On Generalized Fiducial Inference

Transcription

1 On Generalized Fiducial Inference Jan Hannig University of North Carolina at Chapel Hill Parts of this talk are based on joint work with: Hari Iyer, Thomas C.M. Lee, Paul Patterson, Lidong E, Damian Wandler and Derek Sonderegger Colorado State University Barrett Lectures, 2009 p.1/22

2 Fiducial? Fiducial inference was mentioned only briefly during my graduate studies. I did not remember what it was about. The only think that stuck in my mind was that it is bad. Barrett Lectures, 2009 p.2/22

3 Fiducial? Fiducial inference was mentioned only briefly during my graduate studies. I did not remember what it was about. The only think that stuck in my mind was that it is bad. Oxford English Dictionary adjective TECHNICAL (of a point or line) used as a fixed basis of comparison. ORIGIN from Latin fiducia trust, confidence Merriam-Webster dictionary 1. taken as standard of reference a fiducial mark 2. founded on faith or trust 3. having the nature of a trust : FIDUCIARY Barrett Lectures, 2009 p.2/22

4 Fiducial: A Brief History goal: to construct distributions for model parameters Barrett Lectures, 2009 p.3/22

5 Fiducial: A Brief History goal: to construct distributions for model parameters introduced by Fisher (1930, 1935) Barrett Lectures, 2009 p.3/22

6 Fiducial: A Brief History goal: to construct distributions for model parameters introduced by Fisher (1930, 1935) attempt to overcome what he saw as an issue of the Bayesian approach to inference Barrett Lectures, 2009 p.3/22

7 Fiducial: A Brief History goal: to construct distributions for model parameters introduced by Fisher (1930, 1935) attempt to overcome what he saw as an issue of the Bayesian approach to inference use of a prior distribution when no prior information was available Barrett Lectures, 2009 p.3/22

8 Fiducial: A Brief History goal: to construct distributions for model parameters introduced by Fisher (1930, 1935) attempt to overcome what he saw as an issue of the Bayesian approach to inference use of a prior distribution when no prior information was available related work: Fraser (1960), Dawid and Stone (1982), Dempster (1968, 2008). Barrett Lectures, 2009 p.3/22

9 Fiducial: A Brief History goal: to construct distributions for model parameters introduced by Fisher (1930, 1935) attempt to overcome what he saw as an issue of the Bayesian approach to inference use of a prior distribution when no prior information was available related work: Fraser (1960), Dawid and Stone (1982), Dempster (1968, 2008). it is fair to say that fiducial inference failed to occupy an important place in mainstream statistics Barrett Lectures, 2009 p.3/22

10 Recent Developments Weerahandi (1989, 1993) proposed new concepts of generalized confidence intervals Barrett Lectures, 2009 p.4/22

11 Recent Developments Weerahandi (1989, 1993) proposed new concepts of generalized confidence intervals Hannig, Iyer & Patterson (2006) noted that every published generalized confidence interval was obtainable using the fiducial arguments Barrett Lectures, 2009 p.4/22

12 Recent Developments Weerahandi (1989, 1993) proposed new concepts of generalized confidence intervals Hannig, Iyer & Patterson (2006) noted that every published generalized confidence interval was obtainable using the fiducial arguments and they proved the asymptotic frequentist correctness of such intervals Barrett Lectures, 2009 p.4/22

13 Recent Developments Weerahandi (1989, 1993) proposed new concepts of generalized confidence intervals Hannig, Iyer & Patterson (2006) noted that every published generalized confidence interval was obtainable using the fiducial arguments and they proved the asymptotic frequentist correctness of such intervals Hannig (2008) have developed/modified these ideas further termed the resulting work generalized fiducial inference Barrett Lectures, 2009 p.4/22

14 What was on Fisher s mind? Barrett Lectures, 2009 p.5/22

15 What was on Fisher s mind? a Switching Principle Barrett Lectures, 2009 p.5/22

16 What was on Fisher s mind? a Switching Principle for the celebrated Maximum Likelihood method Barrett Lectures, 2009 p.5/22

