DISCUSSION                    ofBayesian Computation via empirical likelihood     Stefano Cabras, stefano.cabras@uc3m.es  ...
Summary   ◮   Problem:
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;   ◮   we want to obtain t...
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;   ◮   we want to obtain t...
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;   ◮   we want to obtain t...
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;   ◮   we want to obtain t...
Summary   ◮   Problem:         ◮   a statistical model f (y | θ);         ◮   a prior π(θ) on θ;   ◮   we want to obtain t...
... what remains about the f (y | θ) ?
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... what remains about the f (y | θ) ?     ◮   Recall that the Empirical Likelihood is defined, for iid sample,         by ...
... a suggestion               What if we do not even known h(·) ?
... how to elicit h(·) automatically
... how to elicit h(·) automatically
... how to elicit h(·) automatically     ◮   Set h(Y , θ) = Y − g (θ), where                              g (θ) = Ef (y |θ...
... how to elicit h(·) automatically     ◮   Set h(Y , θ) = Y − g (θ), where                              g (θ) = Ef (y |θ...
How to estimate g (θ) ?      1       ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras,   Castellanos, ...
How to estimate g (θ) ?     ◮    Use a once forever pilot-run simulation study:      1      1       ... similar to Fearnhe...
How to estimate g (θ) ?     ◮    Use a once forever pilot-run simulation study:      1    1. Consider a grid (or regular l...
How to estimate g (θ) ?     ◮    Use a once forever pilot-run simulation study:      1    1. Consider a grid (or regular l...
How to estimate g (θ) ?     ◮    Use a once forever pilot-run simulation study:       1    1. Consider a grid (or regular ...
... example: y ∼ N(|θ|, 1)   For a pilot run of M = 1000 we have g (θ) = |θ|.                                       ˆ     ...
... example: y ∼ N(|θ|, 1)   Suppose to draw a n = 100 sample from θ = 2:                             Histogram of y      ...
... example: y ∼ N(|θ|, 1)   The Empirical Likelihood is this               2.5               2.0   Emp. Lik.             ...
1st Point: Do we need necessarily have to use f (y | θ) ?
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal;
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal; ...
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal; ...
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal; ...
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal; ...
1st Point: Do we need necessarily have to use f (y | θ) ?     ◮   The above data maybe drawn from a (e.g.) a Half Normal; ...
2nd Point: Sample free vs Simulation free
2nd Point: Sample free vs Simulation free     ◮   The Empirical Likelihood is ”simulation free” but not ”sample         fr...
2nd Point: Sample free vs Simulation free     ◮   The Empirical Likelihood is ”simulation free” but not ”sample         fr...
2nd Point: Sample free vs Simulation free     ◮   The Empirical Likelihood is ”simulation free” but not ”sample         fr...
2nd Point: Sample free vs Simulation free     ◮   The Empirical Likelihood is ”simulation free” but not ”sample         fr...
2nd Point: Sample free vs Simulation free     ◮   The Empirical Likelihood is ”simulation free” but not ”sample         fr...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   The use of pseudo-likelihoods is not new in...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   Monahan & Boos (Biometrika, 1992) proposed ...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   Monahan & Boos (Biometrika, 1992) proposed ...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   Monahan & Boos (Biometrika, 1992) proposed ...
3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?    ◮   Monahan & Boos (Biometrika, 1992) proposed ...
... Last point: the ABC is still a terrific tool
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:           ◮   Statistical Journals;
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:           ◮   Statistical Journals;      ...
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:           ◮   Statistical Journals;      ...
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:           ◮   Statistical Journals;      ...
... Last point: the ABC is still a terrific tool     ◮   ... a lot of references:           ◮   Statistical Journals;      ...
Discussion of ABC talk by Stefano Cabras, Padova, March 21, 2013
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Discussion of ABC talk by Stefano Cabras, Padova, March 21, 2013

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This discussion was given after my talk by Stefano Cabras, at the Padova workshop on recent advances in statistical inference

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Discussion of ABC talk by Stefano Cabras, Padova, March 21, 2013

  1. 1. DISCUSSION ofBayesian Computation via empirical likelihood Stefano Cabras, stefano.cabras@uc3m.es Universidad Carlos III de Madrid (Spain) Universit` di Cagliari (Italy) a Padova, 21-Mar-2013
  2. 2. Summary ◮ Problem:
  3. 3. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ;
  4. 4. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ; ◮ we want to obtain the posterior πN (θ | y ) ∝ LN (θ)π(θ).
