Typos and errors

A lot of thanks to anyone who found the errors below (their names appear in parentheses). Feel free to contact me if you find others.

Major mistakes

• 100, Exercise 8.7: the fact that the former estimator dominates the latter (in terms of asymptotic variance) is equivalent to the stated equality provided \(\mathbb{Q}(\varphi)>0\) (this condition is omitted in the book). Moreover, when \(w\) is an indicator function, one has equality (both estimators have the same variance asymptotically). (Otmane Sakhi)

• 117, Algorithm 9.6: the last two instructions should be outside the while loop, not inside; i.e. while \(v^m < s\), increment \(m\). And, only when the while loop has ended, we assign \(A^n\) and increment \(s\). (Matt Wand)

• 171, Lemma 1.2: assumption that \(G_t\) is upper bounded is not required. (Adrien Corenflos)

• 182, Lemma 11.10: this lemma is correct only if function \(\varphi\) takes both positive and negative values. In the following calculations, this lemma is applied to centred functions (\(\varphi\) minus its expectations). (Adam Johansen)

• 313, Prop. 16.6 and Algorithm 16.8: the expression for the weights (which define the distribution of \(B_t\)) is correct for the bootstrap filter, but not in general, as a factor \(G_{t+1}(X_t^n, X_{t+1}^{B_{t+1}})\) is missing. (In the bootstrap filter, this factor is constant, since \(G_{t+1}\) does not depend on the first argument in this case.) Alternatively, you can also replace the transition density \(m_{t+1}\) by the one of the model, \(p_{t+1}\), to obtain a correct expression. (Adrien Corenflos) Also, in light of this, the sentence below Prof. 16.6 is a bit daft, and should be ignored.

• 352, two last paragraphs of Python corner: it’s not true that we obtain a \(\times k\) speed-up (where \(k=\) nr of CPU core), see this blog post for more explanation.

Typos

• 15, Eq. (2.3): in the first matrix (before \(X_{t-1}\)), \(0_2\) means the \(2\times 2\) null matrix; the second term should be a vector, not a matrix, i.e. \((0_2, U_t^T)^T\) (Andreas Makris)

• 41, second equation: missing \(dx_{t-1}\) in the integral (Chihiro Kuraya)

• 44, first equation: \(f_t\) should be \(f_s\) (Chihiro Kuraya)

• 56, first equation: \(M_{0:t-1}(dx_{t-1})\) should be \(M_{t-1}(dx_{0:t-1}\)) (Feras Saad)

• 59, first equation: \(H_0(x_0)\) should be \(H_{0:T}(x_0)\) (Feras Saad)

• 63, (5.17), top line: the first probability distribution should be with respect to \(X_t\), not \(X_{t-1};\) i.e. \(P_t(X_t \in dx_t | Y_{0:t} = y_{0:t})\) (Giovanni Diana)

• 68, likelihood of future observations in grey box: \(p(y_{t+2:T}|l)\) should be \(p_T(y_{t+2:T}|x_{t+1}=l)\) (in order to be consistent with notation for LHS) (Yawei Ge).

• 73, summary: mathematically equivalently -> mathematically equivalent (Feras Saad)

• 76, result 2 (near bottom of the page) \(y_{1:t}\) should be \(y_{0:t}\) in the first factor of the RHS.

• 90, MSE of normalised IS (first displayed eq.): term \(\mathbb{Q}(\varphi)^2\) should be multiplied by \(\mathbb{M}(w)^2\) (Jacob Hoover).

• 92, equation below (8.5): missing sup with respect to \(\varphi\) (Adrien Corenflos)

• 92, Theorem 8.5: [w] missing just before big closing parenthesis (Chihiro Kuraya)

• 116, second equation in Lemma 9.2: \(V_n\) should be \(V^n\) (Adrien Corenflos)

• 117, equation on last line: in middle expression, n and n - 1 should be superscripts; the intersection inside \(\lambda(\cdot)\) should be \([C^{n-1}, C^n] \cap [\frac{m-1}{N}, \frac{m}{N}]\). (Chihiro Kuraya)

• 118: same issue in first multi-line equation: replace \(n\) by \(m\) in the interval \([n-1/N, n/N]\), and below the sum (second line). (Chihiro Kuraya)

