# Automatic Derivatives

As of version xx.xx.xx, NIMBLE can automatically provide numerically accurate derivatives of potentially arbitrary order for most calculations in models and/or nimbleFunctions. This feature is in beta-testing mode, meaning it has some rough edges and we expect bugs will be found. This feature supports methods such as Hamiltonian Monte Carlo (HMC, see package nimbleHMC), Laplace approximation, and fast optimization with methods that use function gradients.

You can install the beta-testing version using the remotes package like this:

remotes::install_github("nimble-dev/nimble", ref="AD-rc0", subdir="packages/nimble", quiet=TRUE)

Automatic (or algorithmic) differentiation (AD) refers to the method of carrying derivative information through a set of mathematical operations. When done this way, derivatives are numerically accurate to the precision of the computer. This is distinct from finite element methods, used by packages such as numDeriv (Gilbert and Varadhan 2019) and pracma (Borchers 2022), which approximate derivatives by calculating function values at extremely nearby points. Finite element methods are slower and less accurate than AD. It is also distinct from writing separate functions for each derivative, which can become very complicated and sometimes slower than AD. NIMBLE uses the CppAD package (Bell 2022) as its AD engine, following TMB’s success (Kristensen et al. 2016) in doing so. CppAD supports potentially arbitrary-order derivatives. A general reference on AD is Griewank and Walther (2008).

AD is not a simple feature in NIMBLE. It requires care in models and nimbleFunctions, awareness of what can go wrong, and testing to be sure things are working as you want. Calls to compileNimble that include AD features will result in slower C++ compilation. That said, using a packaged AD algorithm with a model should be as simple as setting buildDerivs=TRUE in the model.

We will introduce NIMBLE’s AD features step by step, from simply turning them on for a model to using them in your own nimbleFunctions. We will show you:

1. how to turn on derivatives in a model and use them for Hamiltonian Monte Carlo (HMC).
2. basics of obtaining derivatives in nimbleFunctions.
3. advanced uses of obtaining derivatives in nimbleFunctions, including double-taping.
4. how to get derivatives involving model calculations.
5. details on what functions are supported and not supported for derivatives.
6. automatic parameter transformations to give any model an unconstrained parameter space for algorithms to work in.
7. an example showing use of nimble’s derivatives for maximum likelihood estimation.
8. an example of nimble’s Laplace approximation.

First, make sure to set the option to enable derivative features. This should be TRUE by default, but just in case:

nimbleOptions(enableDerivs = TRUE)

## How to turn on derivatives in a model and use them for Hamiltonian Monte Carlo (HMC)

NIMBLE’s HMC sampler is provided in a separate package, nimbleHMC. HMC is an MCMC method that works by a physical analogy of frictionless motion on a surface that is the negative log posterior density. It often achieves good mixing, although at high computational cost. Within nimble, HMC is more flexible than in some implementations because HMC can be used on some parts of a model while other samplers are used on (the same or) other parts of the same model. Indeed, the inventor of HMC discussed using it in this way (Neal 2011).

NIMBLE’s Laplace approximation is provided in buildLaplace. An example is given below.

We’ll use a Poisson GLMM as a simple example model. There will be 10 groups (i) of 5 observations (j) each. Each observation has a covariate, X, and each group has a random effect ran_eff. Here is the model code:

model_code <- nimbleCode({
# priors
intercept ~ dnorm(0, sd = 100)
beta ~ dnorm(0, sd = 100)
sigma ~ dunif(0, 10)
# random effects and data
for(i in 1:10) {
# random effects
ran_eff[i] ~ dnorm(0, sd = sigma)
for(j in 1:5) {
# data
y[i,j] ~ dpois(exp(intercept + beta*X[i,j] + ran_eff[i]))
}
}
})

We’ll simulate some values for X.

set.seed(123)
X <- matrix(rnorm(50), nrow = 10)

In order to use one of the algorithms for a specific model, that model must be created with buildDerivs=TRUE.

model <- nimbleModel(model_code, constants = list(X = X), calculate = FALSE,
buildDerivs = TRUE) # Here is the argument needed for AD.
## Defining model
## Building model
## Checking model sizes and dimensions
##   [Note] This model is not fully initialized. This is not an error.
##          To see which variables are not initialized, use model$initializeInfo(). ## For more information on model initialization, see help(modelInitialization). To finish setting up the model, we can set parameter values, simulate data values, and then set those as the data to use. model$intercept <- 0
model$beta <- 0.2 model$sigma <- 0.5
model$calculate() # This will return NA because the model is not fully initialized. ## [1] NA model$simulate(model$getDependencies('ran_eff')) model$calculate() # Now the model is fully initialized: all nodes have valid values.
## [1] -80.74344
model$setData('y') # Now the model has y marked as data, with values from simulation. Now we can configure an MCMC to use HMC. In this case we’ll use HMC for the whole model, i.e. all parameters and latent states. library(nimbleHMC) HMC <- buildHMC(model) ## ===== Monitors ===== ## thin = 1: beta, intercept, sigma ## ===== Samplers ===== ## HMC sampler (1) ## - intercept, beta, sigma, ran_eff[1], ran_eff[2], ran_eff[3], ran_eff[4], ran_eff[5], ran_eff[6], ran_eff[7], ran_eff[8], ran_eff[9], ran_eff[10] Cmodel <- compileNimble(model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. CHMC <- compileNimble(HMC, project = model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. samples <- runMCMC(CHMC, niter = 1000, nburnin = 500) # short run for illustration ## running chain 1 ... ## [Note] HMC sampler (nodes: intercept, beta, sigma, ran_eff[1], ran_eff[2], ran_eff[3], ran_eff[4], ran_eff[5], ran_eff[6], r...) is using 500 warmup iterations ## |-------------|-------------|-------------|-------------| ## |-------------------------------------------------------| ## [Note] HMC sampler (nodes: intercept, beta, sigma, ran_eff[1], ran_eff[2], ran_eff[3], ran_eff[4], ran_eff[5], ran_eff[6], r...) encountered 7 divergent paths We will not investigate results in detail, but here is a summary to see that reasonable results were generated. summary(coda::as.mcmc(samples)) ## ## Iterations = 1:500 ## Thinning interval = 1 ## Number of chains = 1 ## Sample size per chain = 500 ## ## 1. Empirical mean and standard deviation for each variable, ## plus standard error of the mean: ## ## Mean SD Naive SE Time-series SE ## beta 0.1826 0.1396 0.006241 0.006576 ## intercept -0.2162 0.3088 0.013810 0.023374 ## sigma 0.7697 0.2867 0.012821 0.022572 ## ## 2. Quantiles for each variable: ## ## 2.5% 25% 50% 75% 97.5% ## beta -0.09827 0.09303 0.1788 0.280194 0.4464 ## intercept -0.88502 -0.39564 -0.1723 -0.003709 0.3818 ## sigma 0.34567 0.55507 0.7178 0.930289 1.4386 ## Basics of obtaining derivatives in nimbleFunctions. Now that we have seen a derivative-enabled algorithm work, let’s see how to obtain derivatives in methods you write. We’ll start by showing how derivatives work in nimbleFunctions without using a model. The AD system allows you to obtain derivatives of one function or method from another. Let’s get derivatives of the function Y = exp(-d * X) where X is a vector, d is a scalar, and Y is a vector. derivs_demo <- nimbleFunction( setup = function() {}, run = function(d = double(), x = double(1)) { return(exp(-d*x)) returnType(double(1)) }, methods = list( derivsRun = function(d = double(), x = double(1)) { wrt <- 1:(1 + length(x)) # total length of d and x return(derivs(run(d, x), wrt = wrt, order = 0:2)) returnType(ADNimbleList()) } ), buildDerivs = 'run' ) Here are some things to notice: • Having setup code allows the nimbleFunction to have multiple methods (i.e. to behave like a class definition in object-oriented programming). If, as here, you don’t need the setup to do anything, you can instead use setup=TRUE. • Any arguments to run that are real numbers (i.e. regular double precision numbers, but not integers or logicals) will have derivatives tracked when called through derivs. • The “with-respect-to” (wrt) argument gives indices of the arguments for which you want derivatives. In this case, we’re including all elements of d and X. The indices form one sequence over all arguments. • order can contain any of 0 (value), 1 (1st order), or 2 (2nd order) derivatives requested. Higher-order derivatives can be obtained by double-taping, described below. • Basics of nimbleFunction programming are covered in Chapter (ref). These include type declarations (double() for scalar, double(1) for vector), that run is just a method with a favored name, and the distinction between setup code and run code. • derivs can alternatively be called nimDerivs. In fact the former will be converted to the latter internally. Let’s see this work. my_derivs_demo <- derivs_demo() C_my_derivs_demo <- compileNimble(my_derivs_demo) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. d <- 1.2 x <- c(2.1, 2.2) C_my_derivs_demo$derivsRun(d, x)
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] 0.08045961 0.07136127
## Field "jacobian":
##            [,1]        [,2]        [,3]
## [1,] -0.1689652 -0.09655153  0.00000000
## [2,] -0.1569948  0.00000000 -0.08563352
## Field "hessian":
## , , 1
##
##           [,1]      [,2] [,3]
## [1,] 0.3548269 0.1222986    0
## [2,] 0.1222986 0.1158618    0
## [3,] 0.0000000 0.0000000    0
##
## , , 2
##
##           [,1] [,2]      [,3]
## [1,] 0.3453885    0 0.1170325
## [2,] 0.0000000    0 0.0000000
## [3,] 0.1170325    0 0.1027602

