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# Bayesian Core: Chapter 2

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This is the first part of our course associated with Bayesian Core.

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### Bayesian Core: Chapter 2

1. 1. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Bayesian Core: A Practical Approach to Computational Bayesian Statistics Jean-Michel Marin & Christian P. Robert QUT, Brisbane, August 4–8 2008 1 / 785
2. 2. Bayesian Core:A Practical Approach to Computational Bayesian Statistics Outline The normal model 1 Regression and variable selection 2 Generalized linear models 3 Capture-recapture experiments 4 Mixture models 5 Dynamic models 6 Image analysis 7 2 / 785
3. 3. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The normal model The normal model 1 Normal problems The Bayesian toolbox Prior selection Bayesian estimation Conﬁdence regions Testing Monte Carlo integration Prediction 3 / 785
4. 4. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Normal model Sample x1 , . . . , xn from a normal N (µ, σ 2 ) distribution normal sample 0.6 0.5 0.4 Density 0.3 0.2 0.1 0.0 −2 −1 0 1 2 4 / 785
5. 5. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Inference on (µ, σ) based on this sample Estimation of [transforms of] (µ, σ) 5 / 785
6. 6. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Inference on (µ, σ) based on this sample Estimation of [transforms of] (µ, σ) Conﬁdence region [interval] on (µ, σ) 6 / 785
7. 7. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Inference on (µ, σ) based on this sample Estimation of [transforms of] (µ, σ) Conﬁdence region [interval] on (µ, σ) Test on (µ, σ) and comparison with other samples 7 / 785
8. 8. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Datasets Larcenies =normaldata 4 Relative changes in reported 3 larcenies between 1991 and 1995 (relative to 1991) for 2 the 90 most populous US counties (Source: FBI) 1 0 −0.4 −0.2 0.0 0.2 0.4 0.6 8 / 785
9. 9. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Cosmological background =CMBdata Spectral representation of the “cosmological microwave background” (CMB), i.e. electromagnetic radiation from photons back to 300, 000 years after the Big Bang, expressed as diﬀerence in apparent temperature from the mean temperature 9 / 785
10. 10. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Cosmological background =CMBdata Normal estimation 5 4 3 2 1 0 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 CMB 10 / 785
11. 11. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Normal problems Cosmological background =CMBdata Normal estimation 5 4 3 2 1 0 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 CMB 11 / 785
12. 12. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox The Bayesian toolbox Bayes theorem = Inversion of probabilities 12 / 785
13. 13. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox The Bayesian toolbox Bayes theorem = Inversion of probabilities If A and E are events such that P (E) = 0, P (A|E) and P (E|A) are related by P (E|A)P (A) P (A|E) = P (E|A)P (A) + P (E|Ac )P (Ac ) P (E|A)P (A) = P (E) 13 / 785
14. 14. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Who’s Bayes? Reverend Thomas Bayes (ca. 1702–1761) Presbyterian minister in Tunbridge Wells (Kent) from 1731, son of Joshua Bayes, nonconformist minister. Election to the Royal Society based on a tract of 1736 where he defended the views and philosophy of Newton. 14 / 785
15. 15. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Who’s Bayes? Reverend Thomas Bayes (ca. 1702–1761) Presbyterian minister in Tunbridge Wells (Kent) from 1731, son of Joshua Bayes, nonconformist minister. Election to the Royal Society based on a tract of 1736 where he defended the views and philosophy of Newton. Sole probability paper, “Essay Towards Solving a Problem in the Doctrine of Chances”, published posthumously in 1763 by Pierce and containing the seeds of Bayes’ Theorem. 15 / 785
16. 16. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox New perspective Uncertainty on the parameters θ of a model modeled through a probability distribution π on Θ, called prior distribution 16 / 785
17. 17. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox New perspective Uncertainty on the parameters θ of a model modeled through a probability distribution π on Θ, called prior distribution Inference based on the distribution of θ conditional on x, π(θ|x), called posterior distribution f (x|θ)π(θ) π(θ|x) = . f (x|θ)π(θ) dθ 17 / 785
18. 18. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Bayesian model A Bayesian statistical model is made of a likelihood 1 f (x|θ), 18 / 785
19. 19. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Bayesian model A Bayesian statistical model is made of a likelihood 1 f (x|θ), and of a prior distribution on the parameters, 2 π(θ) . 19 / 785
20. 20. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Justiﬁcations Semantic drift from unknown θ to random θ 20 / 785
21. 21. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Justiﬁcations Semantic drift from unknown θ to random θ Actualization of information/knowledge on θ by extracting information/knowledge on θ contained in the observation x 21 / 785
