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This is the third part (Chapter 4) of our course associated with Bayesian Core.

This is the third part (Chapter 4) of our course associated with Bayesian Core.

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- 1. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalized linear models Generalized linear models 3 Generalisation of linear models Metropolis–Hastings algorithms The Probit Model The logit model Loglinear models 313 / 785
- 2. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Generalisation of Linear Models Linear models model connection between a response variable y and a set x of explanatory variables by a linear dependence relation with [approximately] normal perturbations. 314 / 785
- 3. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Generalisation of Linear Models Linear models model connection between a response variable y and a set x of explanatory variables by a linear dependence relation with [approximately] normal perturbations. Many instances where either of these assumptions not appropriate, e.g. when the support of y restricted to R+ or to N. 315 / 785
- 4. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models bank Four measurements on 100 genuine Swiss banknotes and 100 counterfeit ones: x1 length of the bill (in mm), x2 width of the left edge (in mm), x3 width of the right edge (in mm), x4 bottom margin width (in mm). Response variable y: status of the banknote [0 for genuine and 1 for counterfeit] 316 / 785
- 5. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models bank Four measurements on 100 genuine Swiss banknotes and 100 counterfeit ones: x1 length of the bill (in mm), x2 width of the left edge (in mm), x3 width of the right edge (in mm), x4 bottom margin width (in mm). Response variable y: status of the banknote [0 for genuine and 1 for counterfeit] Probabilistic model that predicts counterfeiting based on the four measurements 317 / 785
- 6. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models The impossible linear model Example of the inﬂuence of x4 on y Since y is binary, y|x4 ∼ B(p(x4 )) , c Normal model is impossible 318 / 785
- 7. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models The impossible linear model Example of the inﬂuence of x4 on y Since y is binary, y|x4 ∼ B(p(x4 )) , c Normal model is impossible Linear dependence in p(x) = E[y|x]’s p(x4i ) = β0 + β1 x4i , 319 / 785
- 8. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models The impossible linear model Example of the inﬂuence of x4 on y Since y is binary, y|x4 ∼ B(p(x4 )) , c Normal model is impossible Linear dependence in p(x) = E[y|x]’s p(x4i ) = β0 + β1 x4i , estimated [by MLE] as pi = −2.02 + 0.268 xi4 ˆ which gives pi = .12 for xi4 = 8 and ... ˆ 320 / 785
- 9. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models The impossible linear model Example of the inﬂuence of x4 on y Since y is binary, y|x4 ∼ B(p(x4 )) , c Normal model is impossible Linear dependence in p(x) = E[y|x]’s p(x4i ) = β0 + β1 x4i , estimated [by MLE] as pi = −2.02 + 0.268 xi4 ˆ which gives pi = .12 for xi4 = 8 and ... pi = 1.19 for xi4 = 12!!! ˆ ˆ c Linear dependence is impossible 321 / 785
- 10. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Generalisation of the linear dependence Broader class of models to cover various dependence structures. 322 / 785
- 11. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Generalisation of the linear dependence Broader class of models to cover various dependence structures. Class of generalised linear models (GLM) where y|x, β ∼ f (y|xT β) . i.e., dependence of y on x partly linear 323 / 785
- 12. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Notations Same as in linear regression chapter, with n–sample y = (y1 , . . . , yn ) and corresponding explanatory variables/covariates x11 x12 . . . x1k x21 x22 . . . x2k x32 . . . x3k X = x31 . . . . . . . . . . . . xn1 xn2 . . . xnk 324 / 785
- 13. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Speciﬁcations of GLM’s Deﬁnition (GLM) A GLM is a conditional model speciﬁed by two functions: the density f of y given x parameterised by its expectation 1 parameter µ = µ(x) [and possibly its dispersion parameter ϕ = ϕ(x)] 325 / 785
- 14. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Speciﬁcations of GLM’s Deﬁnition (GLM) A GLM is a conditional model speciﬁed by two functions: the density f of y given x parameterised by its expectation 1 parameter µ = µ(x) [and possibly its dispersion parameter ϕ = ϕ(x)] the link g between the mean µ and the explanatory variables, 2 written customarily as g(µ) = xT β or, equivalently, E[y|x, β] = g −1 (xT β). 326 / 785
