section 4.6, uniform h(q_i) paragraph

  α = 3 and β = (y_i − μ(p))²/{2σ(p)2}

• p. 20 Equation (4.6.3)

  h(q_i) = (β^α/Γ(α)) q_i^{α−1} exp(−βq_i)

• p. 20 Equation (4.6.3), gamma-density parameterization

  qᵢᵅ⁻¹ exp(−βqᵢ)

• p. 21 Section 4.6, paragraph beginning “If h(qᵢ) is uniform”

  α = 3

Only the authors can say what is right here.

Judge's note: Page 21 and the text layer both read a plain "α = 3", while the stacked fraction in α* = α + ½ on page 20 is clearly visible. Reworking: with h uniform on (a,b), the conditional density of q_i is proportional to the N(μ, σ²/q_i) density of y_i, i.e. q_i^{1/2} exp{−q_i (y_i−μ)²/(2σ²)}. Matching q^{α−1} exp(−βq) from (4.6.3) gives α − 1 = 1/2, so α = 3/2, with β as printed. This agrees with the paper's own gamma update α* = α + ½ at α = 1. The printed shape 3 is wrong as stated.

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• p. 32 Haberman's discussion, final displayed scoring equations

  V[E(t|y, φ⁽ᵖ⁾)|φ⁽ᵖ⁾]

• p. 33 Haberman's discussion, opening covariance paragraph

  [E{V(t|φ*)|φ*}]⁻¹

Only the authors can say what is right here.

Judge's note: Read PDF p32 (scoring equations) and p33 (covariance sentence). The scoring iteration on p32 uses [V(t|φ) − E{V(t|y,φ)|φ}]^-1 = [V{E(t|y,φ)|φ}]^-1. On p33 the printed expression is [E{V(t|φ*)|φ*}]^-1; V(t|φ*) does not depend on y, so the outer expectation leaves it unchanged and the expression equals V(t|φ*)^-1. Counterexample checked: n pairs (Y_i, Z_i), all coordinates independent N(φ,1), only Y observed, t = Σ(Y_i+Z_i). V(t|φ) = 2n, V(t|y,φ) = n, so the scoring information is n and the variance of the MLE (mean of Y) is 1/n; the printed formula gives 1/(2n). The printed formula is false as written; probably V and E are transposed and 'y' dropped, for [V{E(t|y,φ*)|φ*}]^-1.

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• p. 33 Haberman contribution, first lines of page

  if scoring is used

• p. 32 Haberman contribution, scoring algorithm display

  and the scoring algorithm may

Only the authors can say what is right here.

Judge's note: I read the scoring equation on PDF p. 32 and the covariance sentence on p. 33. V(t|φ*) does not depend on y, so E{V(t|φ*)|φ*} = V(t|φ*), the complete-data information. By the law of total variance, the scoring matrix used on p. 32 is V(t|φ*) − E{V(t|y,φ*)|φ*} = V{E(t|y,φ*)|φ*}. The two matrices coincide only when no information is missing. The expression appears to have swapped E and V and dropped the conditioning on y, but only the discussant can confirm what was intended.

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• p. 4 Table 1, third column header and entries

  0·126821498

• p. 3 Section 1, last paragraph

  The second column in Table 1 gives the deviation

• p. 3 Section 1, paragraph after the π* formula

  The second column in Table 1 gives the deviation π(p)−π*

Judge's note: Viewed Table 1 on page 4: no minus signs are printed in the deviation column, and every iterate (0.5 up to 0.626821484) is below π*. The signed difference in the header is negative in every row. Neither the header nor the page 3 text says absolute value. The ratio column is unaffected. This is a column label/sign issue; the magnitudes are right.

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• p. 12 End of Section 3, paragraph 'When finding the posterior mode'

  in equation (3·15) by

• p. 9 Lemma 2, equation (3.15)

  = D10 H(φ|φ) = 0

• p. 10 Theorem 4, equation (3.19)

  DM(φ*) = D20 H(φ*|φ*) [D20 Q(φ*|φ*)]−1

• p. 12 Section 3, final paragraph (printed p. 11)

  in equation (3.15) by D20 Q(φ*|φ*)+D2G(φ*)

Judge's note: Viewed page 12 (pointer), page 9 ((3.15): E[∂/∂φ log k | y,φ] = D10H(φ|φ) = 0, in Lemma 2) and page 10 ((3.19): DM(φ*) = D20H(φ*|φ*)[D20Q(φ*|φ*)]⁻¹). Section 3 itself cites (3.19) for the rate. A discussant also cites '(3.15) (or (3.27))', which suggests leftover draft numbering, but in the published numbering the pointer lands on the wrong equation. One reviewer note calls (3.15) the Theorem 3 condition; it is in fact Lemma 2, which does not change the verdict.