17 What was on Fisher s mind? a Switching Principle for the celebrated Maximum Likelihood method density is f(x,θ), where X is variable and θ is fixed Barrett Lectures, 2009 p.5/22

18 What was on Fisher s mind? a Switching Principle for the celebrated Maximum Likelihood method density is f(x,θ), where X is variable and θ is fixed likelihood is f(x,θ), where X is fixed and θ is variable Barrett Lectures, 2009 p.5/22

19 What was on Fisher s mind? a Switching Principle for the celebrated Maximum Likelihood method density is f(x,θ), where X is variable and θ is fixed likelihood is f(x,θ), where X is fixed and θ is variable as we will see, generalized fiducial inference is also based on this idea Barrett Lectures, 2009 p.5/22

20 What was on Fisher s mind? a Switching Principle for the celebrated Maximum Likelihood method density is f(x,θ), where X is variable and θ is fixed likelihood is f(x,θ), where X is fixed and θ is variable as we will see, generalized fiducial inference is also based on this idea the switching of the roles of X and θ Barrett Lectures, 2009 p.5/22

21 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Barrett Lectures, 2009 p.6/22

22 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have 10 = µ + Z. Barrett Lectures, 2009 p.6/22

23 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have µ = 10 Z. Barrett Lectures, 2009 p.6/22

24 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have µ = 10 Z. Though the value of Z is unknown, we know the distribution of Z. Barrett Lectures, 2009 p.6/22

25 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have µ = 10 Z. Though the value of Z is unknown, we know the distribution of Z. Fiducial argument: P(µ = 3 ± dx) = P(10 Z = 3 ± dx) = P(Z = 7 ± dx) dx Barrett Lectures, 2009 p.6/22

26 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have µ = 10 Z. Though the value of Z is unknown, we know the distribution of Z. Fiducial argument: P(µ = 3 ± dx) = P(10 Z = 3 ± dx) = P(Z = 7 ± dx) dx This introduces a distribution on µ. Barrett Lectures, 2009 p.6/22

27 Simplistic Example 1 Consider X = µ + Z, where Z N(0, 1). Observe X = 10. Then we have µ = 10 Z. Though the value of Z is unknown, we know the distribution of Z. Fiducial argument: P(µ = 3 ± dx) = P(10 Z = 3 ± dx) = P(Z = 7 ± dx) dx This introduces a distribution on µ. We can simulate this distribution using R µ = 10 Z, where Z N(0, 1) independent of Z. Barrett Lectures, 2009 p.6/22

28 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Barrett Lectures, 2009 p.7/22

29 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z 1,...,µ = x n Z n. Barrett Lectures, 2009 p.7/22

30 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z1,...,µ = x n Zn. Each equation would lead to a different µ! Barrett Lectures, 2009 p.7/22

31 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z1,...,µ = x n Zn. Each equation would lead to a different µ! Need to condition the distribution of (Z1,...,Z n) on the event that all the equations have the same µ. Barrett Lectures, 2009 p.7/22

32 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z1,...,µ = x n Zn. Each equation would lead to a different µ! Need to condition the distribution of (Z1,...,Z n) on the event that all the equations have the same µ. The fiducial distribution can be defined as x 1 Z1 x 2 = x 1 Z1 + Z2,...,x n = x 1 Z1 + Zn. Barrett Lectures, 2009 p.7/22

33 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z1,...,µ = x n Zn. Each equation would lead to a different µ! Need to condition the distribution of (Z1,...,Z n) on the event that all the equations have the same µ. The fiducial distribution can be defined as x 1 Z1 x 2 = x 1 Z1 + Z2,...,x n = x 1 Z1 + Zn. After simplification the fiducial distribution is N( x, 1/n). Barrett Lectures, 2009 p.7/22

34 Simplistic Example 2 Consider X i = µ + Z i where Z i are i.i.d. N(0, 1). Observe (x 1,...,x n ). We cannot simply follow the previous idea of setting µ = x 1 Z1,...,µ = x n Zn. Each equation would lead to a different µ! Need to condition the distribution of (Z1,...,Z n) on the event that all the equations have the same µ. The fiducial distribution can be defined as x 1 Z1 x 2 = x 1 Z1 + Z2,...,x n = x 1 Z1 + Zn. After simplification the fiducial distribution is N( x, 1/n). We have non-uniqueness due to Borel paradox. Barrett Lectures, 2009 p.7/22