- 15. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Speciﬁcations of GLM’s Deﬁnition (GLM) A GLM is a conditional model speciﬁed by two functions: the density f of y given x parameterised by its expectation 1 parameter µ = µ(x) [and possibly its dispersion parameter ϕ = ϕ(x)] the link g between the mean µ and the explanatory variables, 2 written customarily as g(µ) = xT β or, equivalently, E[y|x, β] = g −1 (xT β). For identiﬁability reasons, g needs to be bijective. 327 / 785
- 16. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Likelihood Obvious representation of the likelihood n f yi |xiT β, ϕ ℓ(β, ϕ|y, X) = i=1 with parameters β ∈ Rk and ϕ > 0. 328 / 785
- 17. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Examples Ordinary linear regression Case of GLM where g(x) = x, ϕ = σ 2 , and y|X, β, σ 2 ∼ Nn (Xβ, σ 2 ). 329 / 785
- 18. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Examples (2) Case of binary and binomial data, when yi |xi ∼ B(ni , p(xi )) with known ni Logit [or logistic regression] model Link is logit transform on probability of success g(pi ) = log(pi /(1 − pi )) , with likelihood Y „ ni « „ exp(xiT β) «yi „ «ni −yi n 1 yi 1 + exp(xiT β) 1 + exp(xiT β) i=1 (n )ﬃ n Y“ ”ni −yi X yi xiT β 1 + exp(xiT β) ∝ exp i=1 i=1 330 / 785
- 19. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Canonical link Special link function g that appears in the natural exponential family representation of the density g ⋆ (µ) = θ if f (y|µ) ∝ exp{T (y) · θ − Ψ(θ)} 331 / 785
- 20. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Canonical link Special link function g that appears in the natural exponential family representation of the density g ⋆ (µ) = θ if f (y|µ) ∝ exp{T (y) · θ − Ψ(θ)} Example Logit link is canonical for the binomial model, since ni pi f (yi |pi ) = exp yi log + ni log(1 − pi ) , yi 1 − pi and thus θi = log pi /(1 − pi ) 332 / 785
- 21. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Examples (3) Customary to use the canonical link, but only customary ... Probit model Probit link function given by g(µi ) = Φ−1 (µi ) where Φ standard normal cdf Likelihood n Φ(xiT β)yi (1 − Φ(xiT β))ni −yi . ℓ(β|y, X) ∝ i=1 Full processing 333 / 785
- 22. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Log-linear models Standard approach to describe associations between several categorical variables, i.e, variables with ﬁnite support Suﬃcient statistic: contingency table, made of the cross-classiﬁed counts for the diﬀerent categorical variables. Full entry to loglinear models 334 / 785
- 23. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Log-linear models Standard approach to describe associations between several categorical variables, i.e, variables with ﬁnite support Suﬃcient statistic: contingency table, made of the cross-classiﬁed counts for the diﬀerent categorical variables. Full entry to loglinear models Example (Titanic survivors) Child Adult Survivor Class Male Female Male Female 1st 0 0 118 4 2nd 0 0 154 13 No 3rd 35 17 387 89 Crew 0 0 670 3 1st 5 1 57 140 2nd 11 13 14 80 Yes 3rd 13 14 75 76 Crew 0 0 192 20 335 / 785
- 24. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Poisson regression model Each count yi is Poisson with mean µi = µ(xi ) 1 Link function connecting R+ with R, e.g. logarithm 2 g(µi ) = log(µi ). 336 / 785
- 25. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Generalisation of linear models Poisson regression model Each count yi is Poisson with mean µi = µ(xi ) 1 Link function connecting R+ with R, e.g. logarithm 2 g(µi ) = log(µi ). Corresponding likelihood n 1 exp yi xiT β − exp(xiT β) . ℓ(β|y, X) = yi ! i=1 337 / 785
- 26. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Metropolis–Hastings algorithms Convergence assessment Posterior inference in GLMs harder than for linear models 338 / 785
- 27. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Metropolis–Hastings algorithms Convergence assessment Posterior inference in GLMs harder than for linear models c Working with a GLM requires speciﬁc numerical or simulation tools [E.g., GLIM in classical analyses] 339 / 785
- 28. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Metropolis–Hastings algorithms Convergence assessment Posterior inference in GLMs harder than for linear models c Working with a GLM requires speciﬁc numerical or simulation tools [E.g., GLIM in classical analyses] Opportunity to introduce universal MCMC method: Metropolis–Hastings algorithm 340 / 785
- 29. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Generic MCMC sampler Metropolis–Hastings algorithms are generic/down-the-shelf MCMC algorithms Only require likelihood up to a constant [diﬀerence with Gibbs sampler] can be tuned with a wide range of possibilities [diﬀerence with Gibbs sampler & blocking] natural extensions of standard simulation algorithms: based on the choice of a proposal distribution [diﬀerence in Markov proposal q(x, y) and acceptance] 341 / 785