We can see that using order = 0:2 results in the the value (“0th” order result, i.e. the value returned by run(d, x)), the Jacobian (matrix of first order derivatives), and the Hessian (array of second order derivatives). The run function here has taken three inputs (in the order d, x[1], x[2]) and returned two outputs (the first and second elements of exp(-d*x)).

The i-th Jacobian row contains the first derivatives of the i-th output with respect to d, x[1], and x[2], in that order. The first and second indices of the Hessian array follow the same ordering as the columns of the Jacobian. The third index of the Hessian array is for the output values. For example, hessian[3,1,2] is the second derivative of the second output value (third index = 2) with respect to x[2] (first index = 3) and d (second index = 1).

(Although it may seem inconsistent to have the output index be first for Jacobian and last for Hessian, it is consistent with some standards for how these objects are created in other packages and used mathematically.)

When a function being called for derivatives (run in this case) has non-scalar arguments (x in this case), the indexing of inputs goes in order of arguments, and in column-major order within arguments. For example, if we had arguments x = double(1) and z = double(2) (a matrix), the inputs would be ordered as x[1], x[2], …, z[1, 1], z[2, 1], …, z[1, 2], z[2, 2], etc.

### Checking derivatives with uncompiled execution

Derivatives can also be calculated in uncompiled execution, but they will be much slower and less accurate: slower because they are run in R, and less accurate because they use finite element methods (from packages pracma and/or numDeriv). Uncompiled execution is mostly useful for checking that compiled derivatives are working correctly, because although they are slower and less accurate, they are also much simpler internally and thus provide good checks on compiled results. For example:

my_derivs_demo$derivsRun(d, x) ## nimbleList object of type NIMBLE_ADCLASS ## Field "value": ## [1] 0.08045961 0.07136127 ## Field "jacobian": ## [,1] [,2] [,3] ## [1,] -0.1689652 -0.09655153 0.00000000 ## [2,] -0.1569948 0.00000000 -0.08563352 ## Field "hessian": ## , , 1 ## ## [,1] [,2] [,3] ## [1,] 0.3548269 0.1222986 0 ## [2,] 0.1222986 0.1158618 0 ## [3,] 0.0000000 0.0000000 0 ## ## , , 2 ## ## [,1] [,2] [,3] ## [1,] 0.3453885 0 0.1170325 ## [2,] 0.0000000 0 0.0000000 ## [3,] 0.1170325 0 0.1027602 We can see that the results are very close, but typically not identical, to those from the compiled version. ### Holding some local variables out of derivative tracking. Sometimes one wants to omit tracking of certain variables in derivative calculations. Here is an example. Say that we write exp(-d*x) using a for-loop instead of a vectorized operation as above. It wouldn’t make sense to track derivatives for the for-loop index (i), and indeed it would cause compileNimble to fail. The code to use a for-loop while telling derivs to ignore the for-loop index is: derivs_demo2 <- nimbleFunction( setup = function() {}, run = function(d = double(), x = double(1)) { ans <- numeric(length = length(x)) for(i in 1:length(x)) ans[i] <- exp(-d*x[i]) return(ans) returnType(double(1)) }, methods = list( derivsRun = function(d = double(), x = double(1)) { wrt <- 1:(1 + length(x)) # total length of d and x return(derivs(run(d, x), wrt = wrt, order = 0:2)) returnType(ADNimbleList()) } ), buildDerivs = list(run = list(ignore = 'i')) ) We can see that it gives identical results as above, looking at only the Jacobian to keep the output short. my_derivs_demo2 <- derivs_demo2() C_my_derivs_demo2 <- compileNimble(my_derivs_demo2) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. d <- 1.2 x <- c(2.1, 2.2) C_my_derivs_demo2$derivsRun(d, x)$jacobian ## [,1] [,2] [,3] ## [1,] -0.1689652 -0.09655153 0.00000000 ## [2,] -0.1569948 0.00000000 -0.08563352 One might think it should be obvious that i should not be involved in derivatives, but sometimes math is actually done with a for-loop index. Another way to ensure derivatives won’t be tracked for i would be to write i <- 1L, before the loop. By assigning a definite integer to i, it will be established as definitely of integer type throughout the function and thus not have derivatives tracked. (If you are not familiar with the L in R, try storage.mode(1L) vs storage.mode(1).) Note the more elaborate value for buildDerivs. Above, this was just a character vector. Here it is a named list, with each element itself being a list of control values. The control value ignore is a character vector of variable names to ignore in derivative tracking. buildDerivs = 'run' is equivalent to buildDerivs = list(run = list()). ### Using AD with multiple nimbleFunctions Derivatives will be tracked for whatever series of calculations occur in a method, possibly including calls to other methods or nimbleFunctions that have buildDerivs set. Let’s look at an example where we have a separate function to return the element-wise square root of an input vector. The net calculation for derivatives will be sqrt(exp(-d*x))). nf_sqrt <- nimbleFunction( run = function(x = double(1)) { return(sqrt(x)) returnType(double(1)) }, buildDerivs = TRUE ) derivs_demo3 <- nimbleFunction( setup = function() {}, run = function(d = double(), x = double(1)) { ans <- exp(-d*x) ans <- nf_sqrt(ans) return(ans) returnType(double(1)) }, methods = list( derivsRun = function(d = double(), x = double(1)) { wrt <- 1:(1 + length(x)) # total length of d and x return(derivs(run(d, x), wrt = wrt, order = 0:2)) returnType(ADNimbleList()) } ), buildDerivs = 'run' ) And then let’s see it work: my_derivs_demo3 <- derivs_demo3() C_my_derivs_demo3 <- compileNimble(my_derivs_demo3) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. d <- 1.2 x <- c(2.1, 2.2) C_my_derivs_demo3$derivsRun(d, x)$jacobian ## [,1] [,2] [,3] ## [1,] -0.2978367 -0.1701924 0.0000000 ## [2,] -0.2938488 0.0000000 -0.1602812 Note that for a nimbleFunction without setup code, one can say buildDerivs=TRUE, buildDerivs = 'run', or buildDerivs = list(run = list()). ### Understanding more about how AD works: taping of operations. At this point, it will be helpful to grasp more of how AD works and its implementation in nimble via the CppAD library (Bell 2022). AD methods work by following a set of calculations multiple times, sometimes in reverse order. For example, the derivative of $$y = \exp(x^2)$$ can be calculated (by the chain rule of calculus) as the derivative of $$y = \exp(z)$$ (evaluated at $$z = x^2$$) times the derivative of $$z = x^2$$ (evaluated at $$x$$). This can be calculated by first going through the sequence of steps for the value, i.e. (i) $$z = x^2$$, (ii) $$y = \exp(z)$$, and then again for the derivatives, i.e. (i) $$dz = 2x (dx)$$, (ii) $$dy = \exp(z) dz$$. These steps determine the instantaneous change $$dy$$ that results from an instantaneous change $$dx$$. (It may be helpful to think of both values as a function of a third variable such as $$t$$. Then we are determining $$dy/dt$$ as a function of $$dx/dt$$.) The derivative calculations typically need the values. For example, the derivative of $$\exp(z)$$ is (also) $$\exp(z)$$, for which the value of $$z$$ is needed, which in this case is $$x^2$$, one of the steps in calculating the value of $$\exp(x^2)$$. (In fact, in this case the derivative of $$\exp(z)$$ can more simply use the value of $$y$$ itself.) Hence, values are calculated before derivatives. What was just described is called forward mode AD. The derivatives can also be calculated by working through $$\exp(x^2)$$ in reverse mode. This is often less familiar to graduates of basic calculus courses. In reverse mode, one determines what instantaneous change $$dx$$ would give an instantaneous change $$dy$$. For example, given a value of $$dy$$, we can calculate $$1/dz = \exp(z) / dy$$ and then $$(1/dx) = (x^2) / dz$$. In general, choice of forward mode versus reverse mode has to do with the lengths of inputs and outputs and possibly specific operations involved. In nimble, some reasonable defaults are used. In CppAD, the metaphor of pre-digital technology – magnetic tapes – is used for the set of operations in a calculation. When a line with nimDerivs is first run, the operations in the function it calls are taped and then re-used – played – in forward and/or reverse orders to obtain derivatives. Hence we will refer to the AD tape in explaining some features below. It is also possible to reset (i.e. re-record) the AD tape used for a particular nimDerivs call. When taped operations are played, the resulting operations can themselves be taped. We call this meta-taping or double-taping. It is useful because it can sometimes boost efficiency. ### Resetting a nimDerivs call The first time a call to nimDerivs runs, it records a tape of the operations in the its first argument (run(d, x) in this case). On subsequent calls, it re-uses that tape without re-recording it. That means subsequent calls are usually much faster. It also means that subsequent calls must use the same sized arguments as the recorded call. For example, we can use C_my_derivs_demo$derivsRun with new arguments of the same size(s):

C_my_derivs_demo$derivsRun(-0.4, c(3.2, 5.1)) ## nimbleList object of type NIMBLE_ADCLASS ## Field "value": ## [1] 3.596640 7.690609 ## Field "jacobian": ## [,1] [,2] [,3] ## [1,] -11.50925 1.438656 0.000000 ## [2,] -39.22211 0.000000 3.076244 ## Field "hessian": ## , , 1 ## ## [,1] [,2] [,3] ## [1,] 36.829591 -8.2003386 0 ## [2,] -8.200339 0.5754624 0 ## [3,] 0.000000 0.0000000 0 ## ## , , 2 ## ## [,1] [,2] [,3] ## [1,] 200.03275 0 -23.379452 ## [2,] 0.00000 0 0.000000 ## [3,] -23.37945 0 1.230497 However, if we call C_my_derivs_demo$derivsRun with length(x) different from 2, the result will be garbage. If we need to change the size of the arguments, we need to re-record the tape. This is done with the reset argument.

Here is a slightly more general version of the derivs_demo allowing a user to reset the tape. It also takes order and wrt as arguments instead of hard-coding them.

derivs_demo4 <- nimbleFunction(
setup = function() {},
run = function(d = double(), x = double(1)) {
return(exp(-d*x))
returnType(double(1))
},
methods = list(
derivsRun = function(d = double(), x = double(1),
wrt = integer(1), order = integer(1),
reset = logical(0, default=FALSE)) {
return(derivs(run(d, x), wrt = wrt, order = order, reset = reset))
}
),
buildDerivs = 'run'
)

Now we will illustrate the use of reset. To make shorter output, we’ll request only the Jacobian.