35 Example 3 Fat data Borel paradox was caused by the fact that probability of observing our data could be 0. Barrett Lectures, 2009 p.8/22

36 Example 3 Fat data Borel paradox was caused by the fact that probability of observing our data could be 0. Due to instrument limitations we never observe our data exactly. Barrett Lectures, 2009 p.8/22

37 Example 3 Fat data Borel paradox was caused by the fact that probability of observing our data could be 0. Due to instrument limitations we never observe our data exactly. We only know that they are in some interval, i.e., we do not observe x i = π but rather < x i < Barrett Lectures, 2009 p.8/22

38 Example 3 Fat data Borel paradox was caused by the fact that probability of observing our data could be 0. Due to instrument limitations we never observe our data exactly. We only know that they are in some interval, i.e., we do not observe x i = π but rather < x i < Let X = µ + σz. Barrett Lectures, 2009 p.8/22

39 Example 3 Fat data Borel paradox was caused by the fact that probability of observing our data could be 0. Due to instrument limitations we never observe our data exactly. We only know that they are in some interval, i.e., we do not observe x i = π but rather < x i < Let X = µ + σz. If we observe a i < X i < b i we need to generate Z keeping only those values that agree with a i < µ + σz i < b i for all i. Barrett Lectures, 2009 p.8/22

40 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < 0.5. Barrett Lectures, 2009 p.9/22

41 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = Barrett Lectures, 2009 p.9/22

42 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = , Z 2 = Barrett Lectures, 2009 p.9/22

43 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = , Z 2 = , Z 3 = Barrett Lectures, 2009 p.9/22

44 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = Barrett Lectures, 2009 p.9/22

45 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = , Z 2 = Barrett Lectures, 2009 p.9/22

46 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = , Z 2 = , Z 3 = Barrett Lectures, 2009 p.9/22

47 Example 3 Fat data Say we observe 2.0 < X 1 < 2.1, 0.6 < X 2 < 0.7, 0.4 < X 3 < mu sigma Z 1 = , Z 2 = , Z 3 = Denote the intersection by Q. Barrett Lectures, 2009 p.9/22

48 Example 3 Fat data Set Q((a,b),u) = {(µ,σ) : a i < µ + σz i < b i }. Barrett Lectures, 2009 p.10/22

49 Example 3 Fat data Set Q((a,b),u) = {(µ,σ) : a i < µ + σz i < b i }. The fiducial distribution could be defined as Q((a,b),Z ) {Q((a,b),Z ) } Barrett Lectures, 2009 p.10/22

50 Example 3 Fat data Set Q((a,b),u) = {(µ,σ) : a i < µ + σz i < b i }. The fiducial distribution could be defined as Q((a,b),Z ) {Q((a,b),Z ) } P(Q ) P(X (a,b)) > 0, so there is no Borel paradox in the definition of fiducial distribution (1). Barrett Lectures, 2009 p.10/22

51 Example 3 Fat data Set Q((a,b),u) = {(µ,σ) : a i < µ + σz i < b i }. The fiducial distribution could be defined as Q((a,b),Z ) {Q((a,b),Z ) } P(Q ) P(X (a,b)) > 0, so there is no Borel paradox in the definition of fiducial distribution (1). Q typically contains more than one element. We can either use Dempster-Shafer calculus to interpret its meaning, or additionally choose (randomly) an element from Q. Barrett Lectures, 2009 p.10/22

52 Example 3 Fat Data Let X = µ + σz. We observe (2.0,2.1), (0.6,0.7), (0.4, 0.5), (1.4,1.5), (0.7,0.8), (0.8,0.9), (1.2,1.3), (1.2,1.3), (1.1,1.2), (1.5,1.6), (1.4,1.5), (0.4,0.5), (1.2,1.3), (0.7,0.8), (0.5,0.6). Barrett Lectures, 2009 p.11/22