- 30. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Why Metropolis? Originally introduced by Metropolis, Rosenbluth, Rosenbluth, Teller and Teller in a setup of optimization on a discrete state-space. All authors involved in Los Alamos during and after WWII: 342 / 785
- 31. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Why Metropolis? Originally introduced by Metropolis, Rosenbluth, Rosenbluth, Teller and Teller in a setup of optimization on a discrete state-space. All authors involved in Los Alamos during and after WWII: Physicist and mathematician, Nicholas Metropolis is considered (with Stanislaw Ulam) to be the father of Monte Carlo methods. Also a physicist, Marshall Rosenbluth worked on the development of the hydrogen (H) bomb Edward Teller was one of the ﬁrst scientists to work on the Manhattan Project that led to the production of the A bomb. Also managed to design with Ulam the H bomb. 343 / 785
- 32. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Generic Metropolis–Hastings sampler For target π and proposal kernel q(x, y) Initialization: Choose an arbitrary x(0) 344 / 785
- 33. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Generic Metropolis–Hastings sampler For target π and proposal kernel q(x, y) Initialization: Choose an arbitrary x(0) Iteration t: 1 Given x(t−1) , generate x ∼ q(x(t−1) , x) ˜ 2 Calculate π(˜)/q(x(t−1) , x) x ˜ ρ(x(t−1) , x) = min ˜ ,1 (t−1) )/q(˜, x(t−1) ) π(x x With probability ρ(x(t−1) , x) accept x and set x(t) = x; ˜ ˜ ˜ 3 otherwise reject x and set x(t) = x(t−1) . ˜ 345 / 785
- 34. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Universality Algorithm only needs to simulate from q which can be chosen [almost!] arbitrarily, i.e. unrelated with π [q also called instrumental distribution] Note: π and q known up to proportionality terms ok since proportionality constants cancel in ρ. 346 / 785
- 35. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Validation Markov chain theory Target π is stationary distribution of Markov chain (x(t) )t because probability ρ(x, y) satisﬁes detailed balance equation π(x)q(x, y)ρ(x, y) = π(y)q(y, x)ρ(y, x) [Integrate out x to see that π is stationary] 347 / 785
- 36. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Validation Markov chain theory Target π is stationary distribution of Markov chain (x(t) )t because probability ρ(x, y) satisﬁes detailed balance equation π(x)q(x, y)ρ(x, y) = π(y)q(y, x)ρ(y, x) [Integrate out x to see that π is stationary] For convergence/ergodicity, Markov chain must be irreducible: q has positive probability of reaching all areas with positive π probability in a ﬁnite number of steps. 348 / 785
- 37. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of proposal Theoretical guarantees of convergence very high, but choice of q is crucial in practice. 349 / 785
- 38. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of proposal Theoretical guarantees of convergence very high, but choice of q is crucial in practice. Poor choice of q may result in very high rejection rates, with very few moves of the Markov chain (x(t) )t hardly moves, or in 350 / 785
- 39. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of proposal Theoretical guarantees of convergence very high, but choice of q is crucial in practice. Poor choice of q may result in very high rejection rates, with very few moves of the Markov chain (x(t) )t hardly moves, or in a myopic exploration of the support of π, that is, in a dependence on the starting value x(0) , with the chain stuck in a neighbourhood mode to x(0) . 351 / 785
- 40. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of proposal Theoretical guarantees of convergence very high, but choice of q is crucial in practice. Poor choice of q may result in very high rejection rates, with very few moves of the Markov chain (x(t) )t hardly moves, or in a myopic exploration of the support of π, that is, in a dependence on the starting value x(0) , with the chain stuck in a neighbourhood mode to x(0) . Note: hybrid MCMC Simultaneous use of diﬀerent kernels valid and recommended 352 / 785
- 41. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms The independence sampler Pick proposal q that is independent of its ﬁrst argument, q(x, y) = q(y) ρ simpliﬁes into π(y)/q(y) ρ(x, y) = min 1, . π(x)/q(x) Special case: q ∝ π Reduces to ρ(x, y) = 1 and iid sampling 353 / 785
- 42. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms The independence sampler Pick proposal q that is independent of its ﬁrst argument, q(x, y) = q(y) ρ simpliﬁes into π(y)/q(y) ρ(x, y) = min 1, . π(x)/q(x) Special case: q ∝ π Reduces to ρ(x, y) = 1 and iid sampling Analogy with Accept-Reject algorithm where max π/q replaced with the current value π(x(t−1) )/q(x(t−1) ) but sequence of accepted x(t) ’s not i.i.d. 354 / 785
