Bayesian Workflow book: Errata

Errata (excluding minor typos) for the book.

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1st printing

• p. 28, in the caption of the rats figure, “\Pr(y_j=1)=\mathrm{logit}^{-1}(a + bx_j)” \rightarrow “\Pr(\mathrm{death} | x)=\mathrm{logit}^{-1}(a + bx)”

• p. 29, fit0 <- cmdstan_model("bioassay0.stan", pedantic=TRUE) should be mod0 <- cmdstan_model("bioassay0.stan", pedantic=TRUE) (thanks to J. J. Ramsey)

• p. 30, “\mathrm{E}(y|x=0)” \rightarrow “the probability of death at x=0”

• p. 31, “along with the curves, \operatorname{logit}^{-1}(a + bx_j)” \rightarrow “along with the curves, \operatorname{logit}^{-1}(a + bx)”

• p. 31, the compile of the model with priors is missing and should be added before it is used: mod1 <- cmdstan_model("bioassay1.stan", pedantic=TRUE). The sampling line should then read fit1 <- mod1$sample(data=bioassay_data, refresh=0) and print(fit_1) should be print(fit1) (thanks to J. J. Ramsey)

• p. 32, “Solving \mathrm{logit}^{-1}(\alpha+\beta x) = 0.5 yields x_{\mathrm{LD50}}=-\alpha/\beta” \rightarrow “\mathrm{logit}^{-1}(a+b x) = 0.5 and x_{\mathrm{LD50}}=-a/b”

• p. 38, in the caption, “y=\mathrm{logit}^{-1}(-1.40 + 0.33x)” \rightarrow “\Pr(y=1)=\mathrm{logit}^{-1}(-1.40 + 0.33x)”

• p. 38, “\int\!\!\int p(a,b|y)\,da\,db=\infty” \rightarrow “\int\!\!\int p(y|a,b)\,p(a,b)\,da\,db=\infty”

• p. 39, “x_i=0” \rightarrow “x_j=0”

• p. 44, “means and variances” \rightarrow “means and standard deviations”

• p. 44, in the paragraph beginning “Next we assign an exponential(1/5) distribution to \sigma_b”, each \sigma_b should be \sigma_a (thanks to Greg Fedewa)

• p. 47, “most of the slopes b_j” \rightarrow “most of the slopes b_k”

• p. 48, corr_matrix[2] Omega_ab = Omega_ab = multiply_lower_tri_self_transpose(L_ab); \rightarrow corr_matrix[2] Omega_ab = multiply_lower_tri_self_transpose(L_ab);

• p. 51, in the caption, “an ability parameter a for each student j and a difficulty parameter for each item k” \rightarrow “an ability parameter \alpha_j for each student j and a difficulty parameter \beta_k for each item k”

• p. 51, “the estimated abilities, \alpha_i” \rightarrow “the estimated abilities, \alpha_j”

• p. 52, “approximately a constant 0.25 + 0.75\cdot 0.5 = 0.65” \rightarrow “approximately a constant 0.25 + 0.75\cdot 0.5 = 0.625”

• p. 52, “In this parameterization of the gamma distribution” \rightarrow “In this parameterization of the exponential distribution”

• p. 53, in the caption, “a discrimination parameter d_k” \rightarrow “a discrimination parameter \gamma_k”

• p. 54, \beta_{2}3 should be \beta_{23} (thanks to Greg Fedewa)

• p. 77, “p|y\sim\mathrm{beta}(y+1, n-y+1)” \rightarrow “\theta|y\sim\mathrm{beta}(y+1, n-y+1)”

• p. 83, “\tau=\sigma\sqrt{\nu_{\nu-2}}” \rightarrow “\tau=\sigma\sqrt{\nu/(\nu-2)}”

• p. 87, “sample 10 balls without replacement” \rightarrow “sample 10 balls with replacement”

• p. 93, “a unit logistic prior distribution for \theta” \rightarrow “a unit logistic prior distribution for \phi”

• p. 94, in the caption, “\phi=\mathrm{logit}^{-1}(p)” \rightarrow “\phi=\mathrm{logit}(p)”

• p. 97, in the caption, “a \mathrm{Cauchy}(0,1) prior on the bias” \rightarrow a \mathrm{Cauchy}(0,0.1) prior on the bias”

• p. 103, “posterior simulations \theta^s … from p(y|x,\theta)” \rightarrow “posterior simulations \theta^s … from p(\theta|x,y)”