• 119^4 (display in proof of Prop 9.4): \(W^n\) should be \(W\). (Chihiro Kuraya)

• 119_5: estimate(d) -> estimates (Adrien Corenflos)

• 122, Ex 9.10: the first constraint is \(E(W)=x\) (missing “=x”). (Omiros)

• 126: Guldas et al (2017), “volume 28” -> “volume 48, Issue 28”. (Feras Saad)

• 130, Algorithm 10.1: resampling step should be based on weights \(W_{t-1}^{1:N}\), not \(W_t^{1:N}\) (Chihiro Kuraya)

• 131^4: done in the first too (two) lines… (Yawei Ge).

• 131, grey box: space missing after “approximates”. (Omiros)

• 139, first box, 3rd bullet: “P_t >> M_t”, should be the reverse: “P_t << M_t”; also, \(P_t(X_t \in dx_{0:t} | \ldots )\) should be \(P_t(X_{0:t} \in dx_{0:t} | \ldots )\) (Feras Saad)

• 139: the following Radon-Nykodym derive (s) exist (Adrien Corenflos)

• 144, last line of Example 10.5: the second term should be \(\sigma^2[y_t^2 e^{-\mu^\star(x_{t-1})} - 1] /2\) (Gonzalo Mena)

• 152, (10.4): missing tilde on the \(X_{t-1}\) in the second term.

• 176: Proposition 11.2 should be Proposition 11.5. (Adam Johansen).

• 180, snd line of Lemma 11.7: \(\phi\) should be \(\varphi\) (Chihiro Kuraya)

• 190: in both lines below the display, h should be k in \(z_{t-1}\) (Suzie Brown)

• 195, Algorithm 12.1: functions \(\Phi_0^n\) and \(\Phi_t^n\) should read \(\Phi_0^N\) and \(\Phi_t^N\); i.e. we compute recursively \(\Phi_t^N(X_t^n)\). (Mathieu Gerber)

• 239: in the 9th line of Alg 13.3, \(H_{t-1}^{1:N}\) should be \(H_t^{1:N}\) (Chihiro Kuraya)

• 258: \(\theta^{n-1}\) should be \(\theta_{n-1}\) in function \(\theta\) -> \(Q(\theta, \theta^{n-1})\) (Feras Saad)

• 264, second displayed equation: last product should be from \(t=0\) to \(T\) (not \(t\))

• 267, (14.9): replace \(\theta\) by \(\theta_{n-1}\) in the gradient

• 282, first line: give(n) proposal (Adrien Corenflos)

• 283, last paragraph: \(O(d^2)\) should be \(O(d_{\theta}^2)\) (Feras Saad)

• 285, 4th line before Sec 15.5: full conditional should be \(q_k(\theta(k)|\theta(-k))\) rather than \(q(\theta(k)|\theta(-k))\) (Chihiro Kuraya)

• 287, caption of Fig. 15.2: (t)he first \(10^5\) simulations

• 290: simply adds an extra step that regenerate(s) the data. (Feras Saad)

• 295, Example 6.1: better behaved tha(n) (Feras Saad)

• 300, Prop 16.1: missing tilde on the \(M\) of the proposed kernel (Feras Saad)

• 301^1: \((z/c)\) instead of \((Z/c)\) in the expression of \(L(\theta, z)\) (Chihiro Kuraya)

• 308-309: capital letters \(X\) and \(A\) should be respectively \(x\) and \(a\) in last equation on page 308, and in the expression of \(W_t^n\) in the first line of p. 309. Moreover, dot should be a comma at the end of the equation at the bottom of p. 308. (Feras Saad)

• 332, second paragraph of 17.2.1: covariance matrix should be set to \(\lambda^2\hat{\Sigma}_{t-1}\); note the missing square (Mathieu Gerber) In fact, \(\lambda\) should be replaced by \(\delta\) on that page, since that quantity was denoted by \(\delta\) in Chapter 15, and since letter \(\lambda\) means something else in the rest of the chapter (the tempering coefficient).

• 337, Algorithm 17.3, second line of the else block: replace \(\Theta_t^n\) by \(\Theta_{t-1}^{A_t^n}\) (Adrien Corenflos)

• 342, (17.5): \(p_t^\theta\) should be \(p_T^\theta\)