22. 22. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Justiﬁcations Semantic drift from unknown θ to random θ Actualization of information/knowledge on θ by extracting information/knowledge on θ contained in the observation x Allows incorporation of imperfect/imprecise information in the decision process 22 / 785
23. 23. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Justiﬁcations Semantic drift from unknown θ to random θ Actualization of information/knowledge on θ by extracting information/knowledge on θ contained in the observation x Allows incorporation of imperfect/imprecise information in the decision process Unique mathematical way to condition upon the observations (conditional perspective) 23 / 785
24. 24. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Example (Normal illustration (σ 2 = 1)) Assume π(θ) = exp{−θ} Iθ>0 24 / 785
25. 25. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Example (Normal illustration (σ 2 = 1)) Assume π(θ) = exp{−θ} Iθ>0 Then π(θ|x1 , . . . , xn ) ∝ exp{−θ} exp{−n(θ − x)2 /2} Iθ>0 ∝ exp −nθ2 /2 + θ(nx − 1) Iθ>0 ∝ exp −n(θ − (x − 1/n))2 /2 Iθ>0 25 / 785
26. 26. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Example (Normal illustration (2)) Truncated normal distribution N + ((x − 1/n), 1/n) mu=−0.08935864 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 26 / 785
27. 27. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Prior and posterior distributions Given f (x|θ) and π(θ), several distributions of interest: the joint distribution of (θ, x), 1 ϕ(θ, x) = f (x|θ)π(θ) ; 27 / 785
28. 28. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Prior and posterior distributions Given f (x|θ) and π(θ), several distributions of interest: the joint distribution of (θ, x), 1 ϕ(θ, x) = f (x|θ)π(θ) ; the marginal distribution of x, 2 m(x) = ϕ(θ, x) dθ = f (x|θ)π(θ) dθ ; 28 / 785
29. 29. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox the posterior distribution of θ, 3 f (x|θ)π(θ) π(θ|x) = f (x|θ)π(θ) dθ f (x|θ)π(θ) = ; m(x) the predictive distribution of y, when y ∼ g(y|θ, x), 4 g(y|x) = g(y|θ, x)π(θ|x)dθ . 29 / 785
30. 30. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Posterior distribution center of to Bayesian inference π(θ|x) ∝ f (x|θ) π(θ) Operates conditional upon the observations 30 / 785
31. 31. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Posterior distribution center of to Bayesian inference π(θ|x) ∝ f (x|θ) π(θ) Operates conditional upon the observations Integrate simultaneously prior information/knowledge and information brought by x 31 / 785
32. 32. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Posterior distribution center of to Bayesian inference π(θ|x) ∝ f (x|θ) π(θ) Operates conditional upon the observations Integrate simultaneously prior information/knowledge and information brought by x Avoids averaging over the unobserved values of x 32 / 785
33. 33. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Posterior distribution center of to Bayesian inference π(θ|x) ∝ f (x|θ) π(θ) Operates conditional upon the observations Integrate simultaneously prior information/knowledge and information brought by x Avoids averaging over the unobserved values of x Coherent updating of the information available on θ, independent of the order in which i.i.d. observations are collected 33 / 785
34. 34. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Posterior distribution center of to Bayesian inference π(θ|x) ∝ f (x|θ) π(θ) Operates conditional upon the observations Integrate simultaneously prior information/knowledge and information brought by x Avoids averaging over the unobserved values of x Coherent updating of the information available on θ, independent of the order in which i.i.d. observations are collected Provides a complete inferential scope and an unique motor of inference 34 / 785
35. 35. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Example (Normal-normal case) Consider x|θ ∼ N (θ, 1) and θ ∼ N (a, 10). (x − θ)2 (θ − a)2 π(θ|x) ∝ f (x|θ)π(θ) ∝ exp − − 2 20 11θ2 ∝ exp − + θ(x + a/10) 20 11 ∝ exp − {θ − ((10x + a)/11)}2 20 35 / 785
36. 36. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model The Bayesian toolbox Example (Normal-normal case) Consider x|θ ∼ N (θ, 1) and θ ∼ N (a, 10). (x − θ)2 (θ − a)2 π(θ|x) ∝ f (x|θ)π(θ) ∝ exp − − 2 20 11θ2 ∝ exp − + θ(x + a/10) 20 11 ∝ exp − {θ − ((10x + a)/11)}2 20 and θ|x ∼ N (10x + a) 11, 10 11 36 / 785
37. 37. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Prior selection The prior distribution is the key to Bayesian inference 37 / 785
38. 38. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Prior selection The prior distribution is the key to Bayesian inference But... In practice, it seldom occurs that the available prior information is precise enough to lead to an exact determination of the prior distribution 38 / 785
39. 39. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Prior selection The prior distribution is the key to Bayesian inference But... In practice, it seldom occurs that the available prior information is precise enough to lead to an exact determination of the prior distribution There is no such thing as the prior distribution! 39 / 785