  5. 5. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ; ◮ we want to obtain the posterior πN (θ | y ) ∝ LN (θ)π(θ). ◮ BUT
  6. 6. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ; ◮ we want to obtain the posterior πN (θ | y ) ∝ LN (θ)π(θ). ◮ BUT ◮ IF LN (θ) is not available: ◮ THEN all life ABC;
  7. 7. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ; ◮ we want to obtain the posterior πN (θ | y ) ∝ LN (θ)π(θ). ◮ BUT ◮ IF LN (θ) is not available: ◮ THEN all life ABC; ◮ IF it is not even possible to simulate from f (y | θ):
  8. 8. Summary ◮ Problem: ◮ a statistical model f (y | θ); ◮ a prior π(θ) on θ; ◮ we want to obtain the posterior πN (θ | y ) ∝ LN (θ)π(θ). ◮ BUT ◮ IF LN (θ) is not available: ◮ THEN all life ABC; ◮ IF it is not even possible to simulate from f (y | θ): ◮ THEN replace LN (θ) with LEL (θ) (the proposed BCel procedure): π(θ|y ) ∝ LEL (θ) × π(θ). .
  9. 9. ... what remains about the f (y | θ) ?
  10. 10. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0.
  11. 11. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0. ◮ The relation between θ and obs. Y is model conditioned and expressed by h(Y , θ);
  12. 12. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0. ◮ The relation between θ and obs. Y is model conditioned and expressed by h(Y , θ); ◮ Constraints are model driven and so there is still a timid trace of f (y | θ) in BCel .
  13. 13. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0. ◮ The relation between θ and obs. Y is model conditioned and expressed by h(Y , θ); ◮ Constraints are model driven and so there is still a timid trace of f (y | θ) in BCel . ◮ Examples:
  14. 14. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0. ◮ The relation between θ and obs. Y is model conditioned and expressed by h(Y , θ); ◮ Constraints are model driven and so there is still a timid trace of f (y | θ) in BCel . ◮ Examples: ◮ The coalescent model example is illuminating in suggesting the score of the pairwise likelihood;
  15. 15. ... what remains about the f (y | θ) ? ◮ Recall that the Empirical Likelihood is defined, for iid sample, by means of a set of constraints: Ef (y |θ) [h(Y , θ)] = 0. ◮ The relation between θ and obs. Y is model conditioned and expressed by h(Y , θ); ◮ Constraints are model driven and so there is still a timid trace of f (y | θ) in BCel . ◮ Examples: ◮ The coalescent model example is illuminating in suggesting the score of the pairwise likelihood; ◮ The residuals in GARCH models.
  16. 16. ... a suggestion What if we do not even known h(·) ?
  17. 17. ... how to elicit h(·) automatically
  18. 18. ... how to elicit h(·) automatically
  19. 19. ... how to elicit h(·) automatically ◮ Set h(Y , θ) = Y − g (θ), where g (θ) = Ef (y |θ) (Y |θ), is the regression function of Y |θ;
  20. 20. ... how to elicit h(·) automatically ◮ Set h(Y , θ) = Y − g (θ), where g (θ) = Ef (y |θ) (Y |θ), is the regression function of Y |θ; ◮ g (θ) should be replaced by an estimator g (θ).
  21. 21. How to estimate g (θ) ? 1 ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras, Castellanos, Ruli (Ercim-2012, Oviedo).
  22. 22. How to estimate g (θ) ? ◮ Use a once forever pilot-run simulation study: 1 1 ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras, Castellanos, Ruli (Ercim-2012, Oviedo).
  23. 23. How to estimate g (θ) ? ◮ Use a once forever pilot-run simulation study: 1 1. Consider a grid (or regular lattice) of θ made by M points: θ1 , . . . , θM 1 ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras, Castellanos, Ruli (Ercim-2012, Oviedo).
  24. 24. How to estimate g (θ) ? ◮ Use a once forever pilot-run simulation study: 1 1. Consider a grid (or regular lattice) of θ made by M points: θ1 , . . . , θM 2. Simulate the corresponding y1 , . . . , yM 1 ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras, Castellanos, Ruli (Ercim-2012, Oviedo).
