Assignment #8
Ashutosh Tripathi
2022-05-24
Chapter 9 - Markov Chain Monte Carlo
This chapter has been an informal introduction to Markov chain Monte Carlo (MCMC) estimation. The goal has been to introduce the purpose and approach MCMC algorithms. The major algorithms introduced were the Metropolis, Gibbs sampling, and Hamiltonian Monte Carlo algorithms. Each has its advantages and disadvantages. The ulam function in the rethinking package was introduced. It uses the Stan (mc-stan.org) Hamiltonian Monte Carlo engine to fit models as they are defined in this book. General advice about diagnosing poor MCMC fits was introduced by the use of a couple of pathological examples.
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Questions
9-1. Re-estimate the terrain ruggedness model from the chapter, but now using a uniform prior for the standard deviation, sigma. The uniform prior should be dunif(0,1). Visualize the priors. Use ulam to estimate the posterior. Visualize the posteriors for both models. Does the different prior have any detectible influence on the posterior distribution of sigma? Why or why not?
library(rethinking)
data(rugged)
d <- rugged
d$log_gdp <- log(d$rgdppc_2000)
dd <- d[ complete.cases(d$rgdppc_2000) , ]
dd$log_gdp_std <- dd$log_gdp/ mean(dd$log_gdp)
dd$rugged_std<- dd$rugged/max(dd$rugged)
dd$cid<-ifelse(dd$cont_africa==1,1,2)
m9.1 <- quap(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]* (rugged_std-0.215) ,
a[cid] ~ dnorm(1,0.1),
b[cid] ~ dnorm(0,0.3),
sigma ~ dexp(1)
) ,
data=dd)
precis(m9.1 , depth=2)## mean sd 5.5% 94.5%
## a[1] 0.8865597 0.015675251 0.86150765 0.91161180
## a[2] 1.0505706 0.009936325 1.03469048 1.06645081
## b[1] 0.1325050 0.074202453 0.01391513 0.25109483
## b[2] -0.1425765 0.054747890 -0.23007424 -0.05507884
## sigma 0.1094910 0.005934874 0.10000591 0.11897606pairs(m9.1)
m9.1_unif <- quap(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]* (rugged_std-0.215) ,
a[cid] ~ dnorm(1,0.1),
b[cid] ~ dnorm(0,0.3),
sigma ~ dunif(0,1)
) ,
data=dd)
precis(m9.1 , depth=2)## mean sd 5.5% 94.5%
## a[1] 0.8865597 0.015675251 0.86150765 0.91161180
## a[2] 1.0505706 0.009936325 1.03469048 1.06645081
## b[1] 0.1325050 0.074202453 0.01391513 0.25109483
## b[2] -0.1425765 0.054747890 -0.23007424 -0.05507884
## sigma 0.1094910 0.005934874 0.10000591 0.11897606pairs(m9.1_unif)
It shows that it does not have detectable influence on the posterior distribution of sigma. # In comparison, parameter estimations are very similar. Except for sigma, which has a higher n_eff in the uniform-prior model, the individual model outputs are also quite close, with all parameter estimates in the model with the exponential-prior model on sigma being significantly higher. As a result, we discover that while different priors impact n_eff, posterior distributions are unaffected.
9-2. Modify the terrain ruggedness model again. This time, change the prior for b[cid] to dexp(0.3). What does this do to the posterior distribution? Can you explain it?
m9.3_exp <- quap(
alist(
log_gdp_std ~ dnorm(mu , sigma) ,
mu <- a[cid] + b[cid]* (rugged_std-0.215) ,
a[cid] ~ dnorm(1,0.1),
b[cid] ~ dnorm(0,0.3),
sigma ~ dexp(0.3)
) ,
data=dd)
precis(m9.3_exp , depth=2)## mean sd 5.5% 94.5%
## a[1] 0.8865654 0.01567868 0.86150782 0.9116229
## a[2] 1.0505681 0.00993853 1.03468439 1.0664518
## b[1] 0.1325124 0.07421804 0.01389765 0.2511272
## b[2] -0.1425681 0.05475976 -0.23008473 -0.0550514
## sigma 0.1095155 0.00593820 0.10002512 0.1190059pairs(m9.3_exp)
When compared to the uniform prior, the posterior distribution appears to be slightly right skewed. #the most significant difference is in the b parameter, which appears to have fewer samples and no longer accepts negative values.