40. 40. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Strategies for prior determination Ungrounded prior distributions produce unjustiﬁed posterior inference. —Anonymous, ca. 2006 40 / 785
41. 41. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Strategies for prior determination Ungrounded prior distributions produce unjustiﬁed posterior inference. —Anonymous, ca. 2006 Use a partition of Θ in sets (e.g., intervals), determine the probability of each set, and approach π by an histogram 41 / 785
42. 42. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Strategies for prior determination Ungrounded prior distributions produce unjustiﬁed posterior inference. —Anonymous, ca. 2006 Use a partition of Θ in sets (e.g., intervals), determine the probability of each set, and approach π by an histogram Select signiﬁcant elements of Θ, evaluate their respective likelihoods and deduce a likelihood curve proportional to π 42 / 785
43. 43. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Strategies for prior determination Ungrounded prior distributions produce unjustiﬁed posterior inference. —Anonymous, ca. 2006 Use a partition of Θ in sets (e.g., intervals), determine the probability of each set, and approach π by an histogram Select signiﬁcant elements of Θ, evaluate their respective likelihoods and deduce a likelihood curve proportional to π Use the marginal distribution of x, m(x) = f (x|θ)π(θ) dθ Θ 43 / 785
44. 44. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Strategies for prior determination Ungrounded prior distributions produce unjustiﬁed posterior inference. —Anonymous, ca. 2006 Use a partition of Θ in sets (e.g., intervals), determine the probability of each set, and approach π by an histogram Select signiﬁcant elements of Θ, evaluate their respective likelihoods and deduce a likelihood curve proportional to π Use the marginal distribution of x, m(x) = f (x|θ)π(θ) dθ Θ Empirical and hierarchical Bayes techniques 44 / 785
45. 45. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Conjugate priors Speciﬁc parametric family with analytical properties Conjugate prior A family F of probability distributions on Θ is conjugate for a likelihood function f (x|θ) if, for every π ∈ F, the posterior distribution π(θ|x) also belongs to F. 45 / 785
46. 46. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Conjugate priors Speciﬁc parametric family with analytical properties Conjugate prior A family F of probability distributions on Θ is conjugate for a likelihood function f (x|θ) if, for every π ∈ F, the posterior distribution π(θ|x) also belongs to F. Only of interest when F is parameterised : switching from prior to posterior distribution is reduced to an updating of the corresponding parameters. 46 / 785
47. 47. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Limited/ﬁnite information conveyed by x Preservation of the structure of π(θ) 47 / 785
48. 48. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Limited/ﬁnite information conveyed by x Preservation of the structure of π(θ) Exchangeability motivations Device of virtual past observations 48 / 785
49. 49. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Limited/ﬁnite information conveyed by x Preservation of the structure of π(θ) Exchangeability motivations Device of virtual past observations Linearity of some estimators But mostly... 49 / 785
50. 50. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Limited/ﬁnite information conveyed by x Preservation of the structure of π(θ) Exchangeability motivations Device of virtual past observations Linearity of some estimators But mostly... tractability and simplicity 50 / 785
51. 51. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Limited/ﬁnite information conveyed by x Preservation of the structure of π(θ) Exchangeability motivations Device of virtual past observations Linearity of some estimators But mostly... tractability and simplicity First approximations to adequate priors, backed up by robustness analysis 51 / 785
52. 52. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Exponential families Sampling models of interest Exponential family The family of distributions f (x|θ) = C(θ)h(x) exp{R(θ) · T (x)} is called an exponential family of dimension k. When Θ ⊂ Rk , X ⊂ Rk and f (x|θ) = h(x) exp{θ · x − Ψ(θ)}, the family is said to be natural. 52 / 785
53. 53. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Analytical properties of exponential families Suﬃcient statistics (Pitman–Koopman Lemma) 53 / 785
54. 54. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Analytical properties of exponential families Suﬃcient statistics (Pitman–Koopman Lemma) Common enough structure (normal, Poisson, &tc...) 54 / 785
55. 55. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Analytical properties of exponential families Suﬃcient statistics (Pitman–Koopman Lemma) Common enough structure (normal, Poisson, &tc...) Analyticity (E[x] = ∇Ψ(θ), ...) 55 / 785
56. 56. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Analytical properties of exponential families Suﬃcient statistics (Pitman–Koopman Lemma) Common enough structure (normal, Poisson, &tc...) Analyticity (E[x] = ∇Ψ(θ), ...) Allow for conjugate priors π(θ|µ, λ) = K(µ, λ) eθ.µ−λΨ(θ) λ>0 56 / 785