• p. 106, “then draw \tilde{y}_j \sim \mathrm{normal}(\theta_j,\tilde{\sigma}_j)” \rightarrow “then draw \tilde{y}_j \sim \mathrm{normal}(\tilde{\theta}_j,\tilde{\sigma}_j)”

• p. 107, estimates of \sigma are missing the squares on the residuals \rightarrow \sqrt{\sum_{i=1}^n(y_i - X_i\hat{\beta})^2/n} and \sqrt{\sum_{i=1}^n(y_i - X_i\hat{\beta})^2/(n-k)}

• p. 107, “\mathrm{mad\ sd}(z) = 1.483,\mathrm{median}(|z - \mathrm{median}(z)|)” \rightarrow ““\mathrm{mad\ sd}(z) = 1.483\,\mathrm{median}(|z - \mathrm{median}(z)|)”

• p. 111, “y^* = a + bx_i + e_i + \eta_i” \rightarrow “y^*_i = a + bx_i + e_i + \eta_i”

• p. 113, “the 200 students” \rightarrow “the 500 students”

• p. 120, “for contextually relevant values of x^{\rm hi}, x^{\rm lo}, and v” \rightarrow “for contextually relevant values of u^{\rm hi}, u^{\rm lo}”

• p. 120, “the coefficient of y” \rightarrow “the coefficient of u”

• p. 124, \hat{\theta} = \frac{N_j\sum_{j=1}^J\hat{\theta}j }{\sum_{j=1}^{J}N_j} \rightarrow \hat{\theta} = \frac{\sum_{j=1}^J N_j\hat{\theta}_j }{\sum_{j=1}^{J}N_j}

• p. 124, \hat{\theta}_A = \frac{\sum_{j\in A}N_j\hat{\theta}j }{\sum_{j \in A} N_j} \rightarrow \hat{\theta}_A = \frac{\sum_{j\in A}N_j\hat{\theta}j }{\sum_{j \in A} N_j}

• p. 125, “b^{th} sample” \rightarrow “posterior draw s”

• p. 130, “then \tilde{y}^s|\tilde{u},\tilde{x},\tilde{z}=1,\theta^s” \rightarrow “then \tilde{y}^s|\tilde{u}^s,\tilde{x},\tilde{z}=0,\theta^s”

• p. 130, “thus, (\tilde{y}^\prime)^s|\tilde{u}^s,\tilde{z}=1.” \rightarrow “thus, (\tilde{y}^\prime)^s|\tilde{u}^s,\tilde{x},\tilde{z}=1,\theta^s.”

• p. 130, “The model is \hat{\theta}\sim\mathrm{normal}(\hat{\theta},s)” \rightarrow “The model is \hat{\theta}\sim\mathrm{normal}(\theta,s)”

• p. 131, \theta|y\sim\operatorname{normal}\left(\frac{\frac{1}{s^2}\hat\theta}{\frac{1}{s^2} + \frac{1}{\sigma_0^2}}, \frac{1}{\frac{1}{s^2} + \frac{1}{\sigma_0^2}}\right) \rightarrow \theta|y\sim\operatorname{normal}\left(\frac{\frac{1}{s^2}\hat\theta}{\frac{1}{s^2} + \frac{1}{\sigma_0^2}}, \sqrt{\frac{1}{\frac{1}{s^2} + \frac{1}{\sigma_0^2}}}\right)

• p. 133, \theta/s = 0.42 \operatorname{normal}(0,1.5) + 0.58 \operatorname{normal}(0,3.5) \rightarrow \theta/s \sim 0.42 \operatorname{normal}(0,1.5) + 0.58 \operatorname{normal}(0,3.5)

• p. 135, “type S error (probability the estimate has the same sign as the true effect)” \rightarrow “type S error (probability the estimate has a different sign than the true effect)”

• p. 142, y_1^\mathrm{rep}, \ldots, y_n^\mathrm{rep} \sim p(y^\mathrm{rep} | y ) = \int \! \! p(y^\mathrm{rep} | \theta) d\theta. \rightarrow y_1^{\mathrm{rep}}, \ldots, y_n^{\mathrm{rep}} \sim p(y^{\mathrm{rep}} | y ) = \int \! \! p(y^{\mathrm{rep}} | \theta)\, p(\theta|y)\, d\theta.

