speed.Rmd

# Fasten convergence {#speed}

## Under the hood

Our `nimble` workflow so far

```{r}
knitr::include_graphics("images/nimble_workflow_sofar.png")
```

But `nimble` gives full access to the MCMC engine

```{r}
knitr::include_graphics("images/nimble_workflow.png")
```

Steps to use NIMBLE at full capacity

1. Build the model. It is an R object.
2. Build the MCMC.
3. Compile the model and MCMC.
4. Run the MCMC.
5. Extract the samples.

Know what? `nimbleMCMC` does all of this at once.

Back to CJS models with Dipper data.

Define model

```{r echo = FALSE}
dipper <- read_csv(here::here("dat", "dipper.csv"))
y <- dipper %>%
  select(year_1981:year_1987) %>%
  as.matrix()
first <- apply(y, 1, function(x) min(which(x !=0)))
my.constants <- list(N = nrow(y), T = ncol(y), first = first)
my.data <- list(y = y + 1)
zinits <- y + 1 # non-detection -> alive
zinits[zinits == 2] <- 1 # dead -> alive
initial.values <- function() list(phi = runif(1,0,1),
                                  p = runif(1,0,1),
                                  z = zinits)
parameters.to.save <- c("phi", "p")
n.iter <- 2500
n.burnin <- 1000
n.chains <- 2
```

```{r, eval = TRUE, echo = TRUE}
hmm.phip <- nimbleCode({
  delta[1] <- 1              # Pr(alive t = 1) = 1
  delta[2] <- 0              # Pr(dead t = 1) = 0
    phi ~ dunif(0, 1)     # prior survival
    gamma[1,1] <- phi        # Pr(alive t -> alive t+1)
    gamma[1,2] <- 1 - phi    # Pr(alive t -> dead t+1)
    gamma[2,1] <- 0          # Pr(dead t -> alive t+1)
    gamma[2,2] <- 1          # Pr(dead t -> dead t+1)
    p ~ dunif(0, 1)       # prior detection
    omega[1,1] <- 1 - p    # Pr(alive t -> non-detected t)
    omega[1,2] <- p        # Pr(alive t -> detected t)
    omega[2,1] <- 1        # Pr(dead t -> non-detected t)
    omega[2,2] <- 0        # Pr(dead t -> detected t)
  # likelihood
  for (i in 1:N){
    z[i,first[i]] ~ dcat(delta[1:2])
    for (j in (first[i]+1):T){
      z[i,j] ~ dcat(gamma[z[i,j-1], 1:2])
      y[i,j] ~ dcat(omega[z[i,j], 1:2])
    }
  }
})
```

Run and summarise

```{r, eval = TRUE, echo = TRUE, cache = TRUE, message = TRUE}
mcmc.phip <- nimbleMCMC(code = hmm.phip,
                         constants = my.constants,
                         data = my.data,
                         inits = initial.values,
                         monitors = parameters.to.save,
                         niter = n.iter,
                         nburnin = n.burnin,
                         nchains = n.chains)
```

```{r, eval = TRUE, echo=TRUE, message = TRUE}
MCMCsummary(object = mcmc.phip, round = 2)
```

Detailed Nimble workflow

1. Build the model (R object)

```{r, eval = TRUE, echo = TRUE, message = TRUE}
hmm.phip <- nimbleModel(code = hmm.phip,
                        constants = my.constants,
                        data = my.data,
                        inits = initial.values())
```

2. Build the MCMC

```{r, eval = TRUE, echo = TRUE, message = TRUE}
phip.mcmc.configuration <- configureMCMC(hmm.phip)
phip.mcmc <- buildMCMC(phip.mcmc.configuration)
```

3. Compile the model and MCMC

```{r, eval = TRUE, echo = TRUE, message = TRUE}
phip.model <- compileNimble(hmm.phip)
c.phip.mcmc <- compileNimble(phip.mcmc, project = phip.model)
```

4. Run the MCMC

```{r, eval = TRUE, echo = TRUE, message = TRUE}
samples <- runMCMC(c.phip.mcmc, niter = 1000)
# Alternative:
# c.phip.mcmc$run(1000)
# samples <- as.matrix(c.phip.mcmc$mvSamples)
```

