Chapter 7. Further topics in MCMC

· Example 7.1 (pages 245-6) – Figure 7.1 (page 247) – Table 7.2 (page 246) - Comparing several methods to compute the normalizing constant f(y)

f₀ – normal approximation

f₁ – proposal is the prior

f₄ – proposal is the posterior

f₅ – Newton and Raftery’s

f₆ – generalized harmonic mean

f₇ – annealed importance sampling

f₈ – optimal bridge sampling

f₉ – path sampling

f₁₁ – candidate's estimator from Metropolis output

· Example 7.3 (pages 251-2) – Figure 7.2 (page 252) - Random walk Metropolis-Hastings algorithm

Data presented by Ratkowski (1983) on the temporal evolution of the dry weight of onion bulbs.

Gelfand, Dey and Chang (1992) considered 2 possible non-linear models in the form

y_t ~ N(f(θ,t),σ²)

where

Molde 1 - Logistic model : f(θ,t) = θ₁/(1+θ₂θ₃^t)

Model 2 - Gompertz model : f(θ,t) = θ₁ + exp(θ₂θ₃^t)

with θ₂ > 0 and 0 < θ₃ < 1 and ψ₁ = θ ₁, ψ₂ = log(θ₂) and ψ₃ = log(θ₃/(1-θ₃)).

Non-informative prior: p(ψ,σ²) = 1/σ².

Data  on the temporal evolution of the dry weight of onion bulbs:

y = (16.08,33.83,65.80,97.20,191.55,326.20,386.87,520.53,590.03,651.92,724.93,699.56,689.96,637.56,717.41)

· Example 7.5 (pages 254-5) – Figure 7.3 (page 256) - Table 7.6 (page 255) - Computing AICs, BICs, pseudoBFs, DGs and DICs

Comparing gamma,lognormal,weibull models.

Data on 100 cycles-to-failure times for samples of yarn airplanes. For each individual airplane, it has been suggested that an

exponential model fits the data well (Quesenberry and Kent, 1982 and Leonard and Hsu, 1999) .

· Example 7.10 (pages 274-5) – Figure 7.5 (page 275) - Delayed rejection Metropolis and Random walk Metropolis algorithms to sample from

π(θ) = 0.9f_N(θ;0,1) + 0.1f_N(θ;10,1)

· Example 7.11 (pages 277) – Figure 7.6 (page 278) - Multiple try Metropolis and Random walk Metropolis algorithms to sample from

π(β) ∝ (∑[y_i-β₁-β₁e^-β₂^x_i]²)^-2 f_N(β₁;20,20²) f_N(β₂;0,1.5²)

for β=(β₁,β₂) in [-20,50]x[-2,6] , x = (1,2,3,4,5,7) and y = (8.3,10.3,19,16,15.6,19.8).

This example is related to Example 3.7 (pages 102-3).

· Example 7.12 (pages 280) – Figure 7.7 (page 281) - Simulated annealing to find the mode of

π(β) ∝ ∏ exp{(β₁ +β₂x_i)y_i}∏ {1+exp(β₁ +β₂x_i)}^-n_i

· Example 7.15 (pages 284) – Figure 7.8 (page 285) - Slice sampling to sample from

π(θ) ∝ (2+θ)¹⁴(1-θ)³θ⁵

for θ in [0,1].