my_derivs_demo4 <- derivs_demo4()
C_my_derivs_demo4 <- compileNimble(my_derivs_demo4)
## Compiling
##   [Note] This may take a minute.
##   [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details.
d <- 1.2
x <- c(2.1, 2.2)
# On the first call, reset is ignored because the tape must be recorded.
C_my_derivs_demo4$derivsRun(d, x, 1:3, 1) ## nimbleList object of type NIMBLE_ADCLASS ## Field "value": ## numeric(0) ## Field "jacobian": ## [,1] [,2] [,3] ## [1,] -0.1689652 -0.09655153 0.00000000 ## [2,] -0.1569948 0.00000000 -0.08563352 ## Field "hessian": ## <0 x 0 x 0 array of double> ##  # On the second call, reset=FALSE, so the tape is re-used. C_my_derivs_demo4$derivsRun(-0.4, c(3.2, 5.1), 1:3, 1)
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## numeric(0)
## Field "jacobian":
##           [,1]     [,2]     [,3]
## [1,] -11.50925 1.438656 0.000000
## [2,] -39.22211 0.000000 3.076244
## Field "hessian":
## <0 x 0 x 0 array of double>
## 
# If we need a longer X, we need to say reset=TRUE
C_my_derivs_demo4$derivsRun(1.2, c(2.1, 2.2, 2.3), 1:4, 1, reset=TRUE) ## nimbleList object of type NIMBLE_ADCLASS ## Field "value": ## numeric(0) ## Field "jacobian": ## [,1] [,2] [,3] [,4] ## [1,] -0.1689652 -0.09655153 0.00000000 0.00000000 ## [2,] -0.1569948 0.00000000 -0.08563352 0.00000000 ## [3,] -0.1455711 0.00000000 0.00000000 -0.07595012 ## Field "hessian": ## <0 x 0 x 0 array of double> ##  # Now we can use with reset=FALSE with X of length 3 C_my_derivs_demo4$derivsRun(-0.4, c(3.2, 5.1, 4.5), 1:4, 1)
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## numeric(0)
## Field "jacobian":
##           [,1]     [,2]     [,3]     [,4]
## [1,] -11.50925 1.438656 0.000000 0.000000
## [2,] -39.22211 0.000000 3.076244 0.000000
## [3,] -27.22341 0.000000 0.000000 2.419859
## Field "hessian":
## <0 x 0 x 0 array of double>
## 

Remember, you can make multiple instances of derivs_demo4, so you could keep track of the argument sizes currently recorded for each one.

Other important situations where you need to reset a tape are when values that are otherwise “baked in” to an AD tape need to be changed. By “baked in”, we mean that some values are permanently part of a tape until it is reset.

#### What gets baked into AD tapes until reset=TRUE

Specifically, you need to reset a tape when:

• the extent of any for-loops change. For-loops are not natively recorded, but rather each operation done in execution of the for-loop is recorded. (This is sometimes called “loop unrolling”.) Therefore, in for(i in start:end), the values of start and end are in effect baked into a tape.
• the outcomes of any if-then-else conditions in a nimbleFunction are changed.
• any other member data used in calculations changes. It is possible to create a variable in the setup function and use it in methods such as run. Such variables become member data in the terminology of object-oriented programming. Such variables will not have derivatives tracked if used in a function while it is being taped, so the values will be baked into the tape. This can be useful if you understand what is happening or confusing if not.

Uncompiled execution of derivs ignores the reset argument.

### A note on performance benchmarking

If you are interested in measuring the performance of AD in nimble, please remember that the first call to nimDerivs and any subsequent call with reset=TRUE will be slower, often much slower, that subsequent calls.

Suppose you are interested only in the Jacobian, not in the value, and/or want only some elements of the Jacobian. You might still need the value, but obtaining the value alone from an AD tape will be slower than obtaining it by simply calling the function. On the other hand, AD methods need to calculate the value before calculating first order derivatives, so the value will be calculated anyway. However, in some cases, some of the steps of value calculations aren’t really needed if one only wants, say, first-order derivatives. In addition, if not all elements of the Jacobian are wanted, then some unnecessary calculations will be done internally that one might want to avoid.

A way to cut out unnecessary calculations is to record a tape of a tape, which we call double taping. Let’s see an example before explaining further.

derivs_demo5 <- nimbleFunction(
setup = function() {},
run = function(d = double(), x = double(1)) {
return(exp(-d*x))
returnType(double(1))
},
methods = list(
jacobian_run_wrt_d  = function(d = double(), x = double(1),
wrt = integer(1),
reset = logical(0, default=FALSE)) {
ans <- derivs(run(d, x), wrt = wrt, order = 1, reset = reset)
jac <- ans$jacobian[,1] # derivatives wrt 'd' only return(jac) returnType(double(1)) }, derivsJacobian = function(d = double(), x = double(1), wrt = integer(1), order = integer(1), reset = logical(0, default = FALSE)) { innerWrt <- nimInteger(value = 1, length = 1) ans <- nimDerivs(jacobian_run_wrt_d(d, x, wrt = innerWrt, reset = reset), wrt = wrt, order = order, reset = reset) return(ans) returnType(ADNimbleList()) } ), buildDerivs = c('run', 'jacobian_run_wrt_d') ) What is happening in this code? jacobian_run_wrt_d is a method that returns part of the Jacobian of run(d, x), specifically the first column, which contains derivatives with respect to d. It happens to use AD to do it, but otherwise it is just some function with inputs d and x and a vector output. (Arguments that are integer or logical do not have derivatives tracked.) derivsJacobian calculates derivatives of jacobian_run_wrt_d. This means that the value returned by derivsJacobian will be the value of jacobian_run_wrt_d, which comprises the first derivatives of run with respect to d. The jacobian returned by derivsJacobian will contain second derivatives of run with respect to d and each of d, x[1], and x[2]. And the hessian returned by derivsJacobian will contain some third derivatives of run. Notice that buildDerivs now includes jacobian_run_wrt_d. Let’s see this in use. my_derivs_demo5 <- derivs_demo5() C_my_derivs_demo5 <- compileNimble(my_derivs_demo5) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. d <- 1.2 x <- c(2.1, 2.2) C_my_derivs_demo5$derivsJacobian(d, x, wrt = 1:3, order = 0:2)
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -0.1689652 -0.1569948
## Field "jacobian":
##           [,1]      [,2]      [,3]
## [1,] 0.3548269 0.1222986 0.0000000
## [2,] 0.3453885 0.0000000 0.1170325
## Field "hessian":
## , , 1
##
##             [,1]        [,2] [,3]
## [1,] -0.74513642 -0.08786189    0
## [2,] -0.08786189 -0.05020679    0
## [3,]  0.00000000  0.00000000    0
##
## , , 2
##
##            [,1] [,2]        [,3]
## [1,] -0.7598548    0 -0.10047667
## [2,]  0.0000000    0  0.00000000
## [3,] -0.1004767    0 -0.05480546

We’ll compare these results to those shown above from the same values of d and x. The value here is the same as jacobian[1:2, 1] above. The jacobian here is the same as t(hessian[1, 1:3, 1:2]) or (by symmetry of second derivatives) t(hessian[1:3, 1, 1:2]) above. And the hessian[i,j,k] here contains the third derivative of the k-th output of run(d, x) with respect to d, the i-th input, and the j-th input.

It is also possible to call derivs on (i.e. tape) a function containing a derivs call from which some 0-th and/or 2nd order derivatives are extracted. It is even possible to triple tape by calling derivs on a function calling derivs on a function calling derivs, and so on.

## Derivatives involving model calculations

Obtaining derivatives involving model calculations takes some special considerations, and there are two ways to do it.

We will use the Poisson GLMM above as an example model for which we want derivatives. We can obtain derivatives with respect to any variables for all or any subset of model calculations.