57. 57. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Standard exponential families f (x|θ) π(θ) π(θ|x) Normal Normal N (θ, σ 2 ) N (µ, τ 2 ) N (ρ(σ 2 µ + τ 2 x), ρσ 2 τ 2 ) ρ−1 = σ 2 + τ 2 Poisson Gamma P(θ) G(α, β) G(α + x, β + 1) Gamma Gamma G(ν, θ) G(α, β) G(α + ν, β + x) Binomial Beta B(n, θ) Be(α, β) Be(α + x, β + n − x) 57 / 785
58. 58. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection More... f (x|θ) π(θ) π(θ|x) Negative Binomial Beta N eg(m, θ) Be(α, β) Be(α + m, β + x) Multinomial Dirichlet Mk (θ1 , . . . , θk ) D(α1 , . . . , αk ) D(α1 + x1 , . . . , αk + xk ) Normal Gamma G(α + 0.5, β + (µ − x)2 /2) N (µ, 1/θ) Ga(α, β) 58 / 785
59. 59. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Linearity of the posterior mean If θ ∼ πλ,µ (θ) ∝ eθ·µ−λΨ(θ) with µ ∈ X , then µ Eπ [∇Ψ(θ)] = . λ where ∇Ψ(θ) = (∂Ψ(θ)/∂θ1 , . . . , ∂Ψ(θ)/∂θp ) 59 / 785
60. 60. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Linearity of the posterior mean If θ ∼ πλ,µ (θ) ∝ eθ·µ−λΨ(θ) with µ ∈ X , then µ Eπ [∇Ψ(θ)] = . λ where ∇Ψ(θ) = (∂Ψ(θ)/∂θ1 , . . . , ∂Ψ(θ)/∂θp ) Therefore, if x1 , . . . , xn are i.i.d. f (x|θ), µ + n¯ x Eπ [∇Ψ(θ)|x1 , . . . , xn ] = . λ+n 60 / 785
61. 61. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Example (Normal-normal) In the normal N (θ, σ 2 ) case, conjugate also normal N (µ, τ 2 ) and Eπ [∇Ψ(θ)|x] = Eπ [θ|x] = ρ(σ 2 µ + τ 2 x) where ρ−1 = σ 2 + τ 2 61 / 785
62. 62. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Example (Full normal) In the normal N (µ, σ 2 ) case, when both µ and σ are unknown, there still is a conjugate prior on θ = (µ, σ 2 ), of the form (σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2 62 / 785
63. 63. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Example (Full normal) In the normal N (µ, σ 2 ) case, when both µ and σ are unknown, there still is a conjugate prior on θ = (µ, σ 2 ), of the form (σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2 since π(µ, σ 2 |x1 , . . . , xn ) ∝ (σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2 ×(σ 2 )−n/2 exp − n(µ − x)2 + s2 /2σ 2 x ∝ (σ 2 )−λσ −n/2 exp − (λµ + n)(µ − ξx )2 nλµ (x − ξ)2 +α + s2 + /2σ 2 x n + λµ 63 / 785
64. 64. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection parameters (0,1,1,1) 2.0 1.5 σ 1.0 0.5 −1.0 −0.5 0.0 0.5 1.0 µ 64 / 785
65. 65. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Improper prior distribution Extension from a prior distribution to a prior σ-ﬁnite measure π such that π(θ) dθ = +∞ Θ 65 / 785
66. 66. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Improper prior distribution Extension from a prior distribution to a prior σ-ﬁnite measure π such that π(θ) dθ = +∞ Θ Formal extension: π cannot be interpreted as a probability any longer 66 / 785
67. 67. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Often only way to derive a prior in noninformative/automatic 1 settings 67 / 785
68. 68. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Often only way to derive a prior in noninformative/automatic 1 settings Performances of associated estimators usually good 2 68 / 785
69. 69. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Often only way to derive a prior in noninformative/automatic 1 settings Performances of associated estimators usually good 2 Often occur as limits of proper distributions 3 69 / 785
70. 70. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Often only way to derive a prior in noninformative/automatic 1 settings Performances of associated estimators usually good 2 Often occur as limits of proper distributions 3 More robust answer against possible misspeciﬁcations of the 4 prior 70 / 785
71. 71. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Justiﬁcations Often only way to derive a prior in noninformative/automatic 1 settings Performances of associated estimators usually good 2 Often occur as limits of proper distributions 3 More robust answer against possible misspeciﬁcations of the 4 prior Improper priors (inﬁnitely!) preferable to vague proper priors 5 such as a N (0, 1002 ) distribution [e.g., BUGS] 71 / 785
72. 72. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Validation Extension of the posterior distribution π(θ|x) associated with an improper prior π given by Bayes’s formula f (x|θ)π(θ) π(θ|x) = , Θ f (x|θ)π(θ) dθ when f (x|θ)π(θ) dθ < ∞ Θ 72 / 785
73. 73. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Example (Normal+improper) If x ∼ N (θ, 1) and π(θ) = ̟, constant, the pseudo marginal distribution is +∞ 1 √ exp −(x − θ)2 /2 dθ = ̟ m(x) = ̟ 2π −∞ and the posterior distribution of θ is (x − θ)2 1 π(θ | x) = √ exp − , 2 2π i.e., corresponds to N (x, 1). [independent of ̟] 73 / 785
74. 74. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Meaningless as probability distribution The mistake is to think of them [the non-informative priors] as representing ignorance —Lindley, 1990— 74 / 785
75. 75. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Meaningless as probability distribution The mistake is to think of them [the non-informative priors] as representing ignorance —Lindley, 1990— Example Consider a θ ∼ N (0, τ 2 ) prior. Then P π (θ ∈ [a, b]) −→ 0 when τ → ∞ for any (a, b) 75 / 785