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• p. 34 Discussion, Dr R. Sundberg, second paragraph

  of (3.15) (or (3.27))

• p. 11 Last numbered equation of Section 3

  (3.26)

Only the authors can say what is right here.

Judge's note: Checked PDF p34 (printed p33) for the sentence and PDF p9-p10 for the equations. (3.15) is E[∂ log k/∂φ | y, φ] = D10 H = 0; the last numbered equation of Section 3 is (3.26). The rate matrix DM(φ*) is (3.19) and, for exponential families, (3.26), as the paper itself says on printed p10. The pointers do not resolve as published and probably follow draft numbering; the intended targets are likely (3.19) and (3.26), but only the authors can confirm.

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• p. 17 Section 4.3, paragraph on Haberman (1976)

  missing values (Section 3.1.1)

• p. 12 Subsection heading

  4.1.1. Multinomial sampling

• p. 17 Section 4.3, paragraph on Haberman (1976)

  multinomial missing values (Section 3.1.1)

• p. 17

  multinomial missing values (Section 3.1.1) and finite mixtures

Judge's note: Page 17 prints "(Section 3.1.1)". Page 12 shows the heading "4.1.1. Multinomial sampling" under 4.1 Missing Data. A text search finds no other occurrence of 3.1.1, and Section 3 has no numbered subsections. The pointer is wrong and the intended target is evident.

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• p. 18 Section 4.3, last paragraph

  Baum et al., 1970; Baum, 1972)

• p. 22 References

  BAUM, L. E. (1971). An inequality and associated maximization technique

• p. 18 Section 4.3, last part

  Baum and Eagon, 1967; Baum et al., 1970; Baum, 1972

• p. 18 Section 4.3

  Baum, 1972

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Judge's note: Page 18 prints "(Baum and Eagon, 1967; Baum et al., 1970; Baum, 1972)". The reference list on page 22 has BAUM, L. E. (1971), with no 1972 Baum entry. The mismatch is internal; which year is right cannot be settled from the files.

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• p. 19 Section 4.5, second paragraph

  may be found in Maritz (1964)

• p. 23 References

  MARITZ, J. S. (1970). Empirical Bayes Methods.

• p. 23 References

  MARITZ, J. S. (1970). Empirical Bayes Methods. London: Methuen.

• p. 19 Section 4.5

  empirical Bayes' procedures. Many examples and a discussion of their properties may be found in Maritz (1964)

Only the authors can say what is right here.

Judge's note: Page 19 prints Maritz (1964). Page 23 has a single Maritz entry, dated 1970. The mismatch is internal; the correct year cannot be determined from the files.

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• p. 19 Section 4.5, second paragraph

  Mosteller and Wallace (1965)

• p. 23 References

  D. L. (1964). Inference and Disputed Authorship: The Federalist.

• p. 23 References

  MOSTELLER, F. and WALLACE, D. L. (1964). Inference and Disputed Authorship

• p. 23 References

  D. L. (1964). Inference and Disputed Authorship: The Federalist

Only the authors can say what is right here.

Judge's note: Page 19 prints 1965. Page 23 has a single Mosteller and Wallace entry, dated 1964. The mismatch is internal; the correct year cannot be determined from the files.

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• p. 19 Section 4.5, second paragraph

  Good (1967) and Efron and Morris (1975)

• p. 22 References

  GOOD, I. J. (1965) The Estimation of Probabilities

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Judge's note: Page 19 prints "Good (1967) and Efron and Morris (1975)". The reference list on page 22 has Good (1965) and Good (1956) only, and page 20 cites Good (1956) and Good (1965). Either the year is wrong or an entry is missing; only the authors can say which.

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• p. 33 Professor H. O. Hartley, second paragraph

  This is a consequence of the authors' equation (3.13)

• p. 33 Equation (3.13)

  (3.13)

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Judge's note: Checked PDF p33 (printed p32) for Hartley's sentence and PDF p9 (printed p8) for equation (3.13). Published (3.13) is a convergence bound and has no bearing on the rank of D20 Q or on derivatives of log f; the pointer does not fit as published. It probably follows the numbering of a pre-meeting draft (the paper itself, printed p11, refers to 'D20 Q(φ*|φ*) in equation (3.15)', where (3.15) is the score identity). Which published equation was meant cannot be settled from the files.

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• p. 35 Authors' reply, second paragraph

  (Hemmerle, 1974, 1976)

• p. 38 References in the Discussion

  HEMMERLE, W. J. and LORENS, J. O. (1976).

Only the authors can say what is right here.

Judge's note: Read the citation on PDF p35 (printed p34) and the reference list on PDF p38 (printed p37). There is no single-author 1976 Hemmerle entry there, and the main reference list (PDF p23 viewed) has none either. Either the co-author Lorens was dropped from the citation or a reference is missing; the Hemmerle and Lorens title concerns the W-transform in variance component estimation, not 'stretching' in ANOVA, so the intended work is unclear.

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