- 43. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of q Convergence properties highly dependent on q. q needs to be positive everywhere on the support of π for a good exploration of this support, π/q needs to be bounded. 355 / 785
- 44. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of q Convergence properties highly dependent on q. q needs to be positive everywhere on the support of π for a good exploration of this support, π/q needs to be bounded. Otherwise, the chain takes too long to reach regions with low q/π values. 356 / 785
- 45. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms The random walk sampler Independence sampler requires too much global information about π: opt for a local gathering of information 357 / 785
- 46. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms The random walk sampler Independence sampler requires too much global information about π: opt for a local gathering of information Means exploration of the neighbourhood of the current value x(t) in search of other points of interest. 358 / 785
- 47. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms The random walk sampler Independence sampler requires too much global information about π: opt for a local gathering of information Means exploration of the neighbourhood of the current value x(t) in search of other points of interest. Simplest exploration device is based on random walk dynamics. 359 / 785
- 48. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Random walks Proposal is a symmetric transition density q(x, y) = qRW (y − x) = qRW (x − y) 360 / 785
- 49. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Random walks Proposal is a symmetric transition density q(x, y) = qRW (y − x) = qRW (x − y) Acceptance probability ρ(x, y) reduces to the simpler form π(y) ρ(x, y) = min 1, . π(x) Only depends on the target π [accepts all proposed values that increase π] 361 / 785
- 50. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of qRW Considerable ﬂexibility in the choice of qRW , tails: Normal versus Student’s t scale: size of the neighbourhood 362 / 785
- 51. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of qRW Considerable ﬂexibility in the choice of qRW , tails: Normal versus Student’s t scale: size of the neighbourhood Can also be used for restricted support targets [with a waste of simulations near the boundary] 363 / 785
- 52. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Choice of qRW Considerable ﬂexibility in the choice of qRW , tails: Normal versus Student’s t scale: size of the neighbourhood Can also be used for restricted support targets [with a waste of simulations near the boundary] Can be tuned towards an acceptance probability of 0.234 at the burnin stage [Magic number!] 364 / 785
- 53. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Convergence assessment Capital question: How many iterations do we need to run??? MCMC debuts 365 / 785
- 54. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Convergence assessment Capital question: How many iterations do we need to run??? MCMC debuts Rule # 1 There is no absolute number of simulations, i.e. 1, 000 is neither large, nor small. Rule # 2 It takes [much] longer to check for convergence than for the chain itself to converge. Rule # 3 MCMC is a “what-you-get-is-what-you-see” algorithm: it fails to tell about unexplored parts of the space. Rule # 4 When in doubt, run MCMC chains in parallel and check for consistency. 366 / 785
- 55. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Convergence assessment Capital question: How many iterations do we need to run??? MCMC debuts Rule # 1 There is no absolute number of simulations, i.e. 1, 000 is neither large, nor small. Rule # 2 It takes [much] longer to check for convergence than for the chain itself to converge. Rule # 3 MCMC is a “what-you-get-is-what-you-see” algorithm: it fails to tell about unexplored parts of the space. Rule # 4 When in doubt, run MCMC chains in parallel and check for consistency. Many “quick-&-dirty” solutions in the literature, but not necessarily trustworthy. 367 / 785
- 56. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Prohibited dynamic updating Tuning the proposal in terms of its past performances can only be implemented at burnin, because otherwise this cancels Markovian convergence properties. 368 / 785
- 57. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Prohibited dynamic updating Tuning the proposal in terms of its past performances can only be implemented at burnin, because otherwise this cancels Markovian convergence properties. Use of several MCMC proposals together within a single algorithm using circular or random design is ok. It almost always brings an improvement compared with its individual components (at the cost of increased simulation time) 369 / 785
