Assignment #7
Chapter 9
Hemant Yadav
2022-11-06
Chapter 9 - Markov Chain Monte Carlo
This week 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.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.
Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.
Questions
8-1. 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 distribution of sigma. Visualize the posterior distribution of sigma for both models. Do not use a ‘pairs’ plot. Does the different prior have any detectable influence on the posterior distribution of sigma? Why or why not?
# Leveraging the data "rugged" for the re-estimation of the terrain ruggedness model.
data("rugged")
set.seed(1234)
# Fitting the model comprising of dexp(1) prior on sigma for the determination of different prior to have have any detectable influence on the posterior distribution of sigma.
Rugged_Data <- rugged %>%
as_tibble() %>%
drop_na(rgdppc_2000) %>%
transmute(log_gdp = log(rgdppc_2000),
log_gdp_std = log_gdp/ mean(log_gdp),
rugged_std = rugged / max(rugged),
cid = if_else(cont_africa == 1, 1, 2)) %>%
as.list()
str(Rugged_Data)## List of 4
## $ log_gdp : num [1:170] 7.49 8.22 9.93 9.41 7.79 ...
## $ 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 : num [1:170] 1 2 2 2 2 2 2 2 2 1 ...# Model_1 with utilizing the exponential prior dexp(1) for the standard deviation, sigma and utilization of the "ulam" function to estimate the posterior distribution of sigma. (Ref Model # m9.1 from chapter)
Model_1 <- 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 = Rugged_Data, chains = 1)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.1 seconds.show(Model_1)## Hamiltonian Monte Carlo approximation
## 500 samples from 1 chain
##
## Sampling durations (seconds):
## warmup sample total
## chain:1 0.05 0.03 0.08
##
## Formula:
## 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)# Determination of the updated new model comprising of the updated priors and leveraging uniform prior dunif(0,1) for the standard deviation sigma and utilization of the "ulam" function to estimate the posterior distribution of sigma.
Model_1_Update <- 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 ~ dunif(0, 10)),
data = Rugged_Data, chains = 1)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.1 seconds.precis(Model_1, depth = 2)## mean sd 5.5% 94.5% n_eff Rhat4
## a[1] 0.8861204 0.014808642 0.86125785 0.90900762 693.7060 0.9984677
## a[2] 1.0498805 0.010038731 1.03326675 1.06532315 985.9230 0.9981157
## b[1] 0.1373015 0.075225334 0.01539092 0.25897015 560.6033 0.9982974
## b[2] -0.1420128 0.052981991 -0.22474196 -0.05542156 498.0981 0.9979981
## sigma 0.1115245 0.006056566 0.10236817 0.12147287 526.0168 0.9999408# Determination of the plot for the prior distributions on sigma against each other.
tibble(exponential = rexp(1e3, 1),
uniform = runif(1e3, 0, 10)) %>%
pivot_longer(cols = everything(),
names_to = "type",
values_to = "Sigma") %>%
ggplot(aes(Sigma, fill = type)) +
geom_density(colour = "blue",
alpha = 0.6) +
labs(y = NULL, fill = NULL, caption = "Prior Distribution on Sigma") +
theme_minimal()
# Determination for the visualization of the posterior distribution plots for sigma for both the models and for the comparison of both the models.
