BAYESIAN MODEL CRITICISM

Transcription

1 Monte via Chib s BAYESIAN MODEL CRITICM Hedibert Freitas Lopes The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago, IL hlopes@chicagobooth.edu Based Gamerman and Lopes (2006) Markov Chain Monte : Stochastic Simulation for Inference, Chapman&Hall/CRC.

2 Monte via Chib s 1 2 Monte via Chib s Deviance Information Criterion Outline

3 Monte via Chib s Suppose that the competing s can be enumerated and are represented by the set M = {M 1, M 2,...}, and that the true is in M (Bernardo and Smith, 1994). The posterior probability of M j is given by where Pr(M j y) f (y M j )Pr(M j ) f (y M j ) = f (y θ j, M j )p(θ j M j )dθ j is the prior density of M j and Pr(M j ) is the prior probability of M j.

4 Monte via Chib s The posterior odds of M j relative to M k is given by Pr(M j y) Pr(M k y) }{{} posterior odds = Pr(M j) Pr(M k ) }{{} prior odds f (y M j). f (y M k ) }{{} The can be viewed as the weighted ratio of M j to M k. The main difficulty is the computation of the marginal or normalizing constant f (y M j ).

5 Monte via Chib s Jeffreys (1961) recommends the use of the following rule of thumb to decide between s j and k: B jk > 100 : decisive evidence against k 10 < B jk 100 : strong evidence against k 3 < B jk 10 : substantial evidence against k Therefore, the posterior probability for j can be obtained from 1 Pr(M j y) = M k M B kj Pr(M k ) Pr(M j ).

6 Monte via Chib s A basic ingredient for assessment is given by the density f (y M) = f (y θ, M)p(θ M)dθ, which is the normalizing constant of the posterior distribution. The density can now be viewed as the of M. It is sometimes referred to as, because it is obtained after marginalization of parameters. The density can be written as the expectation of the with respect to the prior: f (y) = E p [f (y θ)].

7 Monte via Chib s Let m and V be the posterior mode and the asymptotic approximation for the posterior covariance matrix. A normal approximation to f (y) is given by f0 (y) = (2π) d/2 V 1/2 p( m)f (y m) Sampling-based approximations for m and V can be constructed from posterior draws θ (1),..., θ (N) : m = arg max θ (j) π(θ (j) ) V = 1 N N j=1 (θ(j) θ)(θ (j) θ), where θ = 1 N N j=1 θ(j).

8 Monte via Chib s A direct Monte estimate is ˆf 1 (y) = 1 N N f (y θ (j) ) j=1 where θ (1),..., θ (N) is a sample from the prior distribution p(θ). This does not work well in cases of disagreement between prior and (Raftery, 1996, and McCulloch and Rossi, 1991). Even for large values of n, this estimate will be influenced by a few sampled values, making it very unstable.

9 Monte via Chib s Monte via An alternative is to perform importance sampling with the aim of boosting sampled values in regions where the integrand is large. This approach is based on sampling from the importance density g(θ), since the density can be rewritten as [ ] f (y θ)p(θ) f (y) = E g. g(θ) This form motivates a new estimate ˆf 2 (y) = 1 N N j=1 f (y θ (j) )p(θ (j) ) g(θ (j) ) where θ (1),..., θ (N) is a sample from g(θ).

10 Sometimes g is only known up to a normalizing constant, i.e. g(θ) = kg (θ), and the value of k must be estimated. Monte via Chib s Noting that k = kp(θ)dθ = leads to the of k given by k = 1 N N j=1 p(θ) g (θ) g(θ)dθ p(θ (j) ) g (θ (j) ) where, again, the θ (j) are sampled from g. Replacing this estimate in f 2 (y) gives N j=1 f3 (y) = f (y θ(j) )p(θ (j) )/g (θ (j) ) N. j=1 p(θ(j) )/g (θ (j) )

