| Location | Printed issue | Corrected statement |
|---|---|---|
| printed 5 PDF 6 | Discrete expectation is written as Σ PX(x), omitting the value being averaged. | E[X] = Σ x PX(x). |
| printed 8 PDF 9 | A conjugate prior is described as having the same form as the observation likelihood. | Conjugacy means the posterior belongs to the same family as the prior. The likelihood as a function of data need not share that family. |
| printed 10 PDF 11 | MMSE is said to equivalently minimize squared bias, (E[θ-θ̂])². | MMSE minimizes E[(θ-θ̂)²]. Squared bias omits variance. Given data, the posterior mean minimizes conditional MSE. |
| printed 12 PDF 13 | Type-I and Type-II error probabilities are called inversely proportional. | A likelihood-ratio threshold induces a trade-off; the two probabilities are not generally reciprocals or inversely proportional. |
| printed 13 PDF 14 | The full-dimensional Gaussian density uses an inverse and determinant after assuming only positive semidefinite covariance. The rotated quadratic ends in yᵀΛ⁻¹yᵀ, and the density exponent uses Λᵀ. | The ordinary density requires positive-definite covariance. With y=Qᵀ(x-μ), the quadratic and exponent use yᵀΛ⁻¹y. |
| printed 17 PDF 18 | QDA loses the class index on an inverse; class estimators use a global n; and the covariance-bias sentence names standard deviation. | Use Σc⁻¹, class count nc, and E[σ̂²ML] = (nc-1)σ²/nc in the scalar case. |
| printed 18 PDF 19 | Cross-covariance repeats x in both factors. | Σxy = E[(X-μx)(Y-μy)ᵀ]. |
| printed 20 PDF 21 | The regression likelihood is duplicated and labeled as a joint distribution p(y,x,θ). | Use p(y|x,w,σ²) = N(y|wᵀx,σ²). Model x separately only if a joint model is intended. |
| printed 22 PDF 23 | Ridge augmentation uses -λI, which produces a λ² penalty, while the solution uses λI. The MAP scale is likewise squared inconsistently. | For penalty λ||w||², augment with √λ I; under isotropic Gaussian noise/prior, λ=σ²/τ². |
| printed 22 PDF 23 | The logistic-regression objective maximizes a mixed-sign expression that is neither log likelihood nor negative cross entropy. | Maximize Σ[y log p + (1-y) log(1-p)], or minimize its negative. |
| printed 24 PDF 25 | The probability of path X₀,...,Xₙ omits endpoint transitions. | Use p(X₀) ∏t=0n-1 p(Xt+1|Xt) for the joint path probability. |
| printed 24-26 PDF 25-27 | Stationarity is conflated with convergence. Periodic, reducible, and absorbing chains are said to have no steady state; every nonunit eigenvalue is claimed strictly inside the unit circle. | All finite chains have stationary distributions. Irreducibility gives uniqueness; finite irreducible aperiodicity gives convergence from every initial law. Absorbing chains can converge. Strict subdominant spectral modulus needs the corresponding ergodicity conditions. |
| printed 24-27 PDF 25-28 | Row and column conventions alternate between incompatible forms; power iteration multiplies by one matrix and normalizes with another. | Choose one orientation. For row probability vectors, πt+1=πtA and stationarity is π=πA. |
| printed 30 PDF 31 | The d-separation conclusion is stated for every joint distribution on the variables. | D-separation implies conditional independence for distributions satisfying the DAG's Markov factorization. Faithfulness is needed for the converse. |
| printed 32 PDF 33 | Undirected separation is given as an unconditional "if and only if," and clique factorization equivalence omits positivity/regularity conditions. | Graph separation implies CI for distributions Markov to the graph. For positive distributions, Hammersley-Clifford gives the standard factorization equivalence. |
| printed 34 PDF 35 | Brute-force elimination for N variables of domain size k is called O(Nk). | Naive enumeration is generally O(kN); factorization and elimination order can reduce the effective scope. |
| printed 36-37 PDF 37-38 | General factor-to-variable initialization and marginal formulas omit or double-count sums/messages, and cyclic message passing is presented without a convergence caveat. | A factor message sums the factor times incoming messages from every other neighboring variable. Tree sum-product is exact. Loopy belief propagation is not generically exact and need not converge. |
| printed 38 PDF 39 | The HMM joint product uses zn inside an i loop; the hidden-state estimator maximizes a prior marginal rather than conditioning on observations. | Use p(z₁)∏ p(zt|zt-1)∏ p(xt|zt) and infer from p(zt|x₁:T). |
| printed 39-40 PDF 40-41 | The forward equation's left index does not match its right side; normalized beta is assigned an unjustified probability ratio; the backward recurrence uses the wrong transition index. | αt(j)=p(xt|j)Σiαt-1(i)p(j|i) and βt(i)=Σjp(j|i)p(xt+1|j)βt+1(j). Positive message scaling is allowed if consistently canceled or tracked. |
| printed 40-41 PDF 41-42 | Viterbi starts by maximizing over the observed sequence and contains malformed message/backpointer indices. | Hold observations fixed. Maximize the joint over hidden paths and retain one maximizing predecessor for each destination state and time. |
| printed 41-42 PDF 42-43 | EM equates the full latent posterior with one-node γ; the prior denominator repeats one term; the transition numerator lacks its time sum; the emission update is incomplete. | Use one- and two-slice marginals: πk=γ₁(k), Aij=Σtξt(i,j)/Σtγt(i), and normalized expected sufficient statistics for the chosen emission family. |
Scope and provenance
The corrections belong to this reviewed companion. The source PDF remains byte-for-byte unchanged. This ledger was checked against all 43 rendered pages and the complete 1,437-line TeX source.
No course prompt, dataset, source figure, answer sheet, or stored lab result is reproduced. The PDF is untagged; accessibility claims apply to the HTML companion and ledger, not the facsimile.