• p. 149, “d\hat{\beta}/dy_i will be far from zero” \rightarrow “d\hat{b}/dy_i will be far from zero”

• p. 161, “\mathrm{SATE} = \sum_{i=1}^N (\mathrm{E}(y|X_i,z=1) - \mathrm{E}(y|X_i,z=0))” \rightarrow \mathrm{SATE} = \frac{1}{N}\sum_{i=1}^N (\mathrm{E}(y|X_i,z=1) - \mathrm{E}(y|X_i,z=0))

• p. 161, “\mathrm{PATE} = \sum_{i=1}^{N^{\rm new}} \mathrm{E}(y|X^{\rm new}_i,z=1) - \mathrm{E}(y|X^{\rm new}_i,z=0))” \rightarrow “\mathrm{PATE} = \frac{1}{N^{\rm new}}\sum_{i=1}^{N^{\rm new}} (\mathrm{E}(y|X^{\rm new}_i,z=1) - \mathrm{E}(y|X^{\rm new}_i,z=0))”

• pp. 164–166, in the elpd-difference column headers “$_1, _k” \rightarrow \mathrm{M}_k, \mathrm{M}_1

• p. 194, “In the 10-dimensional normal example” \rightarrow “In the 100-dimensional normal example”

• p. 201, “the numerator of this expression, \frac{1}{2}\mathrm{E}((\theta^s - \theta^{s-t})^2)” \rightarrow ““the numerator of this expression,”\mathrm{E}((\theta^s - \theta^{s-t})^2)”

• p. 201, “z_i= \frac{r_i - 3/8}{MN-1/4}” \rightarrow “z_i= \Phi^{-1}\left(\frac{r_i - 3/8}{MN+1/4}\right), where \Phi^{-1} is the inverse of the normal cumulative distribution function”

• p. 203, “with 99% probability” \rightarrow “with 99.7% probability”

• p. 217, “a 6% chance” \rightarrow “a 12.5% chance”

• p. 221, “j=1,\dots,J” \rightarrow “k=1,\dots,K”

• p. 230, “\beta is a slope parameter” \rightarrow “b is a slope parameter”, “\alpha parameters” \rightarrow “a parameters”, and “\eta = \alpha/\beta” \rightarrow “\eta = -a/b”

• p. 237, “the posterior standard deviation of \beta_1 has the computational estimate of 0.680” should be 1.447 (0.680 is the posterior standard deviation of \beta_2)

• p. 241, in the variational inference objective, q^* = \mathrm{argmin}_{q \in \mathcal{Q}}\, D(p||q) should be D(q||p), matching the reverse KL divergence discussed in the text (also in three later occurrences of D(p||q))

• p. 244, “p(\theta|y_1,y_2)=p(\phi)p(\xi_1|\phi)p(y_1|\phi,\xi_1)p(\xi_1|\phi)p(y_2|\phi,\xi_2)” \rightarrow “p(\theta|y_1,y_2)\propto p(\phi)p(\xi_1|\phi)p(y_1|\phi,\xi_1)p(\xi_2|\phi)p(y_2|\phi,\xi_2)”

• p. 263, “standard deviation \sigma” \rightarrow “standard deviation \sigma_y”

• p. 264, target += normal_lpdf(y | theta[movie[n]], sigma_y); \rightarrow target += normal_lpdf(y[n] | theta[movie[n]], sigma_y);

• p. 267, “our understanding of \theta_3” \rightarrow “our understanding of \theta_1”

• p. 267, “you could add 100 to each of the a_j’s and -100 to each of the b_k’s” \rightarrow “add 100 to each of the a_j’s and 100 to each of the b_k’s” (the model is y_i \sim \mathrm{normal}(a_{j[i]} - b_{k[i]}, \sigma_y), so predictions are unchanged only when the same constant is added to both)

• p. 267, “and not change any of the predictors at all” \rightarrow “and not change any of the predictions at all”

• p. 268, “a_j = \mu_a + \sigma_{\alpha}\alpha_j” \rightarrow “a_j = \mu_a + \sigma_a\alpha_j”

• p. 270, “open circles for the romantic comedies (those with j=1,\dots,10) and solid circles for the crime movies (j=11,\dots,20)” \rightarrow “open circles for the romantic comedies (those with j=1,\dots,20) and solid circles for the crime movies (j=21,\dots,40)”

• p. 276, “N = 180 observations” \rightarrow “N = 144 observations”

• p. 281, “the 95% interval is now (253, 283), as compared to the previous (246, 289)” \rightarrow “the 95% interval is now (246, 289), as compared to the previous (253, 283)”

• p. 281, “\mu_n = b_{0j_n} + b_{j1} x_n” \rightarrow “\mu_n = b_{0j_n} + b_{1j_n} x_n”