76. 76. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Noninformative prior distributions What if all we know is that we know “nothing” ?! 76 / 785
77. 77. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Noninformative prior distributions What if all we know is that we know “nothing” ?! In the absence of prior information, prior distributions solely derived from the sample distribution f (x|θ) 77 / 785
78. 78. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Noninformative prior distributions What if all we know is that we know “nothing” ?! In the absence of prior information, prior distributions solely derived from the sample distribution f (x|θ) Noninformative priors cannot be expected to represent exactly total ignorance about the problem at hand, but should rather be taken as reference or default priors, upon which everyone could fall back when the prior information is missing. —Kass and Wasserman, 1996— 78 / 785
79. 79. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Laplace’s prior Principle of Insuﬃcient Reason (Laplace) Θ = {θ1 , · · · , θp } π(θi ) = 1/p 79 / 785
80. 80. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Laplace’s prior Principle of Insuﬃcient Reason (Laplace) Θ = {θ1 , · · · , θp } π(θi ) = 1/p Extension to continuous spaces π(θ) ∝ 1 [Lebesgue measure] 80 / 785
81. 81. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Who’s Laplace? Pierre Simon de Laplace (1749–1827) French mathematician and astronomer born in Beaumont en Auge (Normandie) who formalised mathematical astronomy in M´canique C´leste. Survived the e e French revolution, the Napoleon Empire (as a comte!), and the Bourbon restauration (as a marquis!!). 81 / 785
82. 82. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Who’s Laplace? Pierre Simon de Laplace (1749–1827) French mathematician and astronomer born in Beaumont en Auge (Normandie) who formalised mathematical astronomy in M´canique C´leste. Survived the e e French revolution, the Napoleon Empire (as a comte!), and the Bourbon restauration (as a marquis!!). In Essai Philosophique sur les Probabilit´s, Laplace set out a e mathematical system of inductive reasoning based on probability, precursor to Bayesian Statistics. 82 / 785
83. 83. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Laplace’s problem Lack of reparameterization invariance/coherence 1 and ψ = eθ π(θ) ∝ 1, π(ψ) = = 1 (!!) ψ 83 / 785
84. 84. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Laplace’s problem Lack of reparameterization invariance/coherence 1 and ψ = eθ π(θ) ∝ 1, π(ψ) = = 1 (!!) ψ Problems of properness x ∼ N (µ, σ 2 ), π(µ, σ) = 1 2 2 π(µ, σ|x) ∝ e−(x−µ) /2σ σ −1 ⇒ π(σ|x) ∝1 (!!!) 84 / 785
85. 85. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Jeﬀreys’ prior Based on Fisher information ∂ log ℓ ∂ log ℓ I F (θ) = Eθ ∂θt ∂θ Ron Fisher (1890–1962) 85 / 785
86. 86. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Jeﬀreys’ prior Based on Fisher information ∂ log ℓ ∂ log ℓ I F (θ) = Eθ ∂θt ∂θ Ron Fisher (1890–1962) the Jeﬀreys prior distribution is π J (θ) ∝ |I F (θ)|1/2 86 / 785
87. 87. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Who’s Jeﬀreys? Sir Harold Jeﬀreys (1891–1989) English mathematician, statistician, geophysicist, and astronomer. Founder of English Geophysics & originator of the theory that the Earth core is liquid. 87 / 785
88. 88. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Who’s Jeﬀreys? Sir Harold Jeﬀreys (1891–1989) English mathematician, statistician, geophysicist, and astronomer. Founder of English Geophysics & originator of the theory that the Earth core is liquid. Formalised Bayesian methods for the analysis of geophysical data and ended up writing Theory of Probability 88 / 785
89. 89. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Pros & Cons Relates to information theory 89 / 785
90. 90. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Pros & Cons Relates to information theory Agrees with most invariant priors 90 / 785
91. 91. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Pros & Cons Relates to information theory Agrees with most invariant priors Parameterization invariant 91 / 785
92. 92. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Prior selection Pros & Cons Relates to information theory Agrees with most invariant priors Parameterization invariant Suﬀers from dimensionality curse 92 / 785
93. 93. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Evaluating estimators Purpose of most inferential studies: to provide the statistician/client with a decision d ∈ D 93 / 785
94. 94. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Evaluating estimators Purpose of most inferential studies: to provide the statistician/client with a decision d ∈ D Requires an evaluation criterion/loss function for decisions and estimators L(θ, d) 94 / 785
95. 95. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Evaluating estimators Purpose of most inferential studies: to provide the statistician/client with a decision d ∈ D Requires an evaluation criterion/loss function for decisions and estimators L(θ, d) There exists an axiomatic derivation of the existence of a loss function. [DeGroot, 1970] 95 / 785