- 58. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Eﬀective sample size How many iid simulations from π are equivalent to N simulations from the MCMC algorithm? 370 / 785
- 59. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Metropolis–Hastings algorithms Eﬀective sample size How many iid simulations from π are equivalent to N simulations from the MCMC algorithm? Based on estimated k-th order auto-correlation, ρk = corr x(t) , x(t+k) , eﬀective sample size −1/2 T0 ess ρ2 N =n 1+2 ˆk , k=1 Only partial indicator that fails to signal chains stuck in one mode of the target 371 / 785
- 60. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model The Probit Model Likelihood Recall Probit n Φ(xiT β)yi (1 − Φ(xiT β))ni −yi . ℓ(β|y, X) ∝ i=1 372 / 785
- 61. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model The Probit Model Likelihood Recall Probit n Φ(xiT β)yi (1 − Φ(xiT β))ni −yi . ℓ(β|y, X) ∝ i=1 If no prior information available, resort to the ﬂat prior π(β) ∝ 1 and then obtain the posterior distribution n yi ni −yi Φ xiT β 1 − Φ(xiT β) π(β|y, X) ∝ , i=1 nonstandard and simulated using MCMC techniques. 373 / 785
- 62. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model MCMC resolution Metropolis–Hastings random walk sampler works well for binary regression problems with small number of predictors ˆ Uses the maximum likelihood estimate β as starting value and ˆ asymptotic (Fisher) covariance matrix of the MLE, Σ, as scale 374 / 785
- 63. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model MLE proposal R function glm very useful to get the maximum likelihood estimate ˆ of β and its asymptotic covariance matrix Σ. Terminology used in R program mod=summary(glm(y~X-1,family=binomial(link=quot;probitquot;))) ˆ ˆ with mod$coeﬀ[,1] denoting β and mod$cov.unscaled Σ. 375 / 785
- 64. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model MCMC algorithm Probit random-walk Metropolis-Hastings ˆ ˆ Initialization: Set β (0) = β and compute Σ Iteration t: ˜ ˆ Generate β ∼ Nk+1 (β (t−1) , τ Σ) 1 Compute 2 ˜ π(β|y) ˜ ρ(β (t−1) , β) = min 1, π(β (t−1) |y) ˜ ˜ With probability ρ(β (t−1) , β) set β (t) = β; 3 (t) (t−1) otherwise set β = β . 376 / 785
- 65. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model bank Probit modelling with 0.8 −1.0 1.0 0.4 no intercept over the −2.0 0.0 0.0 0 4000 8000 −2.0 −1.5 −1.0 −0.5 0 200 600 1000 four measurements. Three diﬀerent scales 3 0.0 0.4 0.8 2 0.4 1 τ = 1, 0.1, 10: best −1 0.0 mixing behavior is 0 4000 8000 −1 0 1 2 3 0 200 600 1000 associated with τ = 1. 2.5 0.8 0.0 0.4 0.8 −0.5 1.0 0.4 Average of the 0.0 parameters over 9, 000 0 4000 8000 −0.5 0.5 1.5 2.5 0 200 600 1000 iterations gives plug-in 1.8 0.0 0.4 0.8 2.0 estimate 1.2 1.0 0.0 0.6 0 4000 8000 0.6 1.0 1.4 1.8 0 200 600 1000 pi = Φ (−1.2193xi1 + 0.9540xi2 + 0.9795xi3 + 1.1481xi4 ) . ˆ 377 / 785
- 66. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model G-priors for probit models Flat prior on β inappropriate for comparison purposes and Bayes factors. 378 / 785
- 67. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model G-priors for probit models Flat prior on β inappropriate for comparison purposes and Bayes factors. Replace the ﬂat prior with a hierarchical prior, β|σ 2 , X ∼ Nk 0k , σ 2 (X T X)−1 and π(σ 2 |X) ∝ σ −3/2 , (almost) as in normal linear regression 379 / 785
- 68. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model G-priors for probit models Flat prior on β inappropriate for comparison purposes and Bayes factors. Replace the ﬂat prior with a hierarchical prior, β|σ 2 , X ∼ Nk 0k , σ 2 (X T X)−1 and π(σ 2 |X) ∝ σ −3/2 , (almost) as in normal linear regression Warning The matrix X T X is not the Fisher information matrix 380 / 785
- 69. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model G-priors for testing Same argument as before: while π is improper, use of the same variance factor σ 2 in both models means the normalising constant cancels in the Bayes factor. 381 / 785
- 70. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model G-priors for testing Same argument as before: while π is improper, use of the same variance factor σ 2 in both models means the normalising constant cancels in the Bayes factor. Posterior distribution of β “ ”−(2k−1)/4 |X T X|1/2 Γ((2k − 1)/4) β T (X T X)β π −k/2 π(β|y, X) ∝ h i1−yi Y n Φ(xiT β)yi 1 − Φ(xiT β) × i=1 [where k matters!] 382 / 785
- 71. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Marginal approximation Marginal Z“ ”−(2k−1)/4 |X T X|1/2 π −k/2 Γ{(2k − 1)/4} β T (X T X)β f (y|X) ∝ h i1−yi Y n Φ(xiT β)yi 1 − (Φ(xiT β) × dβ, i=1 approximated by M ˛˛ ˛˛−(2k−1)/2 Y h i1−yi |X T X|1/2 X ˛˛ n (m) ˛˛ Φ(xiT β (m) )yi 1 − Φ(xiT β (m) ) ˛˛Xβ ˛˛ π k/2 M m=1 i=1 bb b (m) −β)T V −1 (β (m) −β)/4 b × Γ{(2k − 1)/4} |V |1/2 (4π)k/2 e+(β , where β (m) ∼ Nk (β, 2 V ) with β MCMC approximation of Eπ [β|y, X] and V MCMC approximation of V(β|y, X). 383 / 785