### Method 1: nimDerivs of model$calculate Recall that model$calculate(nodes) returns the sum of the log probabilities of all stochastic nodes in nodes. Deterministic calculations are also executed; they contribute 0 to the sum of probabilities but may be needed for inputs to subsequent calculations. Calculations are done in the order of nodes, which should be a valid order for the model, often obtained from model$getDependencies. The simplest way to get derivatives for model calculations is to use model$calculate as the function taped by derivs (here shown by the alias nimDerivs for illustration).

derivs_nf <- nimbleFunction(
setup = function(model, with_respect_to_nodes, calc_nodes) {},
run = function(order = integer(1),
reset = logical(0, default=FALSE)) {
ans <- nimDerivs(model$calculate(calc_nodes), wrt = with_respect_to_nodes, order = order, reset = reset) return(ans) returnType(ADNimbleList()) } ) In derivs_nf: • The mere presence of model, with_respect_to_nodes, and calc_nodes as setup arguments makes them available as member data for run or other methods. • model will be a model object returned from nimbleModel. • with_respect_to_nodes will be the names of nodes we want derivatives with respect to. • calc_nodes will be the nodes to be calculated, in the order given. • order can contain any of 0 (value), 1 (1st order), or 2 (2nd order) derivatives requested, as above. • reset should be TRUE if the AD tape should be reset (re-recorded), as above. There are additional situations when a tape for model$calculate should be reset, discussed below.

Thus, nimDerivs(model$calculate(calc_nodes), wrt = with_respect_to_nodes, <other args>) takes derivatives of a function whose inputs are the values of with_respect_to_nodes (using their values in the model object) and whose output is the summed log probability of return by model$calculate(calc_nodes). The internal handling of this case is distinct from other calls to nimDerivs.

Now we can make an instance of derivs_nf, compile the model and nimbleFunction instance, and look at the results. We will assume the model was built as above in the HMC example

wrt_nodes <- c('intercept','beta', 'sigma')
calc_nodes <- model$getDependencies(wrt_nodes) derivs_all <- derivs_nf(model, wrt_nodes, calc_nodes) cModel <- compileNimble(model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. cDerivs_all <- compileNimble(derivs_all, project = model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. derivs_result <- cDerivs_all$run(order = 0:2)
derivs_result
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -80.74344
## Field "jacobian":
##           [,1]       [,2]     [,3]
## [1,] -9.358104 -0.3637619 -5.81414
## Field "hessian":
## , , 1
##
##           [,1]      [,2]      [,3]
## [1,] -62.35820 -16.21705   0.00000
## [2,] -16.21705 -69.47074   0.00000
## [3,]   0.00000   0.00000 -45.11516

As above, using order = 0:2 results in the the value (0th order), Jacobian (1st order), and Hessian (2nd order). The function model$calculate is organized here to have inputs that are the current values of intercept, beta, and sigma in the model. It has output that is the summed log probability of the calc_nodes. The Jacobian columns are first derivatives with respect to intercept, beta and sigma, respectively. The first and second indices of the Hessian array follow the same ordering. For example derivs_result$hessian[2,1,1] is the second derivative with respect to beta (first index = 2) and intercept (second index = 1).

In the case of model$calculate, the first index of the Jacobian and the last index of the Hessian are always 1 because derivatives are of the first (and only) output value. The ordering of inputs is similar to that used above, such as for the arguments d and x, but in this case the inputs are model nodes. When non-scalar nodes such as matrix or array nodes are used as with_respect_to_nodes, the resulting elements of Jacobian columns and Hessian first and second indices (corresponding to the inputs) will follow column-major order. For example, for a 2x2 matrix m, the element order would be m[1, 1], m[2, 1], m[1, 2], m[2, 2]. The actual values used as inputs are the current values in the compiled model object. As above, derivatives can also be calculated in uncompiled execution, but they will be much slower and less accurate. Uncompiled execution is mostly useful for checking that compiled derivatives are working correctly. This will use values in the uncompiled model object. derivs_all$run(order = 0:2)
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -80.74344
## Field "jacobian":
##           [,1]       [,2]     [,3]
## [1,] -9.358104 -0.3637619 -5.81414
## Field "hessian":
## , , 1
##
##           [,1]      [,2]      [,3]
## [1,] -62.35821 -16.21705   0.00000
## [2,] -16.21705 -69.47074   0.00000
## [3,]   0.00000   0.00000 -45.11517

We can see that all the results clearly match the compiled results to a reasonable numerical precision.

### Method 2: nimDerivs of a method that calls model$calculate Sometimes one needs derivatives of calculations done in a nimbleFunction as well as in a model. And sometimes one needs to change values in a model before and/or after doing model calculations and have those changes recorded in the derivative tape. For these reasons, it is possible in one method to take derivatives of another method that includes a call to model$calculate. Continuing the above example, say we want to take derivatives with respect to the log of sigma, so the input vector will be treated as intercept, beta, and log(sigma), in that order. (This would be useful so we could run an optimization algorithm on an unconstrained parameter space, for example. Here, since sigma must be positive, any algorithm must respect that constraint, so it can be easier to use algorithms involving log(sigma). See below for NIMBLE’s automated parameter transformation system if you need to do this systematically.) We will convert log(sigma) to sigma before using it in the model, and we want the derivative of that transformation included in the tape (i.e. using the chain rule).

Here is a nimbleFunction to do that.

derivs_nf2 <- nimbleFunction(
setup = function(model, wrt_nodes, calc_nodes) {
derivsInfo <- makeDerivsInfo(model, wrt_nodes, calc_nodes)
updateNodes <- derivsInfo$updateNodes constantNodes <- derivsInfo$constantNodes
n_wrt <- length(wrt_nodes)
# If wrt_nodes might contain non-scalar nodes, the more general way
# to determine the length of all scalar elements is:
# length(model$expandNodeNames(wrt_nodes, returnScalarComponents = TRUE)) }, run = function(x = double(1)) { x_trans <- x # x[1:2] don't need transformation x_trans[3] <- exp(x[3]) # transformation of x[3] values(model, wrt_nodes) <<- x_trans # put inputs into model ans <- model$calculate(calc_nodes)   # calculate model
return(ans)
returnType(double(0))
},
methods = list(
derivsRun = function(x = double(1),
order = integer(1),
reset = logical(0, default=FALSE)) {
wrt <- 1:n_wrt
ans <- nimDerivs(run(x), wrt = wrt, order = order, reset = reset,
model = model, updateNodes = updateNodes,
constantNodes = constantNodes)
return(ans)
}
),
buildDerivs = list(run = list()) # or simply 'run' would work in this case
)

Let’s see how this can be used.

derivs_all2 <- derivs_nf2(model, wrt_nodes, calc_nodes)
cDerivs_all2 <- compileNimble(derivs_all2, project = model)
params <- values(model, wrt_nodes)
params[3] <- log(params[3])
cDerivs_all2$derivsRun(params, order = 0:2) Notice that these results are the same as we saw above except for third elements, which represent derivatives with respect to sigma above and to log(sigma) here. There are several important new points here: • The only valid way to get values into the model that will be recorded on the AD tape is with values(model, nodes) <<- some_values as shown. Other ways such as nimCopy or model[[node]] <<- some_value are not currently supported to work with AD. • The call to nimDerivs of run must be told some information about the model calculations that will be done inside of run. model is of course the model that will be used. constantNodes is a vector of node names whose values are needed for the calculations but are not expected to change until you use reset = TRUE. Values of these nodes will be baked into the AD tape until reset = TRUE. updateNodes is a vector of node names whose values are needed for the calculations, might change between calls even with reset = FALSE, and are neither part of wrt_nodes nor a deterministic part of calc_nodes. The function makeDerivsInfo, as shown in the setup code, determines what is usually needed for constantNodes and updateNodes. • In Method 1 above, the NIMBLE compiler automatically determines constantNodes and updateNodes using makeDerivsInfo based on the inputs to nimDerivs(model$calculate(...),...).