96. 96. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Loss functions Decision procedure δ π usually called estimator (while its value δ π (x) is called estimate of θ) 96 / 785
97. 97. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Loss functions Decision procedure δ π usually called estimator (while its value δ π (x) is called estimate of θ) Impossible to uniformly minimize (in d) the loss function L(θ, d) when θ is unknown 97 / 785
98. 98. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Bayesian estimation Principle Integrate over the space Θ to get the posterior expected loss = Eπ [L(θ, d)|x] = L(θ, d)π(θ|x) dθ, Θ and minimise in d 98 / 785
99. 99. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Bayes estimates Bayes estimator A Bayes estimate associated with a prior distribution π and a loss function L is arg min Eπ [L(θ, d)|x] . d 99 / 785
100. 100. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation The quadratic loss Historically, ﬁrst loss function (Legendre, Gauss, Laplace) L(θ, d) = (θ − d)2 100 / 785
101. 101. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation The quadratic loss Historically, ﬁrst loss function (Legendre, Gauss, Laplace) L(θ, d) = (θ − d)2 The Bayes estimate δ π (x) associated with the prior π and with the quadratic loss is the posterior expectation Θ θf (x|θ)π(θ) dθ δ π (x) = Eπ [θ|x] = . Θ f (x|θ)π(θ) dθ 101 / 785
102. 102. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation The absolute error loss Alternatives to the quadratic loss: L(θ, d) =| θ − d |, or k2 (θ − d) if θ > d, Lk1 ,k2 (θ, d) = k1 (d − θ) otherwise. 102 / 785
103. 103. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation The absolute error loss Alternatives to the quadratic loss: L(θ, d) =| θ − d |, or k2 (θ − d) if θ > d, Lk1 ,k2 (θ, d) = k1 (d − θ) otherwise. Associated Bayes estimate is (k2 /(k1 + k2 )) fractile of π(θ|x) 103 / 785
104. 104. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation MAP estimator With no loss function, consider using the maximum a posteriori (MAP) estimator arg max ℓ(θ|x)π(θ) θ 104 / 785
105. 105. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation MAP estimator With no loss function, consider using the maximum a posteriori (MAP) estimator arg max ℓ(θ|x)π(θ) θ Penalized likelihood estimator 105 / 785
106. 106. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation MAP estimator With no loss function, consider using the maximum a posteriori (MAP) estimator arg max ℓ(θ|x)π(θ) θ Penalized likelihood estimator Further appeal in restricted parameter spaces 106 / 785
107. 107. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Example (Binomial probability) Consider x|θ ∼ B(n, θ). Possible priors: 1 π J (θ) = θ−1/2 (1 − θ)−1/2 , B(1/2, 1/2) π2 (θ) = θ−1 (1 − θ)−1 . π1 (θ) = 1 and 107 / 785
108. 108. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Example (Binomial probability) Consider x|θ ∼ B(n, θ). Possible priors: 1 π J (θ) = θ−1/2 (1 − θ)−1/2 , B(1/2, 1/2) π2 (θ) = θ−1 (1 − θ)−1 . π1 (θ) = 1 and Corresponding MAP estimators: x − 1/2 δ πJ (x) = max ,0 , n−1 δ π1 (x) = x/n, x−1 δ π2 (x) = max ,0 . n−2 108 / 785
109. 109. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Not always appropriate Example (Fixed MAP) Consider 1 −1 1 + (x − θ)2 f (x|θ) = , π 1 and π(θ) = 2 e−|θ| . 109 / 785
110. 110. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Bayesian estimation Not always appropriate Example (Fixed MAP) Consider 1 −1 1 + (x − θ)2 f (x|θ) = , π 1 and π(θ) = 2 e−|θ| . Then the MAP estimate of θ is always δ π (x) = 0 110 / 785
111. 111. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Credible regions Natural conﬁdence region: Highest posterior density (HPD) region π Cα = {θ; π(θ|x) > kα } 111 / 785
112. 112. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Credible regions Natural conﬁdence region: Highest posterior density (HPD) region π Cα = {θ; π(θ|x) > kα } Optimality 0.15 The HPD regions give the highest probabilities of containing θ for a 0.10 given volume 0.05 0.00 −10 −5 0 5 10 µ 112 / 785
113. 113. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Example If the posterior distribution of θ is N (µ(x), ω −2 ) with ω 2 = τ −2 + σ −2 and µ(x) = τ 2 x/(τ 2 + σ 2 ), then Cα = µ(x) − kα ω −1 , µ(x) + kα ω −1 , π where kα is the α/2-quantile of N (0, 1). 113 / 785
114. 114. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Example If the posterior distribution of θ is N (µ(x), ω −2 ) with ω 2 = τ −2 + σ −2 and µ(x) = τ 2 x/(τ 2 + σ 2 ), then Cα = µ(x) − kα ω −1 , µ(x) + kα ω −1 , π where kα is the α/2-quantile of N (0, 1). If τ goes to +∞, π Cα = [x − kα σ, x + kα σ] , the “usual” (classical) conﬁdence interval 114 / 785
115. 115. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Full normal Under [almost!] Jeﬀreys prior prior π(µ, σ 2 ) = 1/σ 2 , posterior distribution of (µ, σ) σ2 µ|σ, x, s2 ¯x ∼ N x, ¯ , n n − 1 s2 ,x σ 2 |¯, sx x2 ∼ IG . 2 2 115 / 785
116. 116. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Full normal Under [almost!] Jeﬀreys prior prior π(µ, σ 2 ) = 1/σ 2 , posterior distribution of (µ, σ) σ2 µ|σ, x, s2 ¯x ∼ N x, ¯ , n n − 1 s2 ,x σ 2 |¯, sx x2 ∼ IG . 2 2 Then n(¯ − µ)2 (n−3)/2 x π(µ|¯, s2 ) ∝ ω 1/2 exp −ω exp{−ωs2 /2} dω xx ω x 2 −n/2 s2 + n(¯ − µ)2 ∝ x x [Tn−1 distribution] 116 / 785