- 72. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Linear hypothesis Linear restriction on β H0 : Rβ = r (r ∈ Rq , R q × k matrix) where β 0 is (k − q) dimensional and X0 and x0 are linear transforms of X and of x of dimensions (n, k − q) and (k − q). Likelihood n 1−yi 0 Φ(xiT β 0 )yi 1 − Φ(xiT β 0 ) ℓ(β |y, X0 ) ∝ , 0 0 i=1 384 / 785
- 73. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Linear test Associated [projected] G-prior β 0 |σ 2 , X0 ∼ Nk−q 0k−q , σ 2 (X0 X0 )−1 T and π(σ 2 |X0 ) ∝ σ −3/2 , 385 / 785
- 74. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Linear test Associated [projected] G-prior β 0 |σ 2 , X0 ∼ Nk−q 0k−q , σ 2 (X0 X0 )−1 T and π(σ 2 |X0 ) ∝ σ −3/2 , Marginal distribution of y of the same type ﬀZ ˛˛ ˛˛ (2(k − q) − 1) ˛˛Xβ 0 ˛˛−(2(k−q)−1)/2 |X0 X0 |1/2 π −(k−q)/2 Γ T f (y|X0 ) ∝ 4 h i1−yi Y n Φ(xiT β 0 )yi 1 − (Φ(xiT β 0 ) dβ 0 . 0 0 i=1 386 / 785
- 75. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model banknote π For H0 : β1 = β2 = 0, B10 = 157.73 [against H0 ] Generic regression-like output: Estimate Post. var. log10(BF) X1 -1.1552 0.0631 4.5844 (****) X2 0.9200 0.3299 -0.2875 X3 0.9121 0.2595 -0.0972 X4 1.0820 0.0287 15.6765 (****) evidence against H0: (****) decisive, (***) strong, (**) subtantial, (*) poor 387 / 785
- 76. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Informative settings If prior information available on p(x), transform into prior distribution on β by technique of imaginary observations: 388 / 785
- 77. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Informative settings If prior information available on p(x), transform into prior distribution on β by technique of imaginary observations: Start with k diﬀerent values of the covariate vector, x1 , . . . , xk ˜ ˜ For each of these values, the practitioner speciﬁes (i) a prior guess gi at the probability pi associated with xi ; (ii) an assessment of (un)certainty about that guess given by a number Ki of equivalent “prior observations”. 389 / 785
- 78. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Informative settings If prior information available on p(x), transform into prior distribution on β by technique of imaginary observations: Start with k diﬀerent values of the covariate vector, x1 , . . . , xk ˜ ˜ For each of these values, the practitioner speciﬁes (i) a prior guess gi at the probability pi associated with xi ; (ii) an assessment of (un)certainty about that guess given by a number Ki of equivalent “prior observations”. On how many imaginary observations did you build this guess? 390 / 785
- 79. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Informative prior k pKi gi −1 (1 − pi )Ki (1−gi )−1 π(p1 , . . . , pk ) ∝ i i=1 translates into [Jacobian rule] k Ki (1−gi )−1 Φ(˜ iT β)Ki gi −1 1 − Φ(˜ iT β) φ(˜ iT β) π(β) ∝ x x x i=1 391 / 785
- 80. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The Probit Model Informative prior k pKi gi −1 (1 − pi )Ki (1−gi )−1 π(p1 , . . . , pk ) ∝ i i=1 translates into [Jacobian rule] k Ki (1−gi )−1 Φ(˜ iT β)Ki gi −1 1 − Φ(˜ iT β) φ(˜ iT β) π(β) ∝ x x x i=1 [Almost] equivalent to using the G-prior 0 #−1 1 quot; X k β ∼ Nk @0k , A xj xjT ˜˜ j=1 392 / 785
- 81. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model The logit model Recall that [for ni = 1] exp(µi ) yi |µi ∼ B(1, µi ), ϕ = 1 and g(µi ) = . 1 + exp(µi ) Thus exp(xiT β) P(yi = 1|β) = 1 + exp(xiT β) with likelihood yi 1−yi n exp(xiT β) exp(xiT β) ℓ(β|y, X) = 1− 1 + exp(xiT β) 1 + exp(xiT β) i=1 393 / 785
- 82. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model Links with probit usual vague prior for β, π(β) ∝ 1 394 / 785
- 83. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model Links with probit usual vague prior for β, π(β) ∝ 1 Posterior given by n yi 1−yi exp(xiT β) exp(xiT β) π(β|y, X) ∝ 1− 1 + exp(xiT β) 1 + exp(xiT β) i=1 [intractable] 395 / 785
- 84. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model Links with probit usual vague prior for β, π(β) ∝ 1 Posterior given by n yi 1−yi exp(xiT β) exp(xiT β) π(β|y, X) ∝ 1− 1 + exp(xiT β) 1 + exp(xiT β) i=1 [intractable] Same Metropolis–Hastings sampler 396 / 785
- 85. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model bank −1 0.6 0.0 0.4 0.8 −3 Same scale factor equal to 0.3 −5 0.0 τ = 1: slight increase in 0 4000 8000 −5 −4 −3 −2 −1 0 200 600 1000 the skewness of the 246 0.0 0.4 0.8 0.2 histograms of the βi ’s. 0.0 −2 0 4000 8000 −2 0 2 4 6 0 200 600 1000 0.4 246 0.0 0.4 0.8 Plug-in estimate of 0.2 0.0 −2 predictive probability of a 0 4000 8000 −2 0 2 4 6 0 200 600 1000 counterfeit 3.5 0.0 0.4 0.8 2.5 0.6 1.5 0.0 0 4000 8000 1.5 2.5 3.5 0 200 600 1000 exp (−2.5888xi1 + 1.9967xi2 + 2.1260xi3 + 2.1879xi4 ) pi = ˆ . 1 + exp (−2.5888xi1 + 1.9967xi2 + 2.1260xi3 + 2.1879xi4 ) 397 / 785