• p. 282, “(b_{0j}, b_{1j}) = (b_0, b_1) + \mathrm{Diag}(\tau_0, \tau_1) \, L \, (z_{0j}, z_{1j})” \rightarrow “(b_{0j}, b_{1j})^t = (b_0, b_1)^t + \mathrm{Diag}(\tau_0, \tau_1) \, L \, (z_{0j}, z_{1j})^t”

• p. 282, “\tau_0 \sim \mathrm{exponential}(1/10) prior” \rightarrow “\tau_1 \sim \mathrm{exponential}(1/10)”

• p. 288, “at least for 0 to 9 days of sleep deprivation” \rightarrow “at least for 0 to 7 days of sleep deprivation”

• p. 293, in the fit_normal code, prior(cauchy(1 2), class=sd) \rightarrow prior(cauchy(1, 2), class=sd)

• p. 297, the fit_betabinomial2 code is missing the closing parenthesis of the brm( call: prior(normal(0,3), class=sd)) \rightarrow prior(normal(0,3), class=sd)))

• p. 298, the fit_betabinomial2b code is missing the closing parenthesis of the brm( call: prior(normal(0,3), class=b)) \rightarrow prior(normal(0,3), class=b)))

• p. 301, in the fit_normal2c code, data=cu_df_c, normal(), \rightarrow data=cu_df_c, gaussian(),, and prior(normal(0 3), class=Intercept) \rightarrow prior(normal(0, 3), class=Intercept)

• p. 303, “than the negative binomial model” \rightarrow “than the beta-binomial model”

• p. 307, in the caption, “histogram of posterior simulations of \gamma” \rightarrow “histogram of posterior simulations of \pi”

• p. 308, in the caption, “power scaled with \alpha=0.5 (blue curve), 1 (black), or 2 (blue)” \rightarrow “\ldots or 2 (red)”

• p. 311, in the caption, “power scaled with \alpha=0.5 (blue curve), 1 (black), or 2 (blue)” \rightarrow “\ldots or 2 (red)”

• p. 316, in exercise 19.1, “3300 observations” \rightarrow “3330 observations”

• p. 318, in the slope-estimation code, the correct code is

  coefs <- summary(fit)$coefficients
  slope[n] <- 100 * coefs[2, "Estimate"]
  se[n] <- 100 * coefs[2, "Std. Error"]