9-3. Re-estimate one of the Stan models from the chapter, but at different numbers of warmup iterations. Be sure to use the same number of sampling iterations in each case. Compare the n_eff values. How much warmup is enough?
##After warmup is increased,n_eff getting closer to number of iterations. 400 warmup iterations are enough .
dat_slim <- list(
log_gdp_std = dd$log_gdp_std,
rugged_std = dd$rugged_std,
cid = as.integer( dd$cid ))
str(dat_slim)## List of 3
## $ log_gdp_std: num [1:170] 0.88 0.965 1.166 1.104 0.915 ...
## $ rugged_std : num [1:170] 0.138 0.553 0.124 0.125 0.433 ...
## $ cid : int [1:170] 1 2 2 2 2 2 2 2 2 1 ...#cmdstan_version()
#set_cmdstan_path(path = NULL)
#cmdstan_path("/var/folders/r8/k2vy8ns12bb5v2w_m80yj0g40000gn/T//RtmpfUwBoK/downloaded_packages")
m9.4 <- ulam(
alist(
log_gdp_std ~ dnorm(mu , sigma) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215) ,
a[cid] ~ dnorm(1 , 0.1) ,
b[cid] ~ dnorm(0 , 0.3) ,
sigma ~ dexp(1)
) , data=dat_slim , chains=4 , cores=4)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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## Total execution time: 0.4 seconds.precis(m9.4, depth=2)## mean sd 5.5% 94.5% n_eff Rhat4
## a[1] 0.8862118 0.014872238 0.86339524 0.9103114 2725.058 0.9996402
## a[2] 1.0504861 0.009863352 1.03492945 1.0661722 3165.540 0.9995140
## b[1] 0.1325891 0.076898590 0.01061253 0.2571183 2443.519 0.9994959
## b[2] -0.1412440 0.055299931 -0.23067292 -0.0512786 3238.966 0.9993216
## sigma 0.1114371 0.006213709 0.10199424 0.1216622 2820.181 0.99850229-4. Run the model below and then inspect the posterior distribution and explain what it is accomplishing.
mp <- ulam(
alist(
a ~ dnorm(0,1),
b ~ dcauchy(0,1)
), data=list(y=1) , chains=1 )## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.0 seconds.traceplot(mp)
Compare the samples for the parameters a and b. Can you explain the different trace plots? If you are unfamiliar with the Cauchy distribution, you should look it up. The key feature to attend to is that it has no expected value. Can you connect this fact to the trace plot?
#Traceplot for parameter A is normally distributed as prior is around 0 and spread is between +2 to -2. Plot b is Cauchy distribution which contains some extreme value go up to over 30 and -50.
9-5. Recall the divorce rate example from Chapter 5. Repeat that analysis, using ulam this time,fitting models m5.1, m5.2, and m5.3. Use compare to compare the models on the basis of WAIC or PSIS. To use WAIC or PSIS with ulam, you need add the argument log_log=TRUE. Explain the model comparison results.
library(tidybayes)
data(WaffleDivorce)
data = WaffleDivorce
data$Divorce_sd=standardize(data$Divorce)
data$Marriage_sd=standardize(data$Marriage)
data$MedianAgeMarriage_sd=standardize(data$MedianAgeMarriage)
d_trim = list(D = data$Divorce_sd, M = data$Marriage_sd, A = data$MedianAgeMarriage_sd)
m5.1 = ulam(
alist(
D ~ dnorm(mu, sigma),
mu <- a + bA * A,
a ~ dnorm(0, 0.2),
bA ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
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alist(
D ~ dnorm(mu, sigma),
mu <- a + bM * M,
a ~ dnorm(0, 0.2),
bM ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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alist(
D ~ dnorm(mu, sigma),
mu <- a + bA * A + bM * M,
a ~ dnorm(0, 0.2),
bA ~ dnorm(0, 0.5),
bM ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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compare( m5.1 , m5.2 , m5.3 , func=WAIC )## WAIC SE dWAIC dSE pWAIC weight
## m5.1 125.9212 12.726083 0.00000 NA 3.752815 0.6734614752
## m5.3 127.3742 12.674467 1.45304 0.8680393 4.649688 0.3256784692
## m5.2 139.2476 9.792579 13.32638 9.3954017 2.908383 0.0008600555