tibble(exponential = extract.samples(Model_1) %>%
pluck("sigma"),
uniform = extract.samples(Model_1_Update) %>%
pluck("sigma")) %>%
pivot_longer(cols = everything(),
names_to = "model",
values_to = "Sigma") %>%
ggplot(aes(Sigma, fill = model)) +
geom_density(colour = "blue",
alpha = 0.6) +
labs(y = NULL, fill = NULL, caption = "Posterior Distribution for Sigma") +
theme_minimal()
# Plotting for Additional joint Posterior Distribution plot for better visualization:
Plot_Dtafrme <- data.frame(
Posteriors = c(
extract.samples(Model_1, n = 1e4)$sigma,
extract.samples(Model_1_Update, n = 1e4)$sigma
),
Name = rep(c("Exp", "Uni"), each = 1e4),
Model = rep(c("Model_1", "Model_1_Update"), each = 1e4)
)
ggplot(Plot_Dtafrme, aes(y = Model, x = Posteriors)) +
stat_halfeye() +
labs(x = "Parameter Estimates", y = "Model") +
theme_bw()
## Result Inference:
# Referencing the prior and posterior distribution plots, the different prior did not appear to have any significant detectable influence on the posterior distribution of sigma, since there is very less significant or minute differences in the posterior distribution for sigma, and probably may be due to the higher magnitude of the data, however on the other side, the uniform prior yielded a much broader and less peaky posterior outcome results findings and thus in the process also adding to more uncertainty or ambiguity to reflect any detectable influence on the posterior distribution of sigma.8-2. Modify the terrain ruggedness model again. This time, change the prior for b[cid] to dexp(0.3). Plot the joint posterior. Do not use a ‘pairs’ plot. What does this do to the posterior distribution? Can you explain it?
# Modification of our prior terrain ruggedness model "Model_1" with the new changed prior for the b[cid] as "dexp(0.3)", and utilization of 'ulm' function.
Modified_Model_1 <- 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] ~ dexp(0.3) ,
sigma ~ dexp(1)
) ,data = Rugged_Data, chains = 1)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.2 seconds.# Determination of the coefficient estimates of our revised modified model and for b1 and b2 output values changes observation.
precis(Modified_Model_1, depth = 2) %>%
as_tibble(rownames = "coefficient") %>%
knitr::kable(digits = 2)| coefficient | mean | sd | 5.5% | 94.5% | n_eff | Rhat4 |
|---|---|---|---|---|---|---|
| a[1] | 0.89 | 0.02 | 0.86 | 0.91 | 527.50 | 1.00 |
| a[2] | 1.05 | 0.01 | 1.03 | 1.06 | 456.81 | 1.00 |
| b[1] | 0.14 | 0.07 | 0.03 | 0.28 | 189.41 | 1.00 |
| b[2] | 0.02 | 0.02 | 0.00 | 0.05 | 480.17 | 1.01 |
| sigma | 0.11 | 0.01 | 0.11 | 0.12 | 449.95 | 1.00 |
# Determination of the plot for assessment of the modified prior distribution with the new changed prior for the b[cid] as "dexp(0.3)".
tibble(exponential = rexp(1e3, 0.3)) %>%
ggplot(aes(exponential)) +
geom_density(colour = "grey",
alpha = 0.8, fill = "brown") +
labs(y = NULL, fill = NULL) +
theme_minimal()
# Determination of Plot for the posterior distribution for modified terrain ruggedness model.
extract.samples(Modified_Model_1) %>%
pluck("b") %>%
.[,1] %>%
as_tibble_col(column_name = "b1") %>%
ggplot(aes(b1)) +
geom_density(colour = "blue",
fill = "grey",
alpha = 0.8) +
labs(y = NULL, caption = "Posterior Distribution for b[2]") + theme_minimal()
## Result Inference:
# Referencing the plot for the posterior distribution, it is clearly evident that the posterior plot was cut off at the scale of zero, and to a greater magnitude reflecting the positive values, and probably due to the reason and fact that prior did not permit to reflect lesser values or number than zero.8-3. Re-estimate one of the Stan models from the chapter, but at different numbers (at least 5) 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?
## Considering the chapters Model # m9.1 or our prior "Model_1" for this re-estimation determination at different numbers of warm up iterations, and leveraging the same number of sampling iterations in each case.
Refit_Model_9.1 <- function(N){
ulam(Model_1, chains = 1, warmup = N, iter = 1000,
cores = parallel::detectCores()) %>%
precis() %>%
as_tibble() %>%
pull(n_eff)
}
# Determination for the iteration over the warm up number leveraging or utilizing the purr::map.
n_effective <- seq(1, 500, length.out = 100) %>%
round(0) %>%
map_dbl(Refit_Model_9.1)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 10
## Chain 1 adapt_window = 51
## Chain 1 term_buffer = 6
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 10
## Chain 1 adapt_window = 55
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
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## Chain 1 adapt_window = 59
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 12
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 13
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 13
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 14
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 15
## Chain 1 adapt_window = 77
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 16
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 16
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 17
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 18
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 19
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 19
## Chain 1 adapt_window = 100
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 20
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
## Chain 1 init_buffer = 21
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## Chain 1 three stages of adaptation as currently configured.