5. Look at results

```{r, eval = TRUE, echo = TRUE, message = TRUE}
summary(samples[,"phi"])
summary(samples[,"p"])
```

## Why is it useful?

Use and debug model in `R`

Makes your life easier when it comes to debugging

Inspect variables

```{r}
hmm.phip$gamma
```

Calculate likelihood

```{r}
hmm.phip$calculate()
```

Example of debugging a model in `R`

Pretend an impossible state was given in inits, making a dead bird alive again.

```{r echo=FALSE}
# Pretend that inits for the 6th bird are invalid, going from dead to alive.
saved_z_6 <- phip.model$z[6,]
phip.model$z[6,] <- c(1, 1, 2, 1, 2, 2, 2)
```

```{r, eval = TRUE, echo = TRUE, message = TRUE}
phip.model$calculate("z")        # We can see there is a problem in z (states).
c(phip.model$calculate("z[5,]"), # Bird 5 is valid.
  phip.model$calculate("z[6,]")) # Bird 6 isn't.
phip.model$z[6,]                 # We have found the problem
```

```{r echo=FALSE}
# Restore to valid valies
phip.model$z[6,] <- saved_z_6
```

## Change/modify/write samplers

Slice samplers instead of Metropolis-Hastings.

Samplers on a log scale, especially for a variance, standard deviation, or precision parameter.

Blocking correlated parameters.

To know all samplers available in Nimble, type in `help(samplers)`.

Source code for samplers and distributions is **in R** and can be copied and modified.

Use [`compareMCMCs` package](https://github.com/nimble-dev/compareMCMCs) to compare options (including Stan and Jags!).

Consider a model with wing length and individual random effect on survival.

```{r echo = FALSE}
dipper <- read_csv(here::here("dat", "dipper.csv"))
y <- dipper %>%
  select(year_1981:year_1987) %>%
  as.matrix()
first <- apply(y, 1, function(x) min(which(x !=0)))
wing.length.st <- as.vector(scale(dipper$wing_length))
my.constants <- list(N = nrow(y),
                     T = ncol(y),
                     first = first,
                     winglength = wing.length.st)
zinits <- y + 1 # non-detection -> alive
zinits[zinits == 2] <- 1 # dead -> alive
initial.values <- function() list(beta = rnorm(2,0,1.5),
                                  sdeps = runif(1,0,3),
                                  p = runif(1,0,1),
                                  z = zinits)
parameters.to.save <- c("beta", "sdeps", "p")
n.iter <- 10000
n.burnin <- 2500
n.chains <- 2
```

```{r}
hmm.phiwlrep <- nimbleCode({
    p ~ dunif(0, 1) # prior detection
    omega[1,1] <- 1 - p    # Pr(alive t -> non-detected t)
    omega[1,2] <- p        # Pr(alive t -> detected t)
    omega[2,1] <- 1        # Pr(dead t -> non-detected t)
    omega[2,2] <- 0        # Pr(dead t -> detected t)
  for (i in 1:N){
    logit(phi[i]) <- beta[1] + beta[2] * winglength[i] + eps[i] #<<
    eps[i] ~ dnorm(mean = 0, sd = sdeps) #<<
    gamma[1,1,i] <- phi[i]      # Pr(alive t -> alive t+1)
    gamma[1,2,i] <- 1 - phi[i]  # Pr(alive t -> dead t+1)
    gamma[2,1,i] <- 0           # Pr(dead t -> alive t+1)
    gamma[2,2,i] <- 1           # Pr(dead t -> dead t+1)
  }
  beta[1] ~ dnorm(mean = 0, sd = 1.5)
  beta[2] ~ dnorm(mean = 0, sd = 1.5)
  sdeps ~ dunif(0, 10)
  delta[1] <- 1          # Pr(alive t = 1) = 1
  delta[2] <- 0          # Pr(dead t = 1) = 0
  # likelihood
  for (i in 1:N){
    z[i,first[i]] ~ dcat(delta[1:2])
    for (j in (first[i]+1):T){
      z[i,j] ~ dcat(gamma[z[i,j-1], 1:2, i]) #<<
      y[i,j] ~ dcat(omega[z[i,j], 1:2])
    }
  }
})
```

```{r echo = FALSE, eval = FALSE}
# mcmc.phiwlrep <- nimbleMCMC(code = hmm.phiwlrep,
#                             constants = my.constants,
#                             data = my.data,
#                             inits = initial.values,
#                             monitors = parameters.to.save,
#                             niter = n.iter,
#                             nburnin = n.burnin,
#                             nchains = n.chains)
hmm.phiwlrep.m <- nimbleModel(code = hmm.phiwlrep,
                              constants = my.constants,
                              data = my.data,
                              inits = initial.values())
Rmcmc <- buildMCMC(hmm.phiwlrep.m)
Cmodel <- compileNimble(hmm.phiwlrep.m)
Cmcmc <- compileNimble(Rmcmc, project=hmm.phiwlrep.m)
times.phiwlrep <- system.time( mcmc.phiwlrep <- runMCMC(Cmcmc, niter = 10000, nchains = 2) )
# Alternative:
# Cmcmc$run(10000)
# samples1 <- as.matrix(Cmcmc$mvSamples)
# Cmcmc$run(10000)
# samples2 <- as.matrix(Cmcmc$mvSamples)
save(times.phiwlrep, mcmc.phiwlrep, file = here::here("dat", "wl.RData"))
```