53 Example 3 Fat Data Let X = µ + σz. We observe (2.0,2.1), (0.6,0.7), (0.4, 0.5), (1.4,1.5), (0.7,0.8), (0.8,0.9), (1.2,1.3), (1.2,1.3), (1.1,1.2), (1.5,1.6), (1.4,1.5), (0.4,0.5), (1.2,1.3), (0.7,0.8), (0.5,0.6). The exact distribution of Q((a,b),Z ) {Q((a,b),Z ) } is complicated. We use Gibbs sampler to get a sample from this distribution. Barrett Lectures, 2009 p.11/22

54 Example 3 Fat Data Let X = µ + σz. We observe (2.0,2.1), (0.6,0.7), (0.4, 0.5), (1.4,1.5), (0.7,0.8), (0.8,0.9), (1.2,1.3), (1.2,1.3), (1.1,1.2), (1.5,1.6), (1.4,1.5), (0.4,0.5), (1.2,1.3), (0.7,0.8), (0.5,0.6). The exact distribution of Q((a,b),Z ) {Q((a,b),Z ) } is complicated. We use Gibbs sampler to get a sample from this distribution. Each Q is a polygon. When sampling an element of Q we take a random vertex. Barrett Lectures, 2009 p.11/22

55 Example 3 Fat Data Let X = µ + σz. We observe (2.0,2.1), (0.6,0.7), (0.4, 0.5), (1.4,1.5), (0.7,0.8), (0.8,0.9), (1.2,1.3), (1.2,1.3), (1.1,1.2), (1.5,1.6), (1.4,1.5), (0.4,0.5), (1.2,1.3), (0.7,0.8), (0.5,0.6). The exact distribution of Q((a,b),Z ) {Q((a,b),Z ) } is complicated. We use Gibbs sampler to get a sample from this distribution. Each Q is a polygon. When sampling an element of Q we take a random vertex. Notice that this approach does not assume that the true value of X is uniform in the observed interval! Barrett Lectures, 2009 p.11/22

56 Example Fat Data A sample from Q((a,b),Z ) {Q((a,b),Z ) }. Barrett Lectures, 2009 p.12/22

57 Example Fat Data A sample from Q((a,b),Z ) {Q((a,b),Z ) } mu sigma 20 observations. Barrett Lectures, 2009 p.12/22

58 Example Fat Data A sample from Q((a,b),Z ) {Q((a,b),Z ) } mu sigma 20 observations, final sampled value shown. Barrett Lectures, 2009 p.12/22

59 Example Fat Data A sample from Q((a,b),Z ) {Q((a,b),Z ) } mu sigma 200 observations, final sampled value shown. Barrett Lectures, 2009 p.12/22

60 Example Fat Data A sample from Q((a,b),Z ) {Q((a,b),Z ) } mu sigma green sample from usual fiducial computed with fully known observation. Barrett Lectures, 2009 p.12/22

61 Questions What is the distribution of Q((a,b),Z ) {Q((a,b),Z ) }? Issues Barrett Lectures, 2009 p.13/22

62 Questions What is the distribution of Q((a,b),Z ) {Q((a,b),Z ) }? Issues Random environment: The observations X are random. But only partially observed. Barrett Lectures, 2009 p.13/22

63 Questions What is the distribution of Q((a,b),Z ) {Q((a,b),Z ) }? Issues Random environment: The observations X are random. But only partially observed. The set we condition on has an extremely small probability. Barrett Lectures, 2009 p.13/22

64 Questions What is the distribution of Q((a,b),Z ) {Q((a,b),Z ) }? Issues Random environment: The observations X are random. But only partially observed. The set we condition on has an extremely small probability. Geometry is complicated. Barrett Lectures, 2009 p.13/22

65 Questions What is the distribution of Q((a,b),Z ) {Q((a,b),Z ) }? Issues Random environment: The observations X are random. But only partially observed. The set we condition on has an extremely small probability. Geometry is complicated. It is an intersection of large number of random, dependent parallelograms. However it has typically surprisingly low number of vertexes. Barrett Lectures, 2009 p.13/22