## Chain 1 Reducing each adaptation stage to 15%/75%/10% of
## Chain 1 the given number of warmup iterations:
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## Chain 1 finished in 0.1 seconds.# Refitting of new hundred models with in or between the warm up range of 1 and 500, and determination of the plot for the result.
n_effective %>%
as_tibble_col(column_name = "n_eff") %>%
add_column(warmup = seq(1, 500, length.out = 100)) %>%
ggplot(aes(warmup, n_eff)) +
geom_line(colour = "blue", size = 0.9) +
labs(x = "Warm-up", y = "Effective-Sample-Number (ENS)",
caption = "Plot for Effective Numbers of Samples (ENS) as Function of Warmup") + theme_minimal()
# Result Inference: It is clearly evident from the warm up iteration output plot for our simplistic model "Model_1" or m9.1 (Ref chapter), that it exhibited a low correlation between the parameters and an increase in the effective number of samples was further observed to rapidly increased, and probably a robust estimate for the warm up of greater in magnitude than 15 to 20 approximately can be assumed to be adequate for our model.8-4. Run the model below and then inspect the posterior distribution and explain what it is accomplishing.
## Running the new "mp" model with the .
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.precis(mp)## mean sd 5.5% 94.5% n_eff Rhat4
## a 0.0293868 0.9513848 -1.502807 1.538506 103.92130 0.9984458
## b 1.5371004 24.7812287 -5.576877 3.495116 88.18848 1.0043824## Determination and inspection for the priors initially and plotting the priors.
tibble(a = rnorm(1e5, 0, 1),
b = rcauchy(1e5, 0, 1)) %>%
pivot_longer(cols = everything(),
names_to = "Parameter",
values_to = "Estimate") %>%
ggplot(aes(Estimate, fill = Parameter)) +
geom_density(colour = "brown") +
scale_fill_manual(values = c("grey", "blue")) +
facet_wrap(~ Parameter, scales = "free") +
labs(y = NULL, fill = NULL, caption = "Plot for Prior Simulation for a and b") +
theme_minimal()
# Prior Plot observation Inference:
# The prior plot for our "a" appears to be normal and as expected, but the prior plot for the "b" is somewhat different in comparison, and perhaps may be because the cauchy distribution placed the majority of the probability mass around the same region area as "a" but with also the addition of existence of some of the extreme outliers.
## Determination and inspection for the posterior and plotting for the posteriors.
extract.samples(mp) %>%
as_tibble() %>%
pivot_longer(cols = everything(),
names_to = "Parameter",
values_to = "Estimate") %>%
ggplot(aes(Estimate, fill = Parameter)) +
geom_density(colour = "black") +
scale_fill_manual(values = c("grey", "blue")) +
facet_wrap(~ Parameter, scales = "free") +
labs(y = NULL, fill = NULL, caption = "Plot for Posterior Distribution for a and b") +
theme_minimal()
# Posterior Plot observation Inference:
# Interestingly the plot for our posteriors also somewhat represented the same output result as our prior, and probably because we didn't establish any condition for the likelihood to update any data, and thus in the process the MCMC model straightforwardly would have sampled from our priors.Compare the samples for the parameters a and b. Plot the trace plots. 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?
## Determination for Trace Plot for our in-scope model "mp".
traceplot(mp, n_col = 2)
# Trace plot result inference:
# The chain in our trace plot for "a" appears to be normal and as expected, since the plot for "a" appears to be normal distribution and with the prior to a larger extent around zero and further the majority of plot area is within the range of negative and positive 2, however the traceplot for our "b" reflected a different outcome and it's evident that the chain for "b" is diverging from the mean significantly in some regions of the plot time to time, and probably indicating convergence issue to an extent, however because the "b" plot is majorly mirroring the cauchy distribution and with having a large proportion of probability at a mean and in addition of also having some extreme outliers, so our output scenario makes sense.8-5. 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.