Trace plot for standard deviation of the random effect (default sampler)

```{r echo = FALSE, message=FALSE, warning=FALSE}
load(here::here("dat", "wl.RData"))
```

```{r, echo = FALSE}
MCMCtrace(mcmc.phiwlrep, params = "sdeps", pdf = FALSE)
```

Change samplers

Good sampling strategies depend on the model and data.  What are the samplers used by default?

```{r echo = FALSE}
hmm.phiwlrep.m <- nimbleModel(code = hmm.phiwlrep,
                              constants = my.constants,
                              data = my.data,
                              inits = initial.values())
```


```{r}
mcmcConf <- configureMCMC(hmm.phiwlrep.m)
```

Remove default sampler, and use slice sampler

```{r}
mcmcConf$removeSamplers('sdeps')
mcmcConf$addSampler(target = 'sdeps',
                    type = "slice") #<<
mcmcConf
```

```{r, echo = FALSE}
Rmcmc <- buildMCMC(mcmcConf)
Cmodel <- compileNimble(hmm.phiwlrep.m)
Cmcmc <- compileNimble(Rmcmc, project=hmm.phiwlrep.m)
times.phiwlrep.with.slice <- system.time( mcmc.phiwlrep.with.slice <- runMCMC(Cmcmc, niter = 10000, nchains = 2))
samples1 <- mcmc.phiwlrep.with.slice[[1]]
samples2 <- mcmc.phiwlrep.with.slice[[2]]
iUse <- 2501:10000
ESS.with.slice <- coda::effectiveSize(list(coda::as.mcmc(samples1[iUse, 'sdeps']),
                                           coda::as.mcmc(samples2[iUse, 'sdeps'])))
# Cmcmc$run(10000)
# samples1 <- as.matrix(Cmcmc$mvSamples)
# Cmcmc$run(10000)
# samples2 <- as.matrix(Cmcmc$mvSamples)
df <- data.frame(iter = c(2501:10000, 2501:10000),
                 samples = c(samples1[2501:10000,"sdeps"], samples2[2501:10000,"sdeps"]),
                 chain = c(rep("chain 1", length(samples1[2501:10000,"sdeps"])),
                           rep("chain 2", length(samples2[2501:10000,"sdeps"]))))
df.beta <- data.frame(iter = c(2501:10000, 2501:10000),
                 beta1 = c(samples1[2501:10000,"beta[1]"], samples2[2501:10000,"beta[1]"]),
                 beta2 = c(samples1[2501:10000,"beta[2]"], samples2[2501:10000,"beta[2]"]),
                 chain = c(rep("chain 1", length(samples1[2501:10000,"sdeps"])),
                           rep("chain 2", length(samples2[2501:10000,"sdeps"]))))
```

Trace plot for standard deviation of the random effect (slice sampler)

```{r, echo = FALSE}
MCMCtrace(mcmc.phiwlrep.with.slice, params = "sdeps", pdf = FALSE)
```

```{r, echo = FALSE}
# df %>%
#   ggplot() +
#   aes(x = iter, y = samples, group = chain, color = chain) +
#   geom_line() +
#   labs(x = "iterations", y = "random effect standard deviation", color = "")
```

## Which is better? MCMC efficiency

+ MCMC efficiency depends on both mixing and computation time.

+ MCMC efficiency = Effective Sample Size (ESS) / computation time.

+ MCMC efficiency is the number of effectively independent posterior samples generated per second.

+ ESS is different for each parameter.  (Computation time is the same for each parameter.)

+ ESS can be estimated from packages `coda` or `mcmcse`. These give statistical estimates, so different runs will give different estimates.

```{r echo = FALSE}
iUse <- 2501:10000
time_frac <- 0.75 # Include 75% of the time to count sampling of recorded sampled, which is not the only valid option.
ESS <- coda::effectiveSize(list(coda::as.mcmc(mcmc.phiwlrep[[1]][iUse, 'sdeps']),
                                        coda::as.mcmc(mcmc.phiwlrep[[2]][iUse, 'sdeps'])))
t1 <- (time_frac * times.phiwlrep[3])
efficiency <- ESS / t1
t.with.slice <- (time_frac * times.phiwlrep.with.slice[3])
efficiency.with.slice <- ESS.with.slice / t.with.slice
rd2 <- function(x) round(x, digits = 2)
```

+ Efficiency with default sampler = `r rd2(ESS)` / `r rd2(t1)` = `r rd2(efficiency)`.

+ Efficiency with slice sampler = `r rd2(ESS.with.slice)` / `r rd2(t.with.slice)` = `r rd2(efficiency.with.slice)`.

Block sampling

+ High correlation in (regression) parameters may make independent samplers inefficient.

```{r, echo = FALSE}
df.beta %>%
  ggplot() +
  aes(x = beta1, y = beta2, group = chain, color = chain) +
  geom_point(alpha = .05) +
  labs(x = "beta1", y = "beta2", color = "")
```

Block sampling (propose candidate values from multivariate distribution) might help.

Remove and replace independent RW samples by block sampling. Then proceed as usual.

```{r message = FALSE, warning=FALSE}
mcmcConf$removeSamplers(c('beta[1]','beta[2]'))
mcmcConf$addSampler(target = c('beta[1]','beta[2]'),
                    type = "RW_block") #<<
```

```{r message = FALSE, warning=FALSE}
mcmcConf
```

## Strategies for improving MCMC

+ Choose better initial values.

+ Customize sampler choice (more in [Chapter 7 of the User's manual](https://r-nimble.org/html_manual/cha-mcmc.html)).

+ Reparameterize, e.g. standardize covariates, deal with parameter redundancy.

+ Rewrite the model.

    + Vectorize to improve computational efficiency (not covered).
    + Avoid long chains of deterministic dependencies.
    + Marginalize to remove parameters
    + Use new functions and new distributions written as nimbleFunctions.

+ Write new samplers that take advantage of particular model structures (not covered).

+ Using multiple cores with parallelization: see how-to at <https://r-nimble.org/nimbleExamples/parallelizing_NIMBLE.html>

## Marginalization

User-defined distributions is another neat feature of Nimble.

Integrate over latent states if those are not the focus of ecological inference (marginalization).

Marginalization often (but not always) improves MCMC. See [Ponisio et al. 2020](https://onlinelibrary.wiley.com/doi/full/10.1002/ece3.6053) for examples.

The [nimbleEcology](https://cran.r-project.org/web/packages/nimbleEcology/vignettes/Introduction_to_nimbleEcology.html) package implements capture-recapture models and HMMs with marginalization.

Our model $(\phi_A, \phi_B, \psi_{AB}, \psi_{BA}, p_A, p_B)$

```{r eval = FALSE}
multisite <- nimbleCode({
...
  # Likelihood
  for (i in 1:N){
    # Define latent state at first capture
    z[i,first[i]] <- y[i,first[i]] - 1
    for (t in (first[i]+1):K){
      # State process: draw S(t) given S(t-1)
      z[i,t] ~ dcat(gamma[z[i,t-1],1:3])
      # Observation process: draw O(t) given S(t)
      y[i,t] ~ dcat(omega[z[i,t],1:3])
    }
  }
})
```

Same model with nimbleEcology

```{r eval = FALSE}
multisite <- nimbleCode({
...
# initial state probs
for(i in 1:N) {
  init[i, 1:4] <- gamma[ y[i, first[i] ] - 1, 1:4 ] # first state propagation
}

# likelihood
for (i in 1:N){
  y[i,(first[i]+1):K] ~ dHMM(init = init[i,1:4],           # count data from first[i] + 1
                             probObs = omega[1:4,1:4],     # observation matrix
                             probTrans = gamma[1:4,1:4],   # transition matrix
                             len = K - first[i],           # nb of occasions
                             checkRowSums = 0)             # do not check whether elements in a row sum tp 1
}
...
```

This runs twice as fast as the standard formulation with explicit latent states.

Marginalizing typically gives better mixing.

Reducing redundant calculations

So far, a row of the dataset is an individual. However, several individuals may share the same encounter history.

The contribution of $M$ individuals with the same encounter history is the likelihood of this particular encounter history raised to the power $M$.

Using this so-called **weighted likelihood** greatly decreases the computational burden.

This idea is used in most computer programs that implement maximum likelihood. In the Bayesian framework, the same idea was proposed in [Turek et al. (2016)](https://doi.org/10.1007/s10651-016-0353-z).

Cannot be done in Jags. Can be done in nimble thanks to nimble functions! The run is *much* faster. Also allows fitting models to big datasets. More details in dedicated Worksheet.


## Summary

+ NIMBLE is under active development.  Contributors are welcome, including those who want to get involved but don't know where.

+ Faster building of models and algorithms.  Ability to save and re-load compiled work.

+ Automatic differentiation of model calculations, enabling Hamiltonian Monte Carlo, other sampling strategies, and Laplace approximation.

+ Tools for building packages that use NIMBLE "under the hood".

## Further reading

+ Turek, D., de Valpine, P. & Paciorek, C.J. [Efficient Markov chain Monte Carlo sampling for hierarchical hidden Markov models](https://doi.org/10.1007/s10651-016-0353-z) *Environ Ecol Stat* 23: 549–564 (2016).

+ Ponisio, L.C., de Valpine, P., Michaud, N., and Turek, D. [One size does not fit all: Customizing MCMC methods for hierarchical models using NIMBLE](https://doi.org/10.1002/ece3.6053) *Ecol Evol.* 10: 2385–2416 (2020).

+ Nimble workshop to come 26-28 May, check out [here](https://r-nimble.org/nimble-virtual-short-course-may-26-28).

+ Nimble workshop material online available [here](https://github.com/nimble-training).

+ Nimble [manual](https://r-nimble.org/html_manual/cha-welcome-nimble.html) and [cheatsheet](https://r-nimble.org/cheatsheets/NimbleCheatSheet.pdf).

## Tip by Cyril Milleret

- For a model that takes long to run, it might be useful to run MCMC in “bites”. This allows to check the results before the model is done running. For example we can save 80 bites each containing 250 iterations.  

```
bite.size <- 250 # number of iterations in each bite to be exported and cleared
bites <- 80 # we run 80 bites containing of 250 iterations each. 
for(nb in 1:80){
   print(nb)
   if(nb==1){# first bite 
      Cmcmc$run(bite.size)
   }
   if(nb>1){# second bite or more
      Cmcmc$run(bite.size, reset = FALSE)
   }
   this.sample <- as.matrix(Cmcmc$mvSamples)# save the 250iterations
   save(this.sample, RunTime, file=paste(”bite_”, nb, "RData”, sep=””))
}
```

Now imagine the MCMC is still running but 25 bites were finished. We can check the results of the 25 first bites (6250 iterations: 

```
bites <- 25
MCMC <- list()
for(nb in 1: bites){
  load(paste(”bite_”, nb, "RData”, sep=””))
  MCMC[[nb]] <- this.sample
}
MCMCsamples <- as.mcmc(do.call(rbind, MCMC)))
```