  25. 25. How to estimate g (θ) ? ◮ Use a once forever pilot-run simulation study: 1 1. Consider a grid (or regular lattice) of θ made by M points: θ1 , . . . , θM 2. Simulate the corresponding y1 , . . . , yM 3. Regress y1 , . . . , yM on θ 1 , . . . , θ M obtaining g (θ). 1 ... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras, Castellanos, Ruli (Ercim-2012, Oviedo).
  26. 26. ... example: y ∼ N(|θ|, 1) For a pilot run of M = 1000 we have g (θ) = |θ|. ˆ Pilot−Run s.s. g(θ) 10 y 5 0 −10 −5 0 5 10 θ
  27. 27. ... example: y ∼ N(|θ|, 1) Suppose to draw a n = 100 sample from θ = 2: Histogram of y 20 15 Frequency 10 5 0 0 1 2 3 4 y
  28. 28. ... example: y ∼ N(|θ|, 1) The Empirical Likelihood is this 2.5 2.0 Emp. Lik. 1.5 1.0 −4 −2 0 2 4 θ
  29. 29. 1st Point: Do we need necessarily have to use f (y | θ) ?
  30. 30. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal;
  31. 31. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal; ◮ How this is reflected in the BCel ?
  32. 32. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal; ◮ How this is reflected in the BCel ? ◮ For a given data y;
  33. 33. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal; ◮ How this is reflected in the BCel ? ◮ For a given data y; ◮ and h(Y , θ) fixed;
  34. 34. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal; ◮ How this is reflected in the BCel ? ◮ For a given data y; ◮ and h(Y , θ) fixed; ◮ the LEL (θ) is the same regardless of f (y | θ).
  35. 35. 1st Point: Do we need necessarily have to use f (y | θ) ? ◮ The above data maybe drawn from a (e.g.) a Half Normal; ◮ How this is reflected in the BCel ? ◮ For a given data y; ◮ and h(Y , θ) fixed; ◮ the LEL (θ) is the same regardless of f (y | θ). Can we ignore f (y | θ) ?
  36. 36. 2nd Point: Sample free vs Simulation free
  37. 37. 2nd Point: Sample free vs Simulation free ◮ The Empirical Likelihood is ”simulation free” but not ”sample free”, i.e.
  38. 38. 2nd Point: Sample free vs Simulation free ◮ The Empirical Likelihood is ”simulation free” but not ”sample free”, i.e. ◮ LEL (θ) → LN (θ) for n → ∞, ◮ implying π(θ|y) → πN (θ | y ) asymptotically in n.
  39. 39. 2nd Point: Sample free vs Simulation free ◮ The Empirical Likelihood is ”simulation free” but not ”sample free”, i.e. ◮ LEL (θ) → LN (θ) for n → ∞, ◮ implying π(θ|y) → πN (θ | y ) asymptotically in n. ◮ The ABC is ”sample free” but not ”simulation free”, i.e.
  40. 40. 2nd Point: Sample free vs Simulation free ◮ The Empirical Likelihood is ”simulation free” but not ”sample free”, i.e. ◮ LEL (θ) → LN (θ) for n → ∞, ◮ implying π(θ|y) → πN (θ | y ) asymptotically in n. ◮ The ABC is ”sample free” but not ”simulation free”, i.e. ◮ π(θ|ρ(s(y), so bs) < ǫ) → πN (θ | y ) as ǫ → 0 ◮ implying convergence in the number of simulations if s(y ) were sufficient.
  41. 41. 2nd Point: Sample free vs Simulation free ◮ The Empirical Likelihood is ”simulation free” but not ”sample free”, i.e. ◮ LEL (θ) → LN (θ) for n → ∞, ◮ implying π(θ|y) → πN (θ | y ) asymptotically in n. ◮ The ABC is ”sample free” but not ”simulation free”, i.e. ◮ π(θ|ρ(s(y), so bs) < ǫ) → πN (θ | y ) as ǫ → 0 ◮ implying convergence in the number of simulations if s(y ) were sufficient. A quick answer recommends use BCel BUT a small sample would recommend ABC ?
  42. 42. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ?
  43. 43. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting:
  44. 44. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods:
  45. 45. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) ◮ Mengersen et al. (PNAS, 2012) ◮ ...
