BSDA-Homework 7 Solved

1. Linear model: drowning data with Stan
The provided data drowning in the bsda package contains the number of people who died from drowning each year in Finland 1980 2019. A statistician is going to t a linear model with Gaussian residual model to these data using time as the predictor and number of drownings as the target variable (see the related linear model example for the Kilpisj rvitemperature data in the example Stan codes). She has two objective questions:

i)       What is the trend of the number of people drowning per year? (We would plot the histogram of the slope of the linear model.)

ii)     What is the prediction for the year 2020? (We would plot the histogram of the posterior predictive distribution for the number of people drowning at x˜ = 2020.)

To access the data, use:

library(bsda) data("drowning")

Corresponding Stan code is provided in Listing 1. However, it is not entirely correct for the problem. First, there are three mistakes. Second, there are no priors de ned for the parameters. In Stan, this corresponds to using uniform priors. Your tasks are the following:

a)     Find the three mistakes in the code and x them. Report the original mistakes and your xes clearly in your report. Include the full corrected Stan code in your report.

Hint: You may nd some of the mistakes in the code using Stan syntax checker. If you copy the Stan code to a le ending .stan and open it in RStudio (you can also choose from RStudio menu File→New File→Stan le to create a new Stan le), the editor will show you some syntax errors. More syntax errors might be detected by clicking ‘Check’ in the bar just above the Stan le in the RStudio editor. Note that some of the errors in the presented Stan code may not be syntax errors.

b)     Determine a suitable weakly-informative prior normal(0,σβ) for the slope beta. It is very unlikely that the mean number of drownings changes more than 50 % in one year. The approximate historical mean yearly number of drownings is 138. Hence, set σβ so that the following holds for the prior probability for beta: Pr(−69 < beta < 69) = 0.99. Determine suitable value for σβ and report the approximate numerical value for it.

c)     Using the obtained σβ, add the desired prior in the Stan code. In the report, in a separate section, indicate clearly how you carried out your prior implementation, e.g. Added line ...in block ... .

d)     In a similar way, add a weakly informative prior for the intercept alpha and explain how you chose the prior.

Hint! Example resulting plots for the problem, with the xes and the desired prior applied, are shown in Figure 1. If you want, you can use these plots as a reference for testing if your modi ed Stan code produces similar results. However, running the inference and comparing the plots is not required.

Note! The example/test plots and results are based on data up to 2016. You should report your result for the whole period 2019.

 

Figure 1: Example plots for the results obtained for the problem in the Question 1 with data until 2016. In the rst subplot, the red lines indicate the resulting 5 %, 50 %, and 95 % posterior quantiles for the transformed parameter mu at each year.

Listing 1: Broken Stan code for question 1

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data {
 
 
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int<lower=0> N;
//      number    of     data          points
 
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vector[N] x;
//         observation        year
 
 
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vector[N] y;
//         observation         number
of
drowned
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             real       xpred;
//         prediction        year
 
 
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}
 
 
 
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parameters          {
 
 
 
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             real       alpha;
 
 
 
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             real       beta;
 
 
 
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               real <upper=0>         sigma;
 
 
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}
 
 
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transformed             parameters       {
 
 
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vector[N] mu = alpha + beta*x;
 
 
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}
 
 
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model {
 
 
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y ~ normal(mu, sigma)
 
 
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}
 
 
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generated             quantities        {
 
 
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             real        ypred =             normal_rng(mu, sigma);
 
 
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}
 
 
2. Hierarchical model: factory data with Stan
Note! Assignment 8 build upon this part of the assignment, so it is important to get this assignment correct before you start with Assignment 8.

The factory data in the bsda package contains quality control measurements from 6 machines in a factory. In the data le, each column contains the measurements for a single machine. Quality control measurements are expensive and time-consuming, so only 5 measurements were done for each machine. In addition to the existing machines, we are interested in the quality of another machine (the seventh machine). To read in the data, just use:

library(bsda) data("factory")

For this problem, you’ll use the following Gaussian models:

a separate model, in which each machine has its own model/parameters

a pooled model, in which all measurements are combined and there is no distinction between machines

a hierarchical model, which has a hierarchical structure as described in BDA3 Section 11.6.

As in the model described in the book, use the same measurement standard deviation σ for all the groups in the hierarchical model. In the separate model, however, use separate measurement standard deviation σj for each group j. You should use weakly informative priors for all your models.

The provided Stan code in Listing 2 given on the next page is an example of the separate model (but with very strange results, why?). This separate model can be summarized mathematically as:

yij ∼ N(µj,σj) µj ∼ N(0,1) σj ∼ Inv-χ2(10)

To run Stan for that model, simply use:

data("factory") sm <- rstan::stan_model(file = "[path to stan model code]")

stan_data <- list( y = factory, N = nrow(factory),

J = ncol(factory)

)

model <- rstan::sampling(sm, data = stan_data) model
## Inference for Stan model: 5cbfa723dd8fb382e0b647b3943db079.

## 4 chains, each with iter=2000; warmup=1000; thin=1;

## post-warmup draws per chain=1000, total post-warmup draws=4000. ##   mean se_mean sd              2.5%       25% 50% ## mu[1]      0.11       0.01 0.98              -1.81      -0.56 0.12       0.77

## mu[2]                  0.10            0.01 1.00          -1.86        -0.56          0.10          0.79

## ...

Note! These are not the results you would expect to turn in your report. You will need to change the code for the separate model as well.

For each of the three models (separate, pooled, hierarchical), your tasks are the following:

a)     Describe the model with mathematical notation (as is done for the separate model above). Also describe in words the di erence between the three models.

b)     Implement the model in Stan and include the code in the report. Use weakly informative priors for all your models.

c)     Using the model (with weakly informative priors) report, comment on and, if applicable, plot histograms for the following distributions:

i)         the posterior distribution of the mean of the quality measurements of the sixth machine.

ii)       the predictive distribution for another quality measurement of the sixth machine.

iii)      the posterior distribution of the mean of the quality measurements of a seventh machine (not in the data).

d)     Report the posterior expectation for µ1 with a 90% credible interval but using a normal(0,10) prior for the µ parameter(s) and a Gamma(1,1) prior for the σ parameter(s). For the hierarchical model, use the normal(0,10) and Gamma(1,1) as hyper-priors.

Listing 2: Stan code for a bad separate model

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data {
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int<lower=0> N;
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int<lower=0> J;
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vector[J] y[N];
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}
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parameters          {
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vector[J] mu;
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vector <lower=0>[J] sigma;
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}
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model {
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      //      priors
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for (j in 1:J){
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mu[j] ~ normal(0, 1);
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                sigma[j] ~               inv_chi_square (10);
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}
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      //        likelihood
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for (j in 1:J)
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y[,j] ~ normal(mu[j], sigma[j]);
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}
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generated             quantities        {
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       real       ypred;
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      //     Compute         predictive             distribution
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      //     for     the       first       machine
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        ypred =           normal_rng(mu[1],             sigma[1]);
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}