Chapter 6.  Metropolis-Hastings algorithms

 

·  Example 6.1 (pages 192-3) – Figure 6.1 (page 194)  - Random walk Metropolis-Hastings algorithm to simulate from

 

                                                                 π(θ) = fN(β;0,100)∏ fN(yi;50+170xi/(θ+xi),126),     for θ in R.

 

            Random walk proposal: q(θ,φ)= fN(φ;θ,0.01)

 

            Dataset:  y = velocity of an enzymatic reacton (in counts/min/min)

                          x = substrate concentration (in ppm)

 

x = (0.02,0.02,0.06,0.06,0.11,0.11,0.22,0.22,0.56,0.56,1.10,1.10)

y = (76,47,97,107,123,139,159,152,191,201,207,200)

 

Figure 6.1 shows the trajectory of the chain with initial value for θ equal to 0.4.

 

·  Example 6.2 (pages 199 and 202-3) – Figures 6.2 - 6.6 (pages 200-204)  - Random walk Metropolis-Hastings versus independence Metropolis-Hastings

 

The target density if a mixture of two 2-dimensional normal densities (Figure 6.2);

 

Part 1 of the example (page 199) implements the random walk Metropolis-Hasting algorithm:

            Figure 6.3 – Chain paths for 6 combinations of initial values and tuning parameters;

            Figure 6.4 – Chain autocorrelations for 6 combinations of initial values and tuning parameters.

 

Part 2 of the example (pages 202-204) implements the independence Metropolis-Hasting algorithm:

            Figure 6.5 – Chain paths for 6 combinations of initial values and tuning parameters;

            Figure 6.6 – Chain autocorrelations for 6 combinations of initial values and tuning parameters.

 

·  Example 6.4 -  Random walk Metropolis-Hastings (single and block moves) and  independence Metropolis-Hastings (block move)

 

Times to failure (f) of  motorettes were tested at different temperatures (t). 

There are 17 uncensored and 23 censored observations.

A constant prior is used for the regression parameters.

 

·  Example 6.4 Part 1 : (pages 210-1) – Figure 6.7 (page 212)  - Original regressor

 

·  Example 6.4 Part 2: (pages 216-7) – Figure 6.9 (page 218)  - Centered regressor

 

·  Example 6.5 (pages 212-3) – Figure 6.8 (page 214)  - Poisson model with change point  revisited.  Metropolis step for change point parameter.

 

·  Exercise 6.9 (page 236)  - Univariate version of Example 6.2 above.