  46. 46. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) ◮ Mengersen et al. (PNAS, 2012) ◮ ... ◮ Modified-Likelihoods:
  47. 47. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) ◮ Mengersen et al. (PNAS, 2012) ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ...
  48. 48. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) ◮ Mengersen et al. (PNAS, 2012) ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ... ◮ Quasi-Likelihoods:
  49. 49. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) ◮ Mengersen et al. (PNAS, 2012) ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) ◮ Greco et al. (JSPI, 2008) ◮ Ventura et al. (JSPI, 2010) ◮ ...
  50. 50. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) : examples and coverages of C.I. ◮ Mengersen et al. (PNAS, 2012) ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) ◮ Greco et al. (JSPI, 2008) ◮ Ventura et al. (JSPI, 2010) ◮ ...
  51. 51. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) : examples and coverages of C.I. ◮ Mengersen et al. (PNAS, 2012) : examples and coverages of C.I. ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) ◮ Greco et al. (JSPI, 2008) ◮ Ventura et al. (JSPI, 2010) ◮ ...
  52. 52. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) : examples and coverages of C.I. ◮ Mengersen et al. (PNAS, 2012) : examples and coverages of C.I. ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) : second order matching properties; ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) ◮ Greco et al. (JSPI, 2008) ◮ Ventura et al. (JSPI, 2010) ◮ ...
  53. 53. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) : examples and coverages of C.I. ◮ Mengersen et al. (PNAS, 2012) : examples and coverages of C.I. ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) : second order matching properties; ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) : examples; ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) ◮ Greco et al. (JSPI, 2008) ◮ Ventura et al. (JSPI, 2010) ◮ ...
  54. 54. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ The use of pseudo-likelihoods is not new in the Bayesian setting: ◮ Empirical Likelihoods: ◮ Lazar (Biometrika, 2003) : examples and coverages of C.I. ◮ Mengersen et al. (PNAS, 2012) : examples and coverages of C.I. ◮ ... ◮ Modified-Likelihoods: ◮ Ventura et al. (JASA, 2009) : second order matching properties; ◮ Chang and Mukerjee (Stat. & Prob. Letters 2006) : examples; ◮ ... ◮ Quasi-Likelihoods: ◮ Lin (Statist. Methodol., 2006) : examples; ◮ Greco et al. (JSPI, 2008) : robustness properties; ◮ Ventura et al. (JSPI, 2010) : examples and coverages of C.I.; ◮ ...
  55. 55. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ Monahan & Boos (Biometrika, 1992) proposed a notion of validity:
  56. 56. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ Monahan & Boos (Biometrika, 1992) proposed a notion of validity: π(θ|y ) should obey the laws of probability in a fashion that is consistent with statements derived from Bayes’rule.
  57. 57. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ Monahan & Boos (Biometrika, 1992) proposed a notion of validity: π(θ|y ) should obey the laws of probability in a fashion that is consistent with statements derived from Bayes’rule. ◮ Very difficult!
  58. 58. 3nd Point: How to validate a pseudo-posteriorπ(θ|y ) ∝ LEL (θ) × π(θ) ? ◮ Monahan & Boos (Biometrika, 1992) proposed a notion of validity: π(θ|y ) should obey the laws of probability in a fashion that is consistent with statements derived from Bayes’rule. ◮ Very difficult! How to validate the pseudo-posterior π(θ|y ) when this is not possible ?
  59. 59. ... Last point: the ABC is still a terrific tool
  60. 60. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references:
  61. 61. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references: ◮ Statistical Journals;
  62. 62. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references: ◮ Statistical Journals; ◮ Twitter;
  63. 63. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references: ◮ Statistical Journals; ◮ Twitter; ◮ Xiang’s blog ( xianblog.wordpress.com )
  64. 64. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references: ◮ Statistical Journals; ◮ Twitter; ◮ Xiang’s blog ( xianblog.wordpress.com ) ◮ ... it is tailored to Approximate LN (θ).
  65. 65. ... Last point: the ABC is still a terrific tool ◮ ... a lot of references: ◮ Statistical Journals; ◮ Twitter; ◮ Xiang’s blog ( xianblog.wordpress.com ) ◮ ... it is tailored to Approximate LN (θ). Where is the A in BCel ?

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