11 Monte via Chib s The harmonic mean (HM) is obtained when g(θ) is the posterior π(θ): f4 (y) = 1 N for θ (1),..., θ (N) from π(θ). N 1 f (y θ (j) ) j=1 This is a very appealing for its simplicity. However, it is strongly affected by small values! 1

12 Monte via Chib s A compromise between f 1 and f 4 would lead to g(θ) = δp(θ) + (1 δ)π(θ) Problem: f (y) needs to be known! Solution via an iterative scheme: f (i) 5 (y) = for i = 1, 2,... and, say, N p(y θ (j) ) j=1 δ b f (i 1) 5 (y)+(1 δ)p(y θ (j) ) N 1 j=1 δ b f (i 1) 5 (y)+(1 δ)p(y θ (j) ) f (0) 5 = f 4.

13 Monte via Chib s For any given density g(θ) it is easy to see that g(θ) f (y)π(θ) f (y θ)p(θ) dθ = 1 so, [ f (y) = ] g(θ) 1 f (y θ)p(θ) π(θ)dθ. Therefore, sampling θ (1),..., θ (1) from π leads to the estimate f6 (y) = 1 N N g(θ (j) ) f (y θ (j) )p(θ (j) ) j=1 1.

14 Monte via Chib s Meng and Wong (1996) introduced the bridge sampling to estimate ratios of normalizing constants by noticing that f (y) = E g {α(θ)p(θ)p(y θ)} E π {α(θ)g(θ)} for any bridge function α(θ) with support encompassing both supports of the posterior density π and the proposal density g. If α(θ) = 1/g(θ) then the bridge reduces to the simple Monte f 1. Similarly, if α(θ) = {p(θ)p(y θ)g(θ)} 1 then the bridge is a variation of the harmonic mean.

15 Monte via Chib s Meng and Wong (1996) showed that the optimal mean square error α function is which depends on f (y) itself. By letting α(θ) = {g(θ) + (N 2 /N 1 )π(θ)} 1, ω j = p(y θ (j) )p(θ (j) )/g(θ (j) ) θ (1),..., θ (N 1) π(θ) ω j = p(y θ (j) )p( θ (j) )/g( θ (j) ) θ (j),..., θ (N 2) g(θ) they devised an iterative scheme: f (i) 7 (y) = 1 N2 ω j N 2 j=1 s 1 ω j +s 2 b(i 1) f 7 (y) 1 N1 1 N 1 j=1 s 1 ω j +s 2 b(i 1) f 7 (y) for i = 1, 2,..., s 1 = N 1 /(N 1 + N 2 ), s 2 = N 2 /(N 1 + N 2 ) and, (0) say, f 7 = f 4.,

16 Monte via Chib s Chib s Chib (1995) introduced an estimate of π(θ) when conditionals are available in closed form: with π(θ) = π(θ 1 ) π(θ i θ 1,..., θ i 1 ) = 1 N N j=1 d π(θ i θ 1,..., θ i 1 ) i=2 π(θ i θ 1,..., θ i 1, θ (j) i+1,..., θ(j) d ) and (θ (j) 1,..., θ(j) ), j = 1,..., n, draws from π(θ). Thus d f8 (y) = f (y θ)p(θ) π(θ) Simple choices of θ are the mode and the mean but any value in that region should be adequate. θ

17 Monte via Chib s Estimate b f0 b f1 b f2 b f3 b f4 b f5 b f6 b f7 b f8 List of s of f (y) Proposal density/method normal approximation p(θ) unnormalized g (θ) unnormalized g(θ) π(θ) δp(θ) + (1 δ)π(θ) generalized harmonic mean optimal bridge sampling candidate s : Gibbs DiCiccio, Kass, Raftery and Wasserman (1997), Han and Carlin (2001) and Lopes and West (2004), among others, compared several of these s.