- 86. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models The logit model G-priors for logit models Same story: Flat prior on β inappropriate for Bayes factors, to be replaced with hierarchical prior, β|σ 2 , X ∼ Nk 0k , σ 2 (X T X)−1 and π(σ 2 |X) ∝ σ −3/2 Example (bank) Estimate Post. var. log10(BF) X1 -2.3970 0.3286 4.8084 (****) X2 1.6978 1.2220 -0.2453 X3 2.1197 1.0094 -0.1529 X4 2.0230 0.1132 15.9530 (****) evidence against H0: (****) decisive, (***) strong, (**) subtantial, (*) poor 398 / 785
- 87. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Loglinear models Introduction to loglinear models Example (airquality) Benchmark in R > air=data(airquality) Repeated measurements over 111 consecutive days of ozone u (in parts per billion) and maximum daily temperature v discretized into dichotomous variables month 5 6 7 8 9 ozone temp [1,31] [57,79] 17 4 2 5 18 (79,97] 0 2332 (31,168] [57,79] 6 1031 (79,97] 1 2 21 12 8 Contingency table with 5 × 2 × 2 = 20 entries 399 / 785
- 88. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Poisson regression Observations/counts y = (y1 , . . . , yn ) are integers, so we can choose yi ∼ P(µi ) Saturated likelihood n 1 yi ℓ(µ|y) = µ exp(−µi ) µi ! i i=1 400 / 785
- 89. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Poisson regression Observations/counts y = (y1 , . . . , yn ) are integers, so we can choose yi ∼ P(µi ) Saturated likelihood n 1 yi ℓ(µ|y) = µ exp(−µi ) µi ! i i=1 GLM constraint via log-linear link iT β log(µi ) = xiT β , yi |xi ∼ P ex 401 / 785
- 90. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Categorical variables Special feature Incidence matrix X = (xi ) such that its elements are all zeros or ones, i.e. covariates are all indicators/dummy variables! 402 / 785
- 91. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Categorical variables Special feature Incidence matrix X = (xi ) such that its elements are all zeros or ones, i.e. covariates are all indicators/dummy variables! Several types of (sub)models are possible depending on relations between categorical variables. 403 / 785
- 92. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Categorical variables Special feature Incidence matrix X = (xi ) such that its elements are all zeros or ones, i.e. covariates are all indicators/dummy variables! Several types of (sub)models are possible depending on relations between categorical variables. Re-special feature Variable selection problem of a speciﬁc kind, in the sense that all indicators related with the same association must either remain or vanish at once. Thus much fewer submodels than in a regular variable selection problem. 404 / 785
- 93. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Parameterisations Example of three variables 1 ≤ u ≤ I, 1 ≤ v ≤ j and 1 ≤ w ≤ K. 405 / 785
- 94. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Parameterisations Example of three variables 1 ≤ u ≤ I, 1 ≤ v ≤ j and 1 ≤ w ≤ K. Simplest non-constant model is I J K u v w log(µτ ) = βb Ib (uτ ) + βb Ib (vτ ) + βb Ib (wτ ) , b=1 b=1 b=1 that is, u v w log(µl(i,j,k) ) = βi + βj + βk , where index l(i, j, k) corresponds to u = i, v = j and w = k. 406 / 785
- 95. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Parameterisations Example of three variables 1 ≤ u ≤ I, 1 ≤ v ≤ j and 1 ≤ w ≤ K. Simplest non-constant model is I J K u v w log(µτ ) = βb Ib (uτ ) + βb Ib (vτ ) + βb Ib (wτ ) , b=1 b=1 b=1 that is, u v w log(µl(i,j,k) ) = βi + βj + βk , where index l(i, j, k) corresponds to u = i, v = j and w = k. Saturated model is uvw log(µl(i,j,k) ) = βijk 407 / 785
- 96. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Log-linear model (over-)parameterisation Representation log(µl(i,j,k) ) = λ + λu + λv + λw + λuv + λuw + λvw + λuvw , i j ij k ik jk ijk as in Anova models. 408 / 785
- 97. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Log-linear model (over-)parameterisation Representation log(µl(i,j,k) ) = λ + λu + λv + λw + λuv + λuw + λvw + λuvw , i j ij k ik jk ijk as in Anova models. λ appears as the overall or reference average eﬀect λu appears as the marginal discrepancy (against the reference i eﬀect λ) when u = i, λuv as the interaction discrepancy (against the added eﬀects ij λ + λu + λv ) when (u, v) = (i, j) i j and so on... 409 / 785
- 98. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Example of submodels if both v and w are irrelevant, then 1 u log(µl(i,j,k) ) = λ + λi , if all three categorical variables are mutually independent, then 2 log(µl(i,j,k) ) = λ + λi + λv + λw , u j k if u and v are associated but are both independent of w, then 3 log(µl(i,j,k) ) = λ + λi + λv + λw + λuv , u j k ij if u and v are conditionally independent given w, then 4 log(µl(i,j,k) ) = λ + λi + λv + λw + λuw + λvw , u j k ik jk if there is no three-factor interaction, then 5 log(µl(i,j,k) ) = λ + λi + λv + λw + λuv + λuw + λvw u j k ij ik jk [the most complete submodel] 410 / 785