• One can use double-taping, but if so both calls to nimDerivs need the model, updateNodes, and constantNodes arguments, which should normally be identical.

### Derivatives in user-defined functions and distributions

It is possible to build derivative support in a user-defined function or distribution used in a model. Simply set buildDerivs=TRUE, as shown in the my_sqrt example above.

#### Advanced topic: more about constantNodes and updateNodes.

In most cases, you can obtain constantNodes and updateNodes from makeDerivsInfo without knowing exactly what they mean. But for advanced uses and possible debugging needs, let’s explore these arguments in more detail. The purpose of constantNodes and updateNodes is to tell the AD system about all nodes that will be needed for taped model calculations. Specifically:

• updateNodes and constantNodes are vectors of nodes names that together comprise any nodes that are necessary for model$calculate(calc_nodes) but are: • not in the wrt argument to nimDerivs (only relevant for Method 1), and • not assigned into the model using values(model, nodes) <<- prior to model$calculate (only relevant for Method 2), and
• not a deterministic node in calc_nodes.
• updateNodes includes node names satisfying those conditions whose values might change between uses of the tape regardless of the reset argument.
• constantNodes includes node names satisfying those conditions whose values will be baked into the tape (will not change) until the next call with reset=TRUE.

To fix ideas, say that we want derivatives with respect to ran_eff[1] for calculations of it and the data that depend on it, including any deterministic nodes. In other words, wrt will be ran_eff[1], and calc_nodes will be:

model$getDependencies('ran_eff[1]') ## [1] "ran_eff[1]" ## [2] "lifted_exp_oPintercept_plus_beta_times_X_oBi_comma_j_cB_plus_ran_eff_oBi_cB_cP_L7[1, 1]" ## [3] "lifted_exp_oPintercept_plus_beta_times_X_oBi_comma_j_cB_plus_ran_eff_oBi_cB_cP_L7[1, 2]" ## [4] "lifted_exp_oPintercept_plus_beta_times_X_oBi_comma_j_cB_plus_ran_eff_oBi_cB_cP_L7[1, 3]" ## [5] "lifted_exp_oPintercept_plus_beta_times_X_oBi_comma_j_cB_plus_ran_eff_oBi_cB_cP_L7[1, 4]" ## [6] "lifted_exp_oPintercept_plus_beta_times_X_oBi_comma_j_cB_plus_ran_eff_oBi_cB_cP_L7[1, 5]" ## [7] "y[1, 1]" ## [8] "y[1, 2]" ## [9] "y[1, 3]" ## [10] "y[1, 4]" ## [11] "y[1, 5]" In this case, the log probability of ran_eff[1] itself requires sigma. Since sigma is not in wrt_nodes, it is not assigned into the model by the step values(model, wrt_nodes) <<- x_trans. It is also not a deterministic node (or any node) in calc_nodes, so it must be included in updateNodes or constantNodes. Since it might change between calls, it should be included in updateNodes. Next, notice that the stochastic node y[1, 1] in calc_nodes means that the log probability of y[1, 1] will be calculated, and this requires the actual value of y[1, 1]. This node is part of calc_nodes but is not a deterministic part of it, so it must be provided in either updateNodes or constantNodes. When data values will not be changed often, it is better to put those nodes in constantNodes, because that will be more efficient than putting them in updateNodes. (If the data values are changed, use reset=TRUE, which will reset values of all constantNodes in the tape.) The function makeDerivsInfo inspects the model and determines the usual needs for updateNodes and constantNodes. By default, data nodes are put in constantNodes. Use dataAsConstantNodes = FALSE in makeDerivsInfo if you want them put in updateNodes. Note that a deterministic node in calc_nodes will have its value calculated as part of the operations recorded in the AD tape, so it does not need to be included in updateNodes or constantNodes. As usual in model-generic programming in NIMBLE, be aware of lifted nodes and their implications. Suppose in the Poisson GLMM we had used a precision parameterization for the random effects, with the changes in this code snippet:  precision ~ dgamma(0.01, 0.01) # <other lines> ran_eff[i] ~ dnorm(0, tau = precision) This would result in a lifted node for the standard deviation, calculated as 1/sqrt(precision). That lifted node is what would actually be used in dnorm for each ran_eff[i]. Now if ran_eff[1] is in wrt_nodes, the lifted node (but not precision) will be in updateNodes (as determined by makeDerivsInfo). If you then change the value of precision, you must be sure to calculate the lifted node before obtaining derivatives. Otherwise the value of the lifted node will correspond to the old value of precision. An example of model-generic programming to update any lifted nodes that depend on precision would be: model$calculate(model$getDependencies('precision', determOnly=TRUE)) In Method 1 above, NIMBLE automatically uses makeDerivsInfo based on the code nimDerivs(model$calculate(<args>), <args>). However, in Method 2, when model$calculate(<args>) is used in a method such as run, then a call to nimDerivs(run(<args>), <args>) requires the model, updateNodes and constantNodes to be provided. In this sense, the two functions (run and derivsRun) must be written in coordination. #### When do you need to reset a tape with model calculations? The additional rule for when tapes need to be reset based on model calculations is: • when values of any constantNodes are changed. Usually these will only be data nodes. ## What operations are and aren’t supported for AD Much of the math supported by NIMBLE will work for AD. Here are some details on what is and isn’t supported, as well as on some optionsn to control how linear algebra operations are handled. Some of what is not supported for AD: • stochastic indexing in models. • cumulative (“p”) and quantile (“q”) functions for distributions. • truncated distributions in models (because they use cumulative distribution functions). • random number generation. • specific distributions including dcat, dcar_normal, dcar_proper, and dinterval. • some specific functions including step, %% (mod), eigen, svd, and bessel_k. • specific model operations including model$calculateDiff, model$simulate, model$getBound, and model$getParam Some details on what is supported include: • round, floor, ceil, and trunc are all supported with derivatives defined to be zero everywhere. • pow(a, b) requires positive a and b. • A new function pow_int(a, b) returns pow(a, round(b)) and thus sets all derivatives with respect to b to zero. This allows valid derivatives with respect to a even if it takes a negative value. For the linear algebra functions %*%, chol, forwardsolve, backsolve, and inverse, there are special extensions provided by NIMBLE for CppAD called (in CppAD terms) “atomics”, short for derived “atomic” classes by which one can extend CppAD. By default, these atomics will be used and often improve efficiency. There may be cases where they decrease efficiency, which might include when the matrix operands are small or contain many zeros. To compare results with and without use of the atomics, they can be turned off with a set of options: nimbleOptions(useADmatMultAtomic = FALSE) # for %*% nimbleOptions(useADcholAtomic = FALSE) # for chol nimbleOptions(useADmatInverseAtomic = FALSE) # for inverse nimbleOptions(useADsolveAtomic = FALSE) # for forwardsolve and backsolve When a linear algebra atomic is turned off, the AD tape simply records each scalar operation that is a step in the linear algebra operation. ## Parameter transformations Many algorithms that in some way explore a parameter space are best used in an unconstrained parameter space. For example, there are a bunch of optimization methods provided in R’s optim, but only one (L-BFGS-B) allows constraints on the parameter space. Similarly, HMC is implemented to work in an unconstrained parameter space. NIMBLE provides a nimbleFunction to automatically create a transformation from original parameters to unconstrained parameters and inverse transformation back to the original parameters. Denoting x as an original parameter (scalar or vector) and g(x) as the transformed parameter, the cases handled include: • If $$x \in (0, \infty)$$, $$g(x) = \log(x)$$. For example, x ~ dweibull in a model means $$x \in (0, \infty)$$. • If $$x \in (0, 1)$$, $$g(x) = \mbox{logit}(x)$$. For example, x ~ dbeta in a model means $$x \in (0, 1)$$ • If $$x \in (a, \infty)$$, $$g(x) = \log(x - a)$$. • If $$x \in (-\infty, b)$$, $$g(x) = -\log(b - x)$$. • If $$x \in (a, b)$$, $$g(x) = \mbox{logit}((x-a)/(b-a))$$. For example x ~ T(<some distribution>, a, b) in a model means $$x \in (a, b)$$. • If x[1:n, 1:n] ~ dwishart or x[1:n, 1:n] ~ dinvwishart in a model, $$g(x) = \mbox{chol}(x)$$ for non-diagonal elements of $$\mbox{chol}(x)$$ and $$g(x) = \log(\mbox{chol}(x))$$ for diagonal elements of $$\mbox{chol}(x)$$, where $$\mbox{chol}(x)$$ is the Cholesky decomposition of $$x$$. Note that $$x$$ in these cases follows a Wishart or inverse Wishart distribution, respectively, and thus is a random covariance matrix. That means it must be positive definite and thus have positive diagonal elements of $$\mbox{chol}(x)$$. • If x[1:n] ~ ddirch in a model, $$g(x[1]) = \mbox{logit}(x[1])$$ and $$g(x[i]) = \mbox{logit}(x[i] / (1-\sum_{k=1}^{i-1}x[i]))$$ for $$i \gt 1$$. In this case, x follows a Dirichlet distribution and thus has the simplex constraint that $$\sum_{i=1}^n x[i] = 1$$. • If x[1:n] ~ dlkj_corr_cholesky, $$g(x)$$ is the LKJ Cholesky transformation. This provides the LKJ prior for a covariance matrix. Note that the scalar transformations are all monotonic increasing functions of the original parameter. Some of these are of limited relevance for AD because truncation of distributions using T() is not currently supported for AD. TBD: Better description of LKJ. The worked example next will illustrate use of the parameter transformation system. # Example: maximum likelihood estimation using optim with gradients from nimDerivs. In this example, we will obtain maximum likelihood estimates of a Poisson GLM. It will be similar to the GLMM above, but without the random effects. Maximization of a likelihood function can be done with methods that don’t require gradients, but it is much faster to use methods that do use gradients. To illustrate the parameter transformation system, we will give p a uniform prior, constrained to values between 0 and 1. The maximum likelihood estimation below will not use the prior probability, but nevertheless the prior will be interpreted for its constraints on valid values in this case. This need for parameter transformation is somewhat artifical in the interest of simplicity. model_code <- nimbleCode({ # priors (will be ignored for MLE) p ~ dunif(0, 1) log_p <- log(p) beta ~ dnorm(0, sd = 100) # random effects and data for(i in 1:50) { # data y[i] ~ dpois(exp(log_p + beta*X[i])) } }) set.seed(123) X <- rnorm(50) # Simulate values for X model <- nimbleModel(model_code, constants = list(X = X), calculate = FALSE, buildDerivs = TRUE) # Create nimble model object ## Defining model ## Building model ## Checking model sizes and dimensions ## [Note] This model is not fully initialized. This is not an error. ## To see which variables are not initialized, use model$initializeInfo().
##          For more information on model initialization, see help(modelInitialization).
model$p <- 0.5 # assign parameter values and simulate data (y) model$beta <- 0.2
model$calculate() ## [1] NA model$simulate(model$getDependencies(c('beta', 'p'), self = FALSE)) model$calculate()
## [1] -53.33015
model$setData('y') # Now the model has y marked as data, with values from simulation. Cmodel <- compileNimble(model) # Make compiled version ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. Now we’ll write a nimbleFunction to provide the negative log likelihood and its gradient using an unconstrained parameter space. The default behavior of optim below is to do a minimization problem, so returning the negative log likelihood means that optim will find the maximum likelihood parameters. logLikelihood_nf <- nimbleFunction( setup = function(model, paramNodes) { # Determine nodes for calculating the log likelihood for parameters given by paramNodes calcNodes <- model$getDependencies(paramNodes, self = FALSE)
# Set up the additional arguments for nimDerivs when the function called includes model$calculate derivsInfo <- makeDerivsInfo(model, paramNodes, calcNodes) updateNodes <- derivsInfo$updateNodes
constantNodes <- derivsInfo$constantNodes # Create a parameter transformation between original and unconstrained parameter spaces. transformer <- parameterTransform(model, paramNodes) }, methods = list( neg_logLikelihood_p = function(p = double(1)) { # Put values in model and calcualte neg log likelihood. values(model, paramNodes) <<- p return(-model$calculate(calcNodes))
returnType(double())
},
neg_logLikelihood = function(ptrans = double(1)) { # Objective function for optim,
# using transformed parameter space
p <- transformer$inverseTransform(ptrans) return(neg_logLikelihood_p(p)) returnType(double()) }, gr_neg_logLikelihood = function(ptrans = double(1)) { # Gradient of neg log likelihood p <- transformer$inverseTransform(ptrans)
d <- derivs(neg_logLikelihood_p(p), wrt = 1:length(p), order = 1,
model = model, updateNodes = updateNodes, constantNodes = constantNodes)
return(d$jacobian[1,]) returnType(double(1)) }, transform = function(p = double(1)) { # Give easy access to transformation ... return(transformer$transform(p))
returnType(double(1))
},
inverse = function(ptrans = double(1)) { # ... and its inverse.
return(transformer$inverseTransform(ptrans)) returnType(double(1)) } ), buildDerivs = 'neg_logLikelihood_p' ) Sometimes you might want a call to transformer$transform or transformer$inverseTransform to be included in a derivative calculation. In this case, we don’t. Next we build and compile an instance of logLikelihood_nf specialized (by the setup function) to our model and its parameters. ll_glm <- logLikelihood_nf(model, c("p", "beta")) Cll_glm <- compileNimble(ll_glm, project = model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. Now let’s see how the parameter transformation works for this model. For beta, it already has an unconstrained range of valid values, so it has no transformation (i.e. it has an indentity transformation). For p, it is constrained beween 0 and 1, so it will be logit-transformed. Cll_glm$transform(c(0.2, 1))
## [1] -1.386294  1.000000
Cll_glm$inverse(c(-1.386294, 1)) # The inverse transformation goes back to the original parameters, within rounding. ## [1] 0.2000001 1.0000000 Now we are ready to use the methods provided in Cll_glm in a call to R’s optim to find the MLE. MLE <- optim(c(0,0), # initial values fn = Cll_glm$neg_logLikelihood, # function to be minimized
gr = Cll_glm$gr_neg_logLikelihood, # gradient of function to be minimized method = "BFGS") # optimization method to use MLE$par
## [1] 0.08257249 0.25577134
-MLE$value ## [1] -47.73243 Those outputs show the MLE in the parameters logit(p), and beta, following by the maximum log likelihood value. Now let’s confirm that it’s working correctly by comparing to results from glm. glm_fit <- glm(I(model$y) ~ X, family = poisson)
coef(glm_fit)
## (Intercept)           X
##  -0.6527680   0.2557751
logLik(glm_fit)
## 'log Lik.' -47.73243 (df=2)

We see that the coefficient for X matches beta (within numerical tolerance), and the maximum log likelihoods match. To understand the intercept, we need to be careful about the parameter transformation used in nimble. The inverse transformation will give p, and the log of that should match the intercept estimated by glm.

Cll_glm$inverse(MLE$par)
## [1] 0.5206314 0.2557713
log(Cll_glm$inverse(MLE$par)[1]) # This should match glm's intercept
## [1] -0.652713

It looks like it worked.

# Example: Laplace approximation

In this example, we will use nimble’s Laplace approximation, which uses derivatives internally, to get maximum (approximate) likelihood estimates for the GLMM model above. What Laplace approximation approximates in this context is the integral over continuous random effects needed to calculate the likelihood. Hence, it gives an approximate likelihood – often quite accurate – that can be used for maximum likelihood estimation. Note that the Laplace approximation uses second derivatives, and the gradient of the Laplace approximation uses third derivatves. These are described in detail by Skaug and Fournier (2006) and Fournier et al. (2012).

We will create the above GLMM model anew here.

model_code <- nimbleCode({
# priors
intercept ~ dnorm(0, sd = 100)
beta ~ dnorm(0, sd = 100)
sigma ~ dunif(0, 10)
# random effects and data
for(i in 1:10) {
# random effects
ran_eff[i] ~ dnorm(0, sd = sigma)
for(j in 1:5) {
# data
y[i,j] ~ dpois(exp(intercept + beta*X[i,j] + ran_eff[i]))
}
}
})
set.seed(123)
X <- matrix(rnorm(50), nrow = 10)
model <- nimbleModel(model_code, constants = list(X = X), calculate = FALSE,
buildDerivs = TRUE) # Here is the argument needed for AD.
## Defining model
## Building model
## Checking model sizes and dimensions
##   [Note] This model is not fully initialized. This is not an error.
##          To see which variables are not initialized, use model$initializeInfo(). ## For more information on model initialization, see help(modelInitialization). model$intercept <- 0
model$beta <- 0.2 model$sigma <- 0.5
model$calculate() ## [1] NA model$simulate(model$getDependencies('ran_eff')) model$calculate()
## [1] -80.74344
model$setData('y') # Now the model has y marked as data, with values from simulation. Cmodel <- compileNimble(model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. Next we create a Laplace approximation specialized to the parameters of interest for this model. The buildLaplace setup code will automatically determine the random effects to be integrated over and the associated nodes to calculate. In fact, if you omit the parameter nodes, it will assume that all top-level nodes in the model should be treated as parameters. If fine-grained control is needed, these various sets of nodes can be input directly into buildLaplace. glmm_laplace <- buildLaplace(model, c('intercept','beta','sigma')) Cglmm_laplace <- compileNimble(glmm_laplace, project = model) ## Compiling ## [Note] This may take a minute. ## [Note] Use 'showCompilerOutput = TRUE' to see C++ compilation details. Now we are ready to use some of the methods provided by buildLaplace. These include calculating the Laplace approximation for some input parameter values, calculating its gradient, and maximizing the Laplace-approximated likelihood. Cglmm_laplace$Laplace(c(0, 0, 1))
## [1] -65.57246
Cglmm_laplace$gr_Laplace(c(0, 0, 1)) ## [1] -1.866842 8.001648 -4.059556 MLE <- Cglmm_laplace$LaplaceMLE(c(0, 0, 1))
MLE$par ## [1] -0.1492312 0.1934101 0.5703647 MLE$value
## [1] -63.44875

The final outputs show the MLE for intercept, beta, and sigma, followed by the maximum (approximate) likelihood.

buildLaplace actually offers several choices in how computations are done, differing in how they use double taping or not. In some cases one or another choice might be more efficient. See help(buildLaplace) if you want to explore it further.

Finally, let’s confirm that it worked by comparing to results from package lme4. In this case, lme4 will be faster because it is highly specialized to GLMMs. Here our interest is in checking that nimble’s Laplace approximation worked correctly in a case where we have an established tool such as lme4.

library(lme4)
y <- as.numeric(model$y) # Rearrage inputs for call to glmer X <- as.numeric(X) group <- rep(1:10, 5) lme4_fit <- glmer(y ~ X + (1 | group), family = poisson) # There is a warning that seems to be innocuous. ## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control\$checkConv, :
## Model failed to converge with max|grad| = 0.0178573 (tol = 0.002, component 1)
summary(lme4_fit)
## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula: y ~ X + (1 | group)
##
##      AIC      BIC   logLik deviance df.resid
##    132.9    138.6    -63.4    126.9       47
##
## Scaled residuals:
##      Min       1Q   Median       3Q      Max
## -1.02386 -0.74101 -0.03475  0.33823  2.46453
##
## Random effects:
##  Groups Name        Variance Std.Dev.
##  group  (Intercept) 0.3285   0.5731
## Number of obs: 50, groups:  group, 10
##
## Fixed effects:
##              Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.153720   0.003913 -39.289   <2e-16 ***
## X            0.193464   0.143528   1.348    0.178
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
##   (Intr)
## X -0.002
## optimizer (Nelder_Mead) convergence code: 0 (OK)
## Model failed to converge with max|grad| = 0.0178573 (tol = 0.002, component 1)
logLik(lme4_fit)
## 'log Lik.' -63.44896 (df=3)

The results match within numerical tolerance typical of optimization problems. Specifically, the coefficients for (Intercept) and X match nimble’s Intercept and beta, the random effects standard deviation for group matches nimble’s sigma, and the maximum likelihood values match.`

# References

Bell, B. 2022. CppAD: A Package for Differentiation of C++ Algorithms.” