An overview of Bayesian computational frameworks for teaching

exclude: true

```r
library(fontawesome)
#reticulate::use_condaenv("base")
```

```python
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import arviz as az
```

```r
local({
  hook_err_old <- knitr::knit_hooks$get("error")  # save the old hook
  knitr::knit_hooks$set(error = function(x, options) {
    # now do whatever you want to do with x, and pass
    # the new x to the old hook
    x = sub("## \n## Detailed traceback:\n.*$", "", x)
    x = sub("Error in py_call_impl\$.*?\$\\: ", "", x)
    #x = stringr::str_wrap(x, width = 100)
    hook_err_old(x, options)
  })
  
  hook_warn_old <- knitr::knit_hooks$get("warning")  # save the old hook
  knitr::knit_hooks$set(warning = function(x, options) {
    x = sub("<string>:1: ", "", x)
    #x = stringr::str_wrap(x, width = 100)
    hook_warn_old(x, options)
  })
  
  hook_msg_old <- knitr::knit_hooks$get("output")  # save the old hook
  knitr::knit_hooks$set(output = function(x, options) {
    if (is.null(options$wrap))
      options$wrap = TRUE
    
    x = stringr::str_replace(x, "(## ).* ([A-Za-z]+Warning:)", "\\1\\2")
    x = stringr::str_split(x, "\n")[[1]]
    #x = stringr::str_wrap(x, width = 120, exdent = 3)
    x = stringr::str_remove_all(x, "\r")
    if (options$wrap)
        x = stringi::stri_wrap(x, width=120, exdent = 3, normalize=FALSE)
    x = paste(x, collapse="\n")
    
    #x = stringr::str_wrap(x, width = 100)
    hook_msg_old(x, options)
  })
})
```

---
class: title_bg
count: false

.title[
An overview of (*some*)<br/>
Bayesian computational frameworks<br/>
for teaching
]

.conference[
  .name[ 
    ISBA 2022, Montreal
  ]
  .bitly[ 
    [bit.ly/rundel_isba2022](https://bit.ly/rundel_isba2022)
  ] 
]

.author[
.name[
Colin Rundel
]
.school[
Duke University
]
]

---

## Personal Context

What I teach:

- Sta 323 - Statistical Computing (`R`)

- Sta 523 - Programming for Statistical Science (`R`)

- STA 663 - Statistical Computing and Computation (`Python`)

- STA 344/444/644 - Spatio-temporal modeling (`R`)

Tools I use:

- The majority of my time is spent with R, with a bit of C++

- I use JAGS and Stan for applied modeling

- Recently, more teaching focused on the Python ecosystem

---

## Bayesian computational frameworks?

A collection of tools that implement a domain specific language for expressing and implementing Bayesian statistical models.

<br/>

For example,
- JAGS

- STAN

- pymc

-  &plus; many others

---

## Some teaching considerations

- Ease of use (installation, syntax, debugging, etc.)

- Blackboxiness / High vs low level

- Generalizability

- Performance / Limitations

- Wider curriculum

---

## Installation + basic usage

- All of the frameworks have external / system dependencies 
  - e.g. libjags, Eigen, theano, etc.

- Generally easy to install binary packages are available
  - source installs can be challenging

- If things break it tends to be spectacular and difficult to troubleshoot
  - OS makes a difference
  - 🔥Burn it down🔥 as a path forward

---

## Example - Bayesian SLR

.pull-left[

```r
d = read.csv("data/lm.csv")
plot(d)
```

![](data:image/png;base64,#ISBA2022_slides_files/figure-html/unnamed-chunk-1-1.png)
]

.pull-right[
$$
`\begin{aligned}
y_i|m,b,\sigma &\sim N(m\cdot x_i + b, \sigma) \\
\\
m &\sim N(0, 10) \\
n &\sim N(0, 10)\\
\sigma &\sim N(0, 5) \\
\end{aligned}`
$$
]

---

## SLR - JAGS

.pull-left[

```r
model = "
model{
  m ~ dnorm(0, 1/100)
  b ~ dnorm(0, 1/100)
  
  sigma ~ dnorm(0, 1/25) T(0,)
  
  for(i in 1:length(y)) {
    mu[i] = m*x[i] + b
    y[i] ~ dnorm(mu[i], 1/(sigma^2))
  }
}
"
```
]

.pull-right[

```r
jags_model = rjags::jags.model(
  textConnection(model), data = d, n.chains=4
)
```

```
## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 11
##    Unobserved stochastic nodes: 3
##    Total graph size: 56
##
## Initializing model
```

```r
update(jags_model, n.iter=1000, progress.bar="none")

post_jags = rjags::coda.samples(
  jags_model, variable.names=c("m","b"),
  n.iter=1000, progress.bar="none"
)
```
]

---

## SLR - Stan

.pull-left[

```r
stan = "
data {
  int<lower=0> N;
  vector[N] x;
  vector[N] y;
}
parameters {
  real m;
  real b;
  real<lower=0> sigma;
}
transformed parameters {
  vector[N] mu = m*x + b;
}
model {
  m ~ normal(0, 10);
  b ~ normal(0, 10);
  sigma ~ normal(0, 5);
  y ~ normal(mu, sigma);
}
"
```
]

.pull-right[

```r
post_stan = rstan::stan(
  model_code = stan, 
  data = list(x=d$x, y=d$y, N=nrow(d)),
  pars = c("m", "b", "sigma"),
  chains = 4, warmup = 1000, iter = 2000,
  refresh = 1000, verbose = FALSE, 
)
```

```
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
## Chain 1:
## Chain 1: Gradient evaluation took 1.7e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.17 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1:
## Chain 1:
## Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
## Chain 1: Iteration: 1000 / 2000 [ 50%]  (Warmup)
## Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
## Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
## Chain 1:
## Chain 1:  Elapsed Time: 0.011 seconds (Warm-up)
## Chain 1:                0.012 seconds (Sampling)
## Chain 1:                0.023 seconds (Total)
## Chain 1:
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
## Chain 2:
## Chain 2: Gradient evaluation took 1e-06 seconds
## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.01 seconds.
## Chain 2: Adjust your expectations accordingly!
## Chain 2:
## Chain 2:
## Chain 2: Iteration:    1 / 2000 [  0%]  (Warmup)
## Chain 2: Iteration: 1000 / 2000 [ 50%]  (Warmup)
## Chain 2: Iteration: 1001 / 2000 [ 50%]  (Sampling)
## Chain 2: Iteration: 2000 / 2000 [100%]  (Sampling)
## Chain 2:
## Chain 2:  Elapsed Time: 0.012 seconds (Warm-up)
## Chain 2:                0.013 seconds (Sampling)
## Chain 2:                0.025 seconds (Total)
## Chain 2:
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
## Chain 3:
## Chain 3: Gradient evaluation took 1e-06 seconds
## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.01 seconds.
## Chain 3: Adjust your expectations accordingly!
## Chain 3:
## Chain 3:
## Chain 3: Iteration:    1 / 2000 [  0%]  (Warmup)
## Chain 3: Iteration: 1000 / 2000 [ 50%]  (Warmup)
## Chain 3: Iteration: 1001 / 2000 [ 50%]  (Sampling)
## Chain 3: Iteration: 2000 / 2000 [100%]  (Sampling)
## Chain 3:
## Chain 3:  Elapsed Time: 0.012 seconds (Warm-up)
## Chain 3:                0.013 seconds (Sampling)
## Chain 3:                0.025 seconds (Total)
## Chain 3:
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
## Chain 4:
## Chain 4: Gradient evaluation took 1e-06 seconds
## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.01 seconds.
## Chain 4: Adjust your expectations accordingly!
## Chain 4:
## Chain 4:
## Chain 4: Iteration:    1 / 2000 [  0%]  (Warmup)
## Chain 4: Iteration: 1000 / 2000 [ 50%]  (Warmup)
## Chain 4: Iteration: 1001 / 2000 [ 50%]  (Sampling)
## Chain 4: Iteration: 2000 / 2000 [100%]  (Sampling)
## Chain 4:
## Chain 4:  Elapsed Time: 0.012 seconds (Warm-up)
## Chain 4:                0.011 seconds (Sampling)
## Chain 4:                0.023 seconds (Total)
## Chain 4:
```
]

---

## SLR - pymc3

.pull-left[

```python
import pymc3 as pm
import arviz as az
```

```python
with pm.Model() as lm:
  m = pm.Normal('m', mu=0, sd=10)
  b = pm.Normal('b', mu=0, sd=10)
  
  mu = m * d.x + b
  sigma = pm.HalfNormal('sigma', sd=5)
  
  y = pm.Normal('y', mu=mu, sd=sigma, observed=d.y)
```
]

.pull-right[

```python
with lm:
  post_pymc = pm.sample(return_inferencedata=True, random_seed=1234)
```

```
## █
## Auto-assigning NUTS sampler...
## Initializing NUTS using jitter+adapt_diag...
## Multiprocess sampling (4 chains in 4 jobs)
## NUTS: [sigma, b, m]
## Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 6 seconds.
## There were 6 divergences after tuning. Increase `target_accept` or reparameterize.
```

]

---

## Modelling results

- JAGS models return a coda `mcmc.list` 
  - basic tabular structure that is easy to work with

- Stan models return a `stanfit` S4 object
  - less directly accessible but provides important basic summaries (e.g. `n_eff`, `Rhat`, etc.)
  - easily convertible to coda (`As.mcmc.list()`)

- pymc3 models return* ArviZ `InferenceData` objects (xarray/NetCDF based)
  - complex schema (everything and the kitchen sink approach)
  - less tabular friendly

<br/>

- All frameworks support quick basic visualizations of results (trace, density, caterpillar, etc.)

---

## Error reporting

- As pymc models are Python code, any syntax errors are reported as Python syntax errors

- JAGS and Stan implement their own parsers which have generally helpful error messages with the former tending to be terser / less detailed,

.pull-left[

```r
rjags::jags.model(
  textConnection("
  model{
    m ~ dnorm(0, 1/100
  }
  ")
)
```

```
## Error in rjags::jags.model(textConnection("\n  model{\n    m ~ dnorm(0, 1/100\n  }\n  ")): 
## Error parsing model file:
## syntax error on line 4 near "}"
```
]

.pull-right[

```r
rstan::stan(
  model_code = "
  model {
    m ~ normal(0, 10;
  }
  "
)
```

```
## Error in stanc(file = file, model_code = model_code, model_name = model_name, : 
## 0
##
## Syntax error in 'string', line 2, column 20 to column 21, 
##   parsing error:
##
## Ill-formed phrase. Found an expression. This can be followed 
##   by a ",", a "}", a ")", a "[", a "]" or an infix or postfix 
##   operator.
```
]

- Runtime errors are a mixed bag

---

## Posterior predictive checks

- Possible with all three frameworks, JAGS and Stan require that extra parameters be included in the model:

.pull-left[
JAGS:
```jags
for(i in 1:length(y)) {
  y_tilde[i] ~ dnorm(mu[i], 1/(sigma^2))
}
```
]

.pull-right[
Stan:
```stan
generated quantities {
  real y_tilde[N] = normal_rng(mu, sigma);
}
```
]

- pymc allows for the PPD to be sampled from an existing model result,

```python
with lm:
    y_tilde = pm.sample_posterior_predictive(
        post_pymc, var_names=["y"], random_seed=1234
    )
```

- Similarly, the prior predictive samples can be generated without rewriting the model

---

## Limitations - GP Reg

.pull-left[

```python
d = pd.read_csv("data/gp.csv")
d.shape
```

```
## (100, 3)
```

```python
D = np.array([ np.abs(xi - d.x) for xi in d.x])
I = (D == 0).astype("double")

fig = plt.figure(figsize=(12, 5))
plt.plot(d.x, d.y, "ok", ".")
plt.show()
```