- 99. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Identiﬁability Representation log(µl(i,j,k) ) = λ + λu + λv + λw + λuv + λuw + λvw + λuvw , i j ij k ik jk ijk not identiﬁable but Bayesian approach handles non-identiﬁable settings and still estimate properly identiﬁable quantities. 411 / 785
- 100. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Identiﬁability Representation log(µl(i,j,k) ) = λ + λu + λv + λw + λuv + λuw + λvw + λuvw , i j ij k ik jk ijk not identiﬁable but Bayesian approach handles non-identiﬁable settings and still estimate properly identiﬁable quantities. Customary to impose identiﬁability constraints on the parameters: set to 0 parameters corresponding to the ﬁrst category of each variable, i.e. remove the indicator of the ﬁrst category. E.g., if u ∈ {1, 2} and v ∈ {1, 2}, constraint could be λu = λv = λuv = λuv = λuv = 0 . 1 1 11 12 21 412 / 785
- 101. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Inference under a ﬂat prior Noninformative prior π(β) ∝ 1 gives posterior distribution n yi exp(xiT β) exp{− exp(xiT β)} π(β|y, X) ∝ i=1 n n yi xiT β − exp(xiT β) = exp i=1 i=1 413 / 785
- 102. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Inference under a ﬂat prior Noninformative prior π(β) ∝ 1 gives posterior distribution n yi exp(xiT β) exp{− exp(xiT β)} π(β|y, X) ∝ i=1 n n yi xiT β − exp(xiT β) = exp i=1 i=1 Use of same random walk M-H algorithm as in probit and logit cases, starting with MLE evaluation > mod=summary(glm(y~-1+X,family=poisson())) 414 / 785
- 103. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models airquality Eﬀect Post. mean Post. var. λ 2.8041 0.0612 λu -1.0684 0.2176 2 λv -5.8652 1.7141 2 λw -1.4401 0.2735 2 λw -2.7178 0.7915 3 Identiﬁable non-saturated model λw -1.1031 0.2295 4 involves 16 parameters λw -0.0036 0.1127 5 λuv Obtained with 10, 000 MCMC 3.3559 0.4490 22 λuw -1.6242 1.2869 iterations with scale factor 22 λuw - 0.3456 0.8432 τ 2 = 0.5 23 λuw -0.2473 0.6658 24 λuw -1.3335 0.7115 25 λvw 4.5493 2.1997 22 λvw 6.8479 2.5881 23 λvw 4.6557 1.7201 24 λvw 3.9558 1.7128 25 415 / 785
- 104. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models airquality: MCMC output 0.0 −2 0.5 3.0 −1.5 −1.0 −6 −3.0 −10 −2.5 2.0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 6 1.0 0.0 −2 5 4 −1.5 0.0 −4 3 −6 −1.0 2 −3.0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 0 0 1 1 −2 −2 −1 −1 −4 −5 −3 −3 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 12 8 8 8 6 6 8 4 4 4 2 4 2 0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 416 / 785
- 105. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models airquality: MCMC output 1400 800 0.0 −2 0.5 800 3.0 −1.5 400 −1.0 −6 400 400 600 −3.0 −10 −2.5 2.0 0 0 0 0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 2.0 3.0 −2.5 −1.0 0.5 −10 −6 −2 −3.0 −1.5 0.0 1000 1000 6 1.0 0.0 800 −2 5 400 4 −1.5 0.0 400 −4 400 400 3 −6 −1.0 2 −3.0 0 0 0 0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 −7 −5 −3 −1 −3.0 −1.0 0.5 −1.0 0.0 1.0 2 4 6 800 800 800 0 0 1 1 400 400 −2 400 400 −2 −1 −1 −4 −5 −3 −3 0 0 0 0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 −5 −2 0 −3 −1 1 3 −3 −1 1 3 −4 −2 0 1200 1000 12 8 500 500 8 8 6 6 600 8 400 4 200 200 4 4 2 4 2 0 0 0 0 0 0 4000 10000 0 4000 10000 0 4000 10000 0 4000 10000 0 4 8 4 8 12 2468 2468 417 / 785
- 106. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Model choice with G-prior G-prior alternative used for probit and logit models still available: (2k − 1) ||Xβ||−(2k−1)/2 π −k/2 π(β|y, X) ∝ |X T X|1/2 Γ 4 T n n yi x i exp(xiT β) × exp β− i=1 i=1 418 / 785
- 107. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models Model choice with G-prior G-prior alternative used for probit and logit models still available: (2k − 1) ||Xβ||−(2k−1)/2 π −k/2 π(β|y, X) ∝ |X T X|1/2 Γ 4 T n n yi x i exp(xiT β) × exp β− i=1 i=1 Same MCMC implementation and similar estimates for airquality 419 / 785
- 108. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models airquality Bayes factors once more approximated by importance sampling based on normal importance functions 420 / 785
- 109. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Generalized linear models Loglinear models airquality Bayes factors once more approximated by importance sampling based on normal importance functions Anova-like output Effect log10(BF) u:v 6.0983 (****) u:w -0.5732 v:w 6.0802 (****) evidence against H0: (****) decisive, (***) strong, (**) subtantial, (*) poor 421 / 785

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