18 Monte via Chib s Example: Cauchy-normal : y 1,..., y n N(θ, σ 2 ), σ 2 known. Cauchy prior: p(θ) = π 1 (1 + θ 2 ) 1. Data: ȳ = 7 and σ 2 /n = 4.5. density for θ: { π(θ) (1 + θ 2 ) 1 exp 1 (θ ȳ)2 2σ2 Error = 100 ˆf (y) f (y) /f (y). For ˆf 5, δ = 0.1. f (y) Estimator Estimate Error (%) f (y) ˆf ˆf ˆf ˆf ˆf ˆf }.

19 Monte via Chib s Suppose that M 2 is described by p(y ω, ψ, M 2 ) and M 1 is a restricted verison of M 2, ie. Suppose also that p(y ψ, M 1 ) p(y ω = ω 0, ψ, M 2 ) π(ψ ω = ω 0, M 2 ) = π(ψ M 1 ) Therefore, it can be proved that the is B 12 = π(ω = ω 0 y, M 2 ) π(ω = ω 0 M 2 ) N 1 N n=1 π(ω = ω 0 ψ (n), y, M 2 ) π(ω = ω 0 M 2 ) where {ψ (1),..., ψ (N) } π(ψ y, M 2 ).

20 Monte via Chib s Example: Normality x Student-t From Verdinelli-Wasserman (1995). Suppose that we have observations x 1,..., x n and we would like to entertain two s: M 1 : x i N(µ, σ 2 ) and M 2 : x i t λ (µ, σ 2 ) Letting ω = 1/λ, M 1 is a particular case of M 2 when ω = ω 0 = 0.0, with ψ = (µ, σ 2 ). Let us also assume that ω U(0, 1), with ω = 1 corresponding to a Cauchy distribution, and that π(µ, σ 2 M 1 ) = π(µ, σ 2, ω M 2 ) σ 2 the formula holds and the is B 12 = π(ω 0 x, M 2 ), i.e., the marginal posterior of ω evaluated at 0.

21 Monte via Chib s Because π(µ, σ 2, ω x, M 2 ) has no closed form solution, they use a algorithm and sample (µ, σ 2, ω) from q(µ, σ 2, ω), ie. q(µ, σ 2, ω) = π(ω)π(σ 2 x, M 1 )π(µ σ 2, x, M 1 ) ω U(0, 1) ( ) n 1 σ 2 (n 1)s2 IG, 2 2 µ σ 2 N( x, σ 2 /n) When n = 100 from N(0, 1), then B 12 = 3.79 (standard error=0.145) When n = 100 from Cauchy(0, 1), then B 12 = (standard error= )

22 Monte via Chib s Suppose that the competing s can be enumerable and are represented by the set M = {M 1, M 2,...}. Under M k, the posterior distribution is p(θ k y, k) p(y θ k, k)p(θ k k) where p(y θ k, k) and p(θ k k) represent the probability and the prior distribution of the parameters of M k, respectively. Then, p(θ k, k y) p(k)p(θ k k, y)

23 Monte via Chib s The RJMCMC methods involve MH-type algorithms that move a simulation analysis between s defined by (k, θ k ) to (k, θ k ) with different defining dimensions k and k. The resulting Markov chain simulations jump between such distinct s and form samples from the joint distribution p(θ k, k). The algorithm are designed to be reversible so as to maintain detailed balance of a irreducible and aperiodic chain that converges to the correct target measure (See Green, 1995, and Gamerman and Lopes, 2006, chapter 7).

24 Monte via Chib s Step 0. Current state: (k, θ k ) Step 1. Sample M k from J(k k ). Step 2. Sample u from q(u θ k, k, k ). The RJMCMC algorithm Step 3. Set (θ k, u ) = g k,k (θ k, u), where g k,k ( ) is a bijection between (θ k, u) and (θ k, u ), where u and u play the role of matching the dimensions of both vectors. Step 4. The acceptance probability of the new, (θ k, k ) can be calculated as the minimum between one and p(y θ k, k )p(θ k )p(k ) J(k k)q(u θ k, k, k) g k,k (θ k, u) p(y θ k, k)p(θ k )p(k) J(k k {z } )q(u θ k, k, k ) (θ k, u) {z } ratio proposal ratio

25 Monte via Chib s p(k y) Looping through steps 1-4 above L times produces a sample {k l, l = 1,..., L} for the indicators and Pr(k y) can be estimated by Pr(k y) = 1 L 1 k (k l ) L l=1 where 1 k (k l ) = 1 if k = k l and zero otherwise.