<img src="data:image/png;base64,#ISBA2022_slides_files/figure-html/unnamed-chunk-14-1.png" width="1152" />
]

.pull-right[

```python
with pm.Model() as gp:
  nugget = pm.HalfCauchy("nugget", beta=5)
  sigma2 = pm.HalfCauchy("sigma2", beta=5)
  ls     = pm.HalfCauchy("ls",     beta=5)

Sigma = I * nugget + sigma2 * np.exp(-0.5 * D**2 * ls**2)
  
  y = pm.MvNormal(
    "y", 
    mu=np.zeros(d.shape[0]), 
    cov=Sigma, observed=d.y
  )
```
]

---

## NUTS

```python
with gp:
    post_nuts = pm.sample(
        return_inferencedata = True,
        chains = 2
    )
```

```
## █
## Multiprocess sampling (2 chains in 4 jobs)
## NUTS: [ls, sigma2, nugget]
## Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 240 seconds.
```

```python
az.summary(post)
```

```
##           mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat
## nugget   0.541  0.087   0.397    0.715  ...    0.002    1754.0    1292.0    1.0
## sigma2   4.096  2.557   1.262    8.273  ...    0.060    1067.0    1004.0    1.0
## ls      10.756  2.383   6.593   15.267  ...    0.049    1068.0    1109.0    1.0
##
## [3 rows x 9 columns]
```

---

## Slice steps

```python
with gp:
    step = pm.Slice([nugget, sigma2, ls])
    post_slice = pm.sample(
        return_inferencedata = True,
        chains = 2,
        step = step
    )
```

```
## █
## Multiprocess sampling (2 chains in 4 jobs)
## CompoundStep
## >Slice: [ls]
## >Slice: [sigma2]
## >Slice: [nugget]
## Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 24 seconds.
```

```python
az.summary(post_slice)
```

```
##           mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat
## nugget   0.542  0.085   0.399    0.705  ...    0.002    1573.0    1510.0    1.0
## sigma2   4.557  3.551   1.082   10.070  ...    0.087     915.0     842.0    1.0
## ls      10.526  2.466   5.815   14.552  ...    0.055     989.0     967.0    1.0
##
## [3 rows x 9 columns]
```

---

## Metropolis-Hastings steps

```python
with gp:
    step = pm.Metropolis([nugget, sigma2, ls])
    post_mh = pm.sample(
        return_inferencedata = True,
        chains = 2,
        step = step
    )
```

```
## █
## Multiprocess sampling (2 chains in 4 jobs)
## CompoundStep
## >Metropolis: [ls]
## >Metropolis: [sigma2]
## >Metropolis: [nugget]
## Sampling 2 chains for 1_000 tune and 1_000 draw iterations (2_000 + 2_000 draws total) took 9 seconds.
## The estimated number of effective samples is smaller than 200 for some parameters.
```

```python
az.summary(post_mh)
```

```
##           mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat
## nugget   0.546  0.096   0.373    0.722  ...    0.004     321.0     351.0   1.01
## sigma2   4.535  3.522   1.081    9.282  ...    0.155     231.0     273.0   1.03
## ls      10.518  2.314   6.730   14.987  ...    0.118     188.0     220.0   1.03
##
## [3 rows x 9 columns]
```

---

## Concluding thoughts

- All of these frameworks are a reasonable choice 
  - Many different axes to optimize over
  - More excellent choices than ever before

- Feeling the grass is greener is real

- Personal choice for Fall 2022 ([Sta 344](https://sta344-fa22.github.io/))?

- Probably BRMS -> Stan

---

## Thank you!

.center[ .large[
Questions or Comments?
] ]

<table class="details">
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    <td>
    <a href="mailto:rundel@gmail.com">rundel@gmail.com</a><br/>
    <a href="mailto:colin.rundel@duke.edu">colin.rundel@duke.edu</a>
    </td>
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