• p. 321, “over 5 standard errors away” \rightarrow “about 5 standard errors away”
• p. 326, posterior_predict(fit, ndraws=1) \rightarrow posterior_predict(bfit_0, ndraws=1)
• p. 327, “weakly informative normal prior for \beta_0” \rightarrow “weakly informative normal prior for \mu_{\beta}”
• p. 329, “Posterior predictive simulations from the two-parameter model are shown in the third row of Figure 21.4” \rightarrow “in the fourth row”
• p. 329, in the loo_compare(bfit_0h, bfit_2) output, the second row label bfit_0 \rightarrow bfit_2
• pp. 334–337, four references to “model 5” \rightarrow “model 4”
• p. 337, “We take the (\alpha_j,\beta_j) simulated parameters” \rightarrow “We take the (a_j,b_j) simulated parameters”
• p. 343, the first Stan program’s model block is missing its closing brace
• pp. 343, 346, 350, in three Stan listings, array<lower=0, upper=1>[N] int adopted; \rightarrow array[N] int<lower=0, upper=1> adopted;
• p. 346, the inline code “target += log((1-P)\textasciicircum days[i])” \rightarrow target += log((1-P)^days[i])
• p. 347, in the caption, “prior simulation” \rightarrow “posterior simulation”
• p. 353, “The 23 teams” \rightarrow “The 32 teams”
• p. 361, in the discrete-model display, “a_j = b \cdot \mathrm{prior\_score} + \sigma_a \alpha” \rightarrow “a_j = b \cdot \mathrm{prior\_score}_j + \sigma_a \alpha_j”, and “y_i = \mathrm{round}(z_i, 1)” \rightarrow “y_i = \mathrm{round}(z_i)”
• p. 365, the integral for the square-root model, “\int_{y_i-0.5}^{y_i+0.5} \frac{1}{2\sqrt{y_i}}\,\mathrm{normal}(y_i \,|\, a_{j_1[i]}-a_{j_2[i]}, \sigma_y)\, dy_i” \rightarrow “\int_{y_i-0.5}^{y_i+0.5} \frac{1}{2\sqrt{|\tilde{y}|}}\,\mathrm{normal}\big(\mathrm{sign}(\tilde{y})\sqrt{|\tilde{y}|} \,\big|\, a_{j_1[i]}-a_{j_2[i]}, \sigma_y\big)\, d\tilde{y}”
• p. 372, “Based on this, the roaches predictor seems to be relevant. Dropping treatment or senior would lead to a large drop in elpd, but the uncertainty is large, and cross validation indicates that these predictors are not necessarily relevant” \rightarrow “the sqrt_roach1 predictor seems to be relevant. Dropping treatment would lead to a large drop in elpd, but the uncertainty is large, and cross validation indicates that neither treatment nor senior is necessarily relevant”
• p. 376, “64% of the observations are 0” \rightarrow “36% of the observations are 0”
• p. 378, loo_compare(list("Negative binomial"=loo(kcvnb), ...)) \rightarrow loo_compare(list("Negative binomial"=kcvnb, ...))
• p. 382, in the data-preparation code, offset=log(roaches$exposure2) \rightarrow offsett=log(roaches$exposure2)
• p. 383, "Negative binomial}=loo_nb)) \rightarrow "Negative binomial"=loo_nb))
• p. 383, ppc_loo_pit_ecdf(y=roaches$y, vyrep=...) \rightarrow yrep=...
• p. 386, in the caption, “coefficients of the inflated negative binomial regression” \rightarrow “of the zero-inflated negative binomial regression”
• p. 387, in the final loo_compare call and its output, “Negative binomial full model” / “Negative binomial without treatment” \rightarrow “ZINB full model” / “ZINB without treatment”
• p. 390, in the caption, “y = \mathrm{logit}^{-1}(a + bx_j)” \rightarrow “y = \mathrm{logit}^{-1}(a + bx)”
• p. 396, vector[J] p = p_angle .* p_distance * (1 - epsilon); \rightarrow vector[J] p = p_angle .* p_distance .* (1 - epsilon);
• p. 412, in the forecast-pseudo-residual definition, “z_t = \Phi^{-1}(\Pr(X_t \leq x_t | X_{1:(t-1)=X_{1:(t-1)}}))” \rightarrow “z_t = \Phi^{-1}(\Pr(X_t \leq x_t | X_{1:(t-1)} = x_{1:(t-1)}))”.
• p. 419, “the estimated year, day-of-year, and day-of-year effects” \rightarrow “the estimated year, day-of-week, and day-of-year effects”
• p. 420, “The factors (1 + \alpha_t), (1 + \beta_t), and (1 + \gamma_t)” \rightarrow “The factors (1 + \alpha t), (1 + \beta t), and (1 + \gamma t)”
• p. 430, “that is, g_3(0)\beta—along with g_3(t)/g_3(0)” \rightarrow “that is, \exp(g_3(0))\beta—along with \exp(g_3(t)-g_3(0))”
• p. 439, “default \operatorname{normal}^+(0, 3) prior (where \operatorname{normal}^+ indicates a normal distribution constrained to be positive)” \rightarrow “default t_3^+(0, 3) prior (where t_\nu^+ indicates a t_\nu-distribution constrained to be positive)”
• p. 442, “When predictors are nearly collinear, it is infer their predictor relevances from the univariate marginal posteriors” \rightarrow “When predictors are nearly collinear, it is difficult to infer their predictor relevances from the univariate marginal posteriors”
• p. 452, “(or 0.41 when fit to the simulated data)” \rightarrow “(or 0.53 when fit to the simulated data)”
• p. 454, “about two-thirds of item intercepts are in the range (-2.33, 2.33)” \rightarrow “(-2.31, 2.31)”
• p. 454, “\mathrm{logit}^{-1}(\mathrm{logit}(0.24) - 2) = 0.05” \rightarrow “\mathrm{logit}^{-1}(\mathrm{logit}(0.24) - 2) = 0.04”
• p. 464, “\log p(q_\mathrm{obs} | k) = C - \frac{1}{2 \sigma} \|q_\mathrm{obs} - q(k)\|^2_2” \rightarrow “\log p(q_\mathrm{obs} | k) = C - \frac{1}{2 \sigma^2} \|q_\mathrm{obs} - q(k)\|^2_2”
• p. 471, “The mixture submodel where the mixing ratio is the same for all variables” \rightarrow “The mixture submodel where the mixing ratio is the same for all observations”
• p. 479, in the combined-model Stan listing, int y[N_obs]; \rightarrow array[N_obs] int y;
• p. 480, “we can safely reject datasets where the variance is more than 1.8 times the mean” \rightarrow “less than 1.8 times the mean”
• p. 482, in exercise 31.1, “for students in the control group who scored 100 on the midterm” \rightarrow “for students in the control group who scored 100 on the pre-test”