66 What can we do? Let d = (d 1, d 2 ) S 2 and define Q d ((a,b),z ) the most extreme point along the direction d. (It is one of the vertexes a.s.) Barrett Lectures, 2009 p.14/22

67 What can we do? Let d = (d 1, d 2 ) S 2 and define Q d ((a,b),z ) the most extreme point along the direction d. (It is one of the vertexes a.s.) The distribution of Q d ((a,b),z ) {Q((a,b),Z ) } is proportional to X i<j c ij i c ij j σ 3 φ cij i µ σ! φ cij j µ σ! Y k i,j «««bi µ ai µ Φ Φ σ σ c ij i is either a i or b i depending on d. Barrett Lectures, 2009 p.14/22

68 What can we do? Let d = (d 1, d 2 ) S 2 and define Q d ((a,b),z ) the most extreme point along the direction d. (It is one of the vertexes a.s.) The distribution of Q d ((a,b),z ) {Q((a,b),Z ) } is proportional to X i<j c ij i c ij j σ 3 φ cij i µ σ! φ cij j µ σ! Y k i,j «««bi µ ai µ Φ Φ σ σ c ij i is either a i or b i depending on d. This allows us to compute Find a limit as b a 0. (More on this later. 2) Barrett Lectures, 2009 p.14/22

69 What can we do? Let d = (d 1, d 2 ) S 2 and define Q d ((a,b),z ) the most extreme point along the direction d. (It is one of the vertexes a.s.) The distribution of Q d ((a,b),z ) {Q((a,b),Z ) } is proportional to X i<j c ij i c ij j σ 3 φ cij i µ σ! φ cij j µ σ! Y k i,j «««bi µ ai µ Φ Φ σ σ c ij i is either a i or b i depending on d. This allows us to compute Find a limit as b a 0. (More on this later. 2) Find a limit as n. (It is consistent and asymptotically normal). Barrett Lectures, 2009 p.14/22

70 What can we do? Let d = (d 1, d 2 ) S 2 and define Q d ((a,b),z ) the most extreme point along the direction d. (It is one of the vertexes a.s.) The distribution of Q d ((a,b),z ) {Q((a,b),Z ) } is proportional to X i<j c ij i c ij j σ 3 φ cij i µ σ! φ cij j µ σ! Y k i,j «««bi µ ai µ Φ Φ σ σ c ij i is either a i or b i depending on d. This allows us to compute Find a limit as b a 0. (More on this later. 2) Find a limit as n. (It is consistent and asymptotically normal). Would love to know n(q Q d ) {Q } Barrett Lectures, 2009 p.14/22

71 Generalized Fiducial Recipe Let X be a random vector with a distribution indexed by a parameter ξ R p. Assume that X = G(U,ξ), where U has some known distribution independent of parameters, e.g, U U(0, 1). Barrett Lectures, 2009 p.15/22

72 Generalized Fiducial Recipe Let X be a random vector with a distribution indexed by a parameter ξ R p. Assume that X = G(U,ξ), where U has some known distribution independent of parameters, e.g, U U(0, 1). Define a set-valued function Q(A,u) = {ξ : G(u,ξ) A}. The function Q(A,U) is an inverse of the function G. Barrett Lectures, 2009 p.15/22

73 Generalized Fiducial Recipe Let X be a random vector with a distribution indexed by a parameter ξ R p. Assume that X = G(U,ξ), where U has some known distribution independent of parameters, e.g, U U(0, 1). Define a set-valued function Q(A,u) = {ξ : G(u,ξ) A}. The function Q(A,U) is an inverse of the function G. Assume that for any measurable S there is a random variable V (S) with support S. Barrett Lectures, 2009 p.15/22

74 Generalized Fiducial Recipe Based on X = G(U,ξ) define a generalized fiducial distribution as the conditional distribution of V (Q(A,U )) { Q(A,U ) }. (1) Here X A is the observed data and U D = U is a random variable independent of X. Barrett Lectures, 2009 p.16/22