117. 117. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Normal credible interval Derived credible interval on µ [¯ − tα/2,n−1 sx x n(n − 1), x + tα/2,n−1 sx ¯ n(n − 1)] 117 / 785
118. 118. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Conﬁdence regions Normal credible interval Derived credible interval on µ [¯ − tα/2,n−1 sx x n(n − 1), x + tα/2,n−1 sx ¯ n(n − 1)] normaldata Corresponding 95% conﬁdence region for µ [−0.070, −0.013, ] Since 0 does not belong to this interval, reporting a signiﬁcant decrease in the number of larcenies between 1991 and 1995 is acceptable 118 / 785
119. 119. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Testing hypotheses Deciding about validity of assumptions or restrictions on the parameter θ from the data, represented as H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0 119 / 785
120. 120. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Testing hypotheses Deciding about validity of assumptions or restrictions on the parameter θ from the data, represented as H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0 Binary outcome of the decision process: accept [coded by 1] or reject [coded by 0] D = {0, 1} 120 / 785
121. 121. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Testing hypotheses Deciding about validity of assumptions or restrictions on the parameter θ from the data, represented as H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0 Binary outcome of the decision process: accept [coded by 1] or reject [coded by 0] D = {0, 1} Bayesian solution formally very close from a likelihood ratio test statistic, but numerical values often strongly diﬀer from classical solutions 121 / 785
122. 122. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing The 0 − 1 loss Rudimentary loss function 1−d if θ ∈ Θ0 L(θ, d) = d otherwise, Associated Bayes estimate  1 if P π (θ ∈ Θ |x) > 1 , 0 π δ (x) = 2 0 otherwise. 122 / 785
123. 123. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing The 0 − 1 loss Rudimentary loss function 1−d if θ ∈ Θ0 L(θ, d) = d otherwise, Associated Bayes estimate  1 if P π (θ ∈ Θ |x) > 1 , 0 π δ (x) = 2 0 otherwise. Intuitive structure 123 / 785
124. 124. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Extension Weighted 0 − 1 (or a0 − a1 ) loss  0 if d = IΘ0 (θ),  L(θ, d) = a0 if θ ∈ Θ0 and d = 0,   a1 if θ ∈ Θ0 and d = 1, 124 / 785
125. 125. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Extension Weighted 0 − 1 (or a0 − a1 ) loss  0 if d = IΘ0 (θ),  L(θ, d) = a0 if θ ∈ Θ0 and d = 0,   a1 if θ ∈ Θ0 and d = 1, Associated Bayes estimator  a1 1 if P π (θ ∈ Θ0 |x) > , a0 + a1 π δ (x) = 0 otherwise. 125 / 785
126. 126. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal-normal) For x ∼ N (θ, σ 2 ) and θ ∼ N (µ, τ 2 ), π(θ|x) is N (µ(x), ω 2 ) with σ2µ + τ 2x σ2τ 2 ω2 = µ(x) = and . σ2 + τ 2 σ2 + τ 2 126 / 785
127. 127. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal-normal) For x ∼ N (θ, σ 2 ) and θ ∼ N (µ, τ 2 ), π(θ|x) is N (µ(x), ω 2 ) with σ2µ + τ 2x σ2τ 2 ω2 = µ(x) = and . σ2 + τ 2 σ2 + τ 2 To test H0 : θ < 0, we compute θ − µ(x) −µ(x) P π (θ < 0|x) = P π < ω ω = Φ (−µ(x)/ω) . 127 / 785
128. 128. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal-normal (2)) If za0 ,a1 is the a1 /(a0 + a1 ) quantile, i.e., Φ(za0 ,a1 ) = a1 /(a0 + a1 ) , H0 is accepted when −µ(x) > za0 ,a1 ω, the upper acceptance bound then being σ2 σ2 x≤− µ − (1 + 2 )ωza0 ,a1 . τ2 τ 128 / 785
129. 129. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Bayes factor Bayesian testing procedure depends on P π (θ ∈ Θ0 |x) or alternatively on the Bayes factor {P π (θ ∈ Θ1 |x)/P π (θ ∈ Θ0 |x)} π B10 = {P π (θ ∈ Θ1 )/P π (θ ∈ Θ0 )} in the absence of loss function parameters a0 and a1 129 / 785
130. 130. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Associated reparameterisations Corresponding models M1 vs. M0 compared via P π (M1 |x) P π (M1 ) π B10 = P π (M0 |x) P π (M0 ) 130 / 785
131. 131. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Associated reparameterisations Corresponding models M1 vs. M0 compared via P π (M1 |x) P π (M1 ) π B10 = P π (M0 |x) P π (M0 ) If we rewrite the prior as π(θ) = Pr(θ ∈ Θ1 ) × π1 (θ) + Pr(θ ∈ Θ0 ) × π0 (θ) then π B10 = f (x|θ1 )π1 (θ1 )dθ1 f (x|θ0 )π0 (θ0 )dθ0 = m1 (x)/m0 (x) [Akin to likelihood ratio] 131 / 785
132. 132. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Jeﬀreys’ scale π if log10 (B10 ) varies between 0 and 0.5, 1 the evidence against H0 is poor, if it is between 0.5 and 1, it is is 2 substantial, if it is between 1 and 2, it is strong, 3 and if it is above 2 it is decisive. 4 132 / 785
133. 133. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Point null diﬃculties If π absolutely continuous, P π (θ = θ0 ) = 0 . . . 133 / 785
134. 134. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Point null diﬃculties If π absolutely continuous, P π (θ = θ0 ) = 0 . . . How can we test H0 : θ = θ0 ?! 134 / 785
135. 135. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing New prior for new hypothesis Testing point null diﬃculties requires a modiﬁcation of the prior distribution so that π(Θ0 ) > 0 and π(Θ1 ) > 0 (hidden information) or π(θ) = P π (θ ∈ Θ0 ) × π0 (θ) + P π (θ ∈ Θ1 ) × π1 (θ) 135 / 785