## Leveraging the Data Waffle model "Waffle Divorce" from Chapter-5.
data(WaffleDivorce)
Data_Waffle <- WaffleDivorce %>%
as_tibble() %>%
transmute(across(c(Divorce, Marriage, MedianAgeMarriage), standardize)) %>%
select(D = Divorce, M = Marriage, A = MedianAgeMarriage)
set.seed(123)
## Leveraging the model "m5.1", "m5.2", and "m5.3" for refitting with ulam(), and "log_lik" argument to be also True for the accurate calculation of the "LOO" and "WAIC" Functions.
# Extracting and running of the prior Model 5.1 utilizing ulam():
m5.1 <- alist(
D ~ dnorm(mu, sigma),
mu <- a + bA * A,
a ~ dnorm(0, 0.2),
bA ~ dnorm(0, 0.5),
sigma ~ dexp(1)) %>%
ulam(data = Data_Waffle, cores = 8, log_lik = TRUE)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.1 seconds.# Extracting and running of the prior Model 5.2 utilizing ulam():
m5.2 <- alist(
D ~ dnorm(mu, sigma),
mu <- a + bM * M,
a ~ dnorm(0, 0.2),
bM ~ dnorm(0, 0.5),
sigma ~ dexp(1)) %>%
ulam(data = Data_Waffle, cores = 8, log_lik = TRUE)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Chain 1 finished in 0.1 seconds.# Extracting and running of the prior Model 5.3 utilizing ulam():
m5.3 <- alist(
D ~ dnorm(mu, sigma),
mu <- a + bM * M + bA * A,
a ~ dnorm(0, 0.2),
c(bM, bA) ~ dnorm(0, 0.5),
sigma ~ dexp(1)) %>%
ulam(data = Data_Waffle, cores = 8, log_lik = TRUE)## Running MCMC with 1 chain, with 1 thread(s) per chain...
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## Comparing the models utilizing the "compare" on the basis of WAIC.
compare(m5.1, m5.2, m5.3, func = WAIC) %>%
as_tibble(rownames = "Model") %>%
knitr::kable(digits = 2)| Model | WAIC | SE | dWAIC | dSE | pWAIC | weight |
|---|---|---|---|---|---|---|
| m5.1 | 125.68 | 12.42 | 0.00 | NA | 3.58 | 0.68 |
| m5.3 | 127.16 | 12.44 | 1.48 | 0.90 | 4.44 | 0.32 |
| m5.2 | 139.25 | 9.80 | 13.57 | 8.95 | 2.96 | 0.00 |
## Comparing the models utilizing the "compare" on the basis of PSIS.
compare(m5.1, m5.2, m5.3, func = PSIS) %>%
as_tibble(rownames = "Model") %>%
knitr::kable(digits = 2)| Model | PSIS | SE | dPSIS | dSE | pPSIS | weight |
|---|---|---|---|---|---|---|
| m5.1 | 125.64 | 12.49 | 0.00 | NA | 3.56 | 0.69 |
| m5.3 | 127.21 | 12.57 | 1.57 | 0.92 | 4.47 | 0.31 |
| m5.2 | 139.28 | 9.89 | 13.64 | 8.88 | 2.97 | 0.00 |
## Model Comparison Result Inference Findings:
# Referring the result out for our select in scope model i.e., Model: m5.1, m5.2, and m5.3, utilizing or on the basis of both the WAIC as well as PSIS function, it appear that our first Model i.e., m5.1 having the predictor of median age at the time of marriage performed the best performance compared to other models, but there is not significant model performance disparity between the model m5.1, and m5.3, however it is also very evident that our third model i.e., "m5.2" performed the least in comparison to model "m5.1", and "m5.3".