75 Generalized Fiducial Recipe Based on X = G(U,ξ) define a generalized fiducial distribution as the conditional distribution of V (Q(A,U )) { Q(A,U ) }. (1) Here X A is the observed data and U = D U is a random variable independent of X. Let R ξ (A) be a random variable with distribution (1). If θ = π(ξ) is of interest use R θ = π(r ξ ). We will call these GFQs. Barrett Lectures, 2009 p.16/22

76 Remarks There are three sources of non-uniquness in (1). Barrett Lectures, 2009 p.17/22

77 Remarks There are three sources of non-uniquness in (1). The choice of structural equation. Barrett Lectures, 2009 p.17/22

78 Remarks There are three sources of non-uniquness in (1). The choice of structural equation. The choice of V (S): Arises if the inverse Q(A, U ) has more then one element. Barrett Lectures, 2009 p.17/22

79 Remarks There are three sources of non-uniquness in (1). The choice of structural equation. The choice of V (S): Arises if the inverse Q(A, U ) has more then one element. The conditioning on Q(A, U ) }: Arises if P {Q(A, U ) } = 0. Barrett Lectures, 2009 p.17/22

80 Remarks There are three sources of non-uniquness in (1). The choice of structural equation. The choice of V (S): Arises if the inverse Q(A, U ) has more then one element. The conditioning on Q(A, U ) }: Arises if P {Q(A, U ) } = 0. This is caused by Borel paradox. Barrett Lectures, 2009 p.17/22

81 Remarks There are three sources of non-uniquness in (1). The choice of structural equation. The choice of V (S): Arises if the inverse Q(A, U ) has more then one element. The conditioning on Q(A, U ) }: Arises if P {Q(A, U ) } = 0. This is caused by Borel paradox. Under suitable conditions the fiducial distribution leads to procedures with asymptotically correct frequentist properties. Barrett Lectures, 2009 p.17/22

82 Limit b a 0 Assume: The model parameter ξ is p-dimensional; Barrett Lectures, 2009 p.18/22

83 Limit b a 0 Assume: The model parameter ξ is p-dimensional; the structural equation factorizes as X 0 = G 0 (ξ, E 0 ) R p and X c = G c (ξ, E c ); Barrett Lectures, 2009 p.18/22

84 Limit b a 0 Assume: The model parameter ξ is p-dimensional; the structural equation factorizes as X 0 = G 0 (ξ, E 0 ) R p and X c = G c (ξ, E c ); G 0 (ξ, ) and G c(ξ, ) are one-to-one and differentiable, G 1 0 (x ( 0, ξ) denotes the inverse; G 0,e 0) is one-to-one and differentiable. Barrett Lectures, 2009 p.18/22

85 Limit b a 0 Assume: The model parameter ξ is p-dimensional; the structural equation factorizes as X 0 = G 0 (ξ, E 0 ) R p and X c = G c (ξ, E c ); G 0 (ξ, ) and G c(ξ, ) are one-to-one and differentiable, G 1 0 (x ( 0, ξ) denotes the inverse; G 0,e 0) is one-to-one and differentiable. The generalized fiducial distribution is then calculated to be r(ξ x) = f X (x ξ)j(x, ξ) Ξ f X(x ξ )J(x, ξ ) dξ, (2) where J(x, ξ) = ( ) n 1 p i det ( d dξ G 1 det d dx i G 1 i (x i,ξ)) i (x i,ξ). Barrett Lectures, 2009 p.18/22

86 Why does it work asymptotically? The reason why generalized fiducial inference works asymptotically in frequentist sense is very similar to the reason why Bayesian inference works Bernstein - von Mises theorem Barrett Lectures, 2009 p.19/22

87 Why does it work asymptotically? The reason why generalized fiducial inference works asymptotically in frequentist sense is very similar to the reason why Bayesian inference works Bernstein - von Mises theorem Roughly speaking, there is a centering T such that conditionally on the data X = x the generalized fiducial quantity R θ N(T(x), σ 2 n). Barrett Lectures, 2009 p.19/22

88 Why does it work asymptotically? The reason why generalized fiducial inference works asymptotically in frequentist sense is very similar to the reason why Bayesian inference works Bernstein - von Mises theorem Roughly speaking, there is a centering T such that conditionally on the data X = x the generalized fiducial quantity R θ N(T(x), σ 2 n). Moreover unconditionally T(X) N(θ, σ 2 n). Barrett Lectures, 2009 p.19/22

89 Why does it work asymptotically? The reason why generalized fiducial inference works asymptotically in frequentist sense is very similar to the reason why Bayesian inference works Bernstein - von Mises theorem Roughly speaking, there is a centering T such that conditionally on the data X = x the generalized fiducial quantity R θ N(T(x), σ 2 n). Moreover unconditionally T(X) N(θ, σ 2 n). The lower CI is approximately (, T(x) + z α σ n ). The coverage of this CI is approximately P (θ < T(X) + z α σ n ) = P ( z α σ n < T(X) θ) = α. Barrett Lectures, 2009 p.19/22

90 Why does it work asymptotically? Theorem (Hannig, 2007). Assume that J(x, ) is continuous in θ, π(θ) = E θ0 J(X, θ) is finite, π(θ 0 > 0, and on some neighborhood of ) θ 0 E θ0 (sup θ (θ0 δ 0,θ 0 +δ 0 ) J(X, θ) <. Then under regularity conditions Z R s2 r(θ,x) e 2/I(θ 0 ) p 2π/I(θ0 ) dθ P θ 0 0. Barrett Lectures, 2009 p.20/22

91 Why does it work asymptotically? Theorem (Hannig, 2007). Assume that J(x, ) is continuous in θ, π(θ) = E θ0 J(X, θ) is finite, π(θ 0 > 0, and on some neighborhood of ) θ 0 E θ0 (sup θ (θ0 δ 0,θ 0 +δ 0 ) J(X, θ) <. Then under regularity conditions Z R s2 r(θ,x) e 2/I(θ 0 ) p 2π/I(θ0 ) dθ P θ 0 0. The rough idea of the proof is to show that J(x, θ) π(θ) uniformly and use Bernstein-von Mises theorem for Bayesian posterior. Barrett Lectures, 2009 p.20/22

92 Why does it work asymptotically? Theorem (Hannig, 2007). Assume that J(x, ) is continuous in θ, π(θ) = E θ0 J(X, θ) is finite, π(θ 0 > 0, and on some neighborhood of ) θ 0 E θ0 (sup θ (θ0 δ 0,θ 0 +δ 0 ) J(X, θ) <. Then under regularity conditions Z R s2 r(θ,x) e 2/I(θ 0 ) p 2π/I(θ0 ) dθ P θ 0 0. The rough idea of the proof is to show that J(x, θ) π(θ) uniformly and use Bernstein-von Mises theorem for Bayesian posterior. There is a technical problem caused by the fact that π(θ) is typically improper. Barrett Lectures, 2009 p.20/22

93 Concluding Remarks Generalized fiducial distributions lead often to attractive solution with asymptotically correct frequentist coverage. Barrett Lectures, 2009 p.21/22

94 Concluding Remarks Generalized fiducial distributions lead often to attractive solution with asymptotically correct frequentist coverage. Many simulation studies show that generalized fiducial solutions have very good small sample properties. Barrett Lectures, 2009 p.21/22

95 Concluding Remarks Generalized fiducial distributions lead often to attractive solution with asymptotically correct frequentist coverage. Many simulation studies show that generalized fiducial solutions have very good small sample properties. Current popularity of generalized inference in some applied circles suggests that if computers were available 70 years ago, fiducial inference might not have been rejected. Barrett Lectures, 2009 p.21/22

96 Quotes Zabell (1992) Fiducial inference stands as R. A. Fisher s one great failure. Efron (1998) Maybe Fisher s biggest blunder will become a big hit in the 21st century! " Barrett Lectures, 2009 p.22/22

97 Quotes Zabell (1992) Fiducial inference stands as R. A. Fisher s one great failure. Efron (1998) Maybe Fisher s biggest blunder will become a big hit in the 21st century! " Barrett Lectures, 2009 p.22/22