26 Monte via Chib s Choice of q(u k, θ k, k ) The choice of the proposal probabilities, J(k k ), and the proposal densities, q(u k, θ k, k ), must be cautiously made, especially in highly parameterized problems. Independent sampler: If all parameters of the proposed are generated from the proposal distribution, then (θ k, u ) = (u, θ k ) and the Jacobian is one. Standard -Hastings: When the proposed k equals the current k, the loop through steps 1-4 corresponds to the traditional -Hastings algorithm.

27 Monte via Chib s Optimal choice If p(θ k y, k) is available in close form for each M k, then q(u θ k, k, k) = p(θ k y, k) and the acceptance probability reduces to the minimum between one and p(k )p(y k ) J(k k) p(k)p(y k) J(k k ) since p(y θ k, k)p(θ k )p(k) = p(θ k, k y)p(y k). The Jacobian equals one and p(y k) is available in close form. If J(k k) = J(k k ), then the acceptance probability is the posterior odds ratio from M k to M k. In this case, the move is automatically accepted when M k has higher posterior probability than M k ; otherwise the posterior odds ratio determines how likely is to move to a lower posterior probability.

28 Monte via Chib s Metropolized Carlin-Chib Let Θ = (θ k, θ k ) be the vector containing the parameters of all competing s. Then the joint posterior of (Θ, k) is p(θ, k y) p(k)p(y θ k, k)p(θ k k)p(θ k θ k, k) where p(θ k θ k, k) are pseudo-prior densities. Carlin and Chib (1995) proposed a Gibbs sampler where the full posterior conditional distributions are { p(y θk, k)p(θ p(θ k y, k, θ k ) k k) if k = k p(θ k k ) if k = k and p(k Θ, y) p(y θ k, k)p(k) p(θ m k) m M

29 Monte via Chib s Notice that the pseudo-prior densities and the RJMCMC s proposal densities have similar functions. As a matter of fact, Carlin and Chib (1995) suggest using pseudo-prior distributions that are close to the posterior distributions within each competing. The main problem with Carlin and Chib s Gibbs sampler is the need of evaluating and drawing from the pseudo-prior distributions at each iteration of the MCMC scheme. This problem can be overwhelmingly exacerbated in large situations where the number of competing s is relatively large.

30 Monte via Chib s Dellaportas, Forster and Ntzoufras (1998) and Godsill (1998) proposed Metropolizing Carlin and Chib s Gibbs sampler: Step 0. Current state: (θ k, k) Step 1. Sample M k from J(k k ); Step 2. Sample θ k from p(θ k k); Step 3. The acceptance probability is min(1, A) A = p(y θ k, k )p(k )J(k k) m M p(θ m k ) p(y θ k, k)p(k)j(k k ) m M p(θ m k) = p(y θ k, k )p(k )J(k k)p(θ k k )p(θ k k ) p(y θ k, k)p(k)j(k k )p(θ k k)p(θ k k). Pseudo-priors and RJMCMC s proposals play similar roles and the closer their are to the competing s posterior probabilities the better the sampler mixing.

31 Monte via Chib s See Hoeting, Madigan, Raftery and Volinsky (1999), Statistical Science, 14, Let M denote the set that indexes all entertained s. Assume that is an outcome of interest, such as the future value y t+k, or an elasticity well defined across s, etc. The posterior distribution for is p( y) = p( m, y)pr(m y) m M for data y and posterior probability Pr(m y) = p(y m)pr(m) where Pr(m) is the prior probability.

32 Monte via Chib s Gelfand and Ghosh (1998) introduced a posterior that, under squared error loss, favors the M j which minimizes D G j = P G j + G G j where P G j = G G j = n V (ỹ t y, M j ) t=1 n [y t E(ỹ t y, M j )] 2 t=1 and (ỹ 1,..., ỹ n ) are predictions/replicates of y. The first term, P j, is a penalty term for complexity. The second term, G j, accounts for goodness of fit.

33 Monte via Chib s More general losses Gelfand and Ghosh (1998) also derived the criteria for more general error loss functions. Expectations E(ỹ t y, M j ) and variances V (ỹ t y, M j ) are computed under posterior densities, ie. E[h(ỹ t ) y, M j ] = h(ỹ t )f (ỹ t y, θ j, M j )π(θ j M j )dθ j dỹ t for h(ỹ t ) = ỹ t and h(ỹ t ) = ỹ 2 t. The above integral can be approximated via Monte.

34 Monte via Chib s Deviance Information Criterion See Spiegelhalter, Best, Carlin and van der Linde (2002), JRSS-B, 64, If θ = E(θ y) and D(θ) = 2 log p(y θ) is the deviance, then the DIC generalizes the AIC DIC = D + p D = goodness of fit + complexity where D = E θ y (D(θ)) and p D = D D(θ ). The p D is the effective number of parameters. Small values of DIC suggests a better-fitting.

35 Monte via Chib s DIC is computationally attractive since its two terms can be easily computed during an MCMC run. Let θ (1),..., θ (M) be an MCMC sample from p(θ y). Then, and D 1 M M D i=1 = 2M 1 M (θ (i)) i=1 log p (y θ (i)) D(θ ) D( θ) = 2 log p(y θ) where θ = M 1 M i=1 θ(i).

36 Monte via Chib s Example: cycles-to-failure times Cycles-to-failure times for airplane yarns Gamma, log-normal and Weibull s: M 1 : y i G(α, β), α, β > 0 M 2 : y i LN(µ, σ 2 ), µ R, σ 2 > 0 M 3 : y i Weibull(γ, δ) γ, δ > 0, for i = 1,..., n.

37 Under M 2, µ and σ 2 are the mean and the variance of log y i, respectively. Monte via Chib s Under M 3, p(y i γ, δ) = γy γ 1 i δ γ e (y i /δ) γ. Flat priors were considered for θ 1 = (log α, log β) θ 2 = (µ, log σ 2 ) θ 3 = (log γ, log δ) It is also easy to see that, E(y θ 1, M 1 ) = α/β V (y θ 1, M 1 ) = α/β 2 E(y θ 2, M 2 ) = exp{µ + 0.5σ 2 } V (y θ 2, M 2 ) = exp{2µ + σ 2 }(e σ2 1) E(y θ 3, M 3 ) = δγ(1/γ)/γ V (y θ 3, M 3 ) = δ 2 [ 2Γ(2/γ) Γ(1/γ) 2 /γ ] /γ

38 Monte via Chib s Weighted resampling schemes, with bivariate normal importance functions, were used to sample from the posterior distributions. Proposals: q i (θ i ) = f N (θ i ; θ i, V i ) θ 1 = (0.15, 0.2) θ 2 = (5.16, 0.26) θ 3 = (0.47, 5.51) V 1 = diag (0.15, 0.2) V 2 = diag (0.087, 0.085) V 3 = diag (0.087, 0.101)

39 Monte via Chib s means, standard deviations and 95% credibility intervals: M1 M2 M3 α: 2.24, 0.21, (1.84, 2.68) β: 0.01, 0.001, (0.008, 0.012) α: 5.16, 0.06, (5.05, 5.27) β: 0.77, 0.04, (0.69, 0.86) α: 1.60, 0.09, (1.42, 1.79) β: , 13.88, (222.47, )

40 Monte via Chib s DIC indicates that both the Gamma and the Weibull s are relatively similar with the Weibull performing slightly better. DIC M 1 Gamma M 2 Log-normal M 3 Weibull