The document discusses Bayesian computation using empirical likelihood, focusing on the challenges of obtaining posterior distributions when certain information, such as the natural logarithm of the likelihood, is not available. It introduces a proposed method, the Bayesian computation via empirical likelihood (BCEL), which utilizes empirical likelihoods to define the relationship between observational data and parameters. Additionally, it highlights the need for validation of the pseudo-posterior and references several studies to support its methodology.

1. DISCUSSION
of
Bayesian 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. Summary
◮ Problem:

3. Summary
◮ Problem:
◮ a statistical model f (y | θ);
◮ a prior π(θ) on θ;

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. 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. 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. 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. 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. ... what remains about the f (y | θ) ?

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. ... 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. ... 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. ... 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. ... 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. ... 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. ... a suggestion
What if we do not even known h(·) ?

17. ... how to elicit h(·) automatically

18. ... how to elicit h(·) automatically

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. ... 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. How to estimate g (θ) ?
1
... similar to Fearnhead, P. and D. Prangle (JRRS-B, 2012) or Cabras,
Castellanos, Ruli (Ercim-2012, Oviedo).

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. 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. 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. 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. ... 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. ... 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. ... 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. 1st Point: Do we need necessarily have to use f (y | θ) ?

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. 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. 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. 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. 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. 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. 2nd Point: Sample free vs Simulation free

37. 2nd Point: Sample free vs Simulation free
◮ The Empirical Likelihood is ”simulation free” but not ”sample
free”, i.e.

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. 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. 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. 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. 3nd Point: How to validate a pseudo-posterior
π(θ|y ) ∝ LEL (θ) × π(θ) ?

43. 3nd Point: How to validate a pseudo-posterior
π(θ|y ) ∝ LEL (θ) × π(θ) ?
◮ The use of pseudo-likelihoods is not new in the Bayesian
setting:

44. 3nd Point: How to validate a pseudo-posterior
π(θ|y ) ∝ LEL (θ) × π(θ) ?
◮ The use of pseudo-likelihoods is not new in the Bayesian
setting:
◮ Empirical Likelihoods:

45. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(θ|y ) ∝ LEL (θ) × π(θ) ?
◮ Monahan & Boos (Biometrika, 1992) proposed a notion of
validity:

56. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. 3nd Point: How to validate a pseudo-posterior
π(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. ... Last point: the ABC is still a terrific tool

60. ... Last point: the ABC is still a terrific tool
◮ ... a lot of references:

61. ... Last point: the ABC is still a terrific tool
◮ ... a lot of references:
◮ Statistical Journals;

62. ... Last point: the ABC is still a terrific tool
◮ ... a lot of references:
◮ Statistical Journals;
◮ Twitter;

63. ... Last point: the ABC is still a terrific tool
◮ ... a lot of references:
◮ Statistical Journals;
◮ Twitter;
◮ Xiang’s blog ( xianblog.wordpress.com )

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. ... 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 ?