136. 136. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing New prior for new hypothesis Testing point null diﬃculties requires a modiﬁcation of the prior distribution so that π(Θ0 ) > 0 and π(Θ1 ) > 0 (hidden information) or π(θ) = P π (θ ∈ Θ0 ) × π0 (θ) + P π (θ ∈ Θ1 ) × π1 (θ) [E.g., when Θ0 = {θ0 }, π0 is Dirac mass at θ0 ] 136 / 785
137. 137. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Posteriors with Dirac masses If H0 : θ = θ0 (= Θ0 ), ρ = P π (θ = θ0 ) and π(θ) = ρIθ0 (θ) + (1 − ρ)π1 (θ) then f (x|θ0 )ρ π(Θ0 |x) = f (x|θ)π(θ) dθ f (x|θ0 )ρ = f (x|θ0 )ρ + (1 − ρ)m1 (x) with m1 (x) = f (x|θ)π1 (θ) dθ. Θ1 137 / 785
138. 138. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal-normal) For x ∼ N (θ, σ 2 ) and θ ∼ N (0, τ 2 ), to test H0 : θ = 0 requires a modiﬁcation of the prior, with 2 /2τ 2 π1 (θ) ∝ e−θ Iθ=0 and π0 (θ) the Dirac mass in 0 138 / 785
139. 139. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal-normal) For x ∼ N (θ, σ 2 ) and θ ∼ N (0, τ 2 ), to test H0 : θ = 0 requires a modiﬁcation of the prior, with 2 /2τ 2 π1 (θ) ∝ e−θ Iθ=0 and π0 (θ) the Dirac mass in 0 Then 2 2 2 e−x /2(σ +τ ) m1 (x) σ √ = 2 2 f (x|0) σ 2 + τ 2 e−x /2σ τ 2 x2 σ2 = exp , σ2 + τ 2 2σ 2 (σ 2 + τ 2 ) 139 / 785
140. 140. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (cont’d) and −1 τ 2 x2 σ2 1−ρ π(θ = 0|x) = 1 + exp . σ2 + τ 2 2σ 2 (σ 2 + τ 2 ) ρ For z = x/σ and ρ = 1/2: z 0 0.68 1.28 1.96 π(θ = 0|z, τ = σ) 0.586 0.557 0.484 0.351 π(θ = 0|z, τ = 3.3σ) 0.768 0.729 0.612 0.366 140 / 785
141. 141. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Banning improper priors Impossibility of using improper priors for testing! 141 / 785
142. 142. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Banning improper priors Impossibility of using improper priors for testing! Reason: When using the representation π(θ) = P π (θ ∈ Θ1 ) × π1 (θ) + P π (θ ∈ Θ0 ) × π0 (θ) π1 and π0 must be normalised 142 / 785
143. 143. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal point null) When x ∼ N (θ, 1) and H0 : θ = 0, for the improper prior π(θ) = 1, the prior is transformed as 1 1 π(θ) = I0 (θ) + · Iθ=0 , 2 2 and 2 /2 e−x π(θ = 0|x) = +∞ e−x2 /2 + −∞ e−(x−θ)2 /2 dθ 1 √ = . 1 + 2πex2 /2 143 / 785
144. 144. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal point null (2)) Consequence: probability of H0 is bounded from above by √ π(θ = 0|x) ≤ 1/(1 + 2π) = 0.285 x 0.0 1.0 1.65 1.96 2.58 π(θ = 0|x) 0.285 0.195 0.089 0.055 0.014 Regular tests: Agreement with the classical p-value (but...) 144 / 785
145. 145. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal one-sided) For x ∼ N (θ, 1), π(θ) = 1, and H0 : θ ≤ 0 to test versus H1 : θ > 0 0 1 2 /2 e−(x−θ) π(θ ≤ 0|x) = √ dθ = Φ(−x). 2π −∞ The generalized Bayes answer is also the p-value 145 / 785
146. 146. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Example (Normal one-sided) For x ∼ N (θ, 1), π(θ) = 1, and H0 : θ ≤ 0 to test versus H1 : θ > 0 0 1 2 /2 e−(x−θ) π(θ ≤ 0|x) = √ dθ = Φ(−x). 2π −∞ The generalized Bayes answer is also the p-value normaldata If π(µ, σ 2 ) = 1/σ 2 , π(µ ≥ 0|x) = 0.0021 since µ|x ∼ T89 (−0.0144, 0.000206). 146 / 785
147. 147. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Jeﬀreys–Lindley paradox Limiting arguments not valid in testing settings: Under a conjugate prior −1 τ 2 x2 σ2 1−ρ π(θ = 0|x) = 1+ exp , σ2 + τ 2 2σ 2 (σ 2 + τ 2 ) ρ which converges to 1 when τ goes to +∞, for every x 147 / 785
148. 148. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Jeﬀreys–Lindley paradox Limiting arguments not valid in testing settings: Under a conjugate prior −1 τ 2 x2 σ2 1−ρ π(θ = 0|x) = 1+ exp , σ2 + τ 2 2σ 2 (σ 2 + τ 2 ) ρ which converges to 1 when τ goes to +∞, for every x Diﬀerence with the “noninformative” answer √ [1 + 2π exp(x2 /2)]−1 [ Invalid answer] 148 / 785
149. 149. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Normalisation diﬃculties If g0 and g1 are σ-ﬁnite measures on the subspaces Θ0 and Θ1 , the choice of the normalizing constants inﬂuences the Bayes factor: If gi replaced by ci gi (i = 0, 1), Bayes factor multiplied by c0 /c1 149 / 785
150. 150. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Testing Normalisation diﬃculties If g0 and g1 are σ-ﬁnite measures on the subspaces Θ0 and Θ1 , the choice of the normalizing constants inﬂuences the Bayes factor: If gi replaced by ci gi (i = 0, 1), Bayes factor multiplied by c0 /c1 Example If the Jeﬀreys prior is uniform and g0 = c0 , g1 = c1 , ρc0 f (x|θ) dθ Θ0 π(θ ∈ Θ0 |x) = ρc0 f (x|θ) dθ + (1 − ρ)c1 f (x|θ) dθ Θ0 Θ1 ρ f (x|θ) dθ Θ0 = ρ f (x|θ) dθ + (1 − ρ)[c1 /c0 ] f (x|θ) dθ Θ0 Θ1 150 / 785
151. 151. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Monte Carlo integration Monte Carlo integration Generic problem of evaluating an integral I = Ef [h(X)] = h(x) f (x) dx X where X is uni- or multidimensional, f is a closed form, partly closed form, or implicit density, and h is a function 151 / 785
152. 152. Bayesian Core:A Practical Approach to Computational Bayesian Statistics The normal model Monte Carlo integration Monte Carlo Principle Use a sample (x1 , . . . , xm ) from the density f to approximate the integral I by the empirical average m 1 hm = h(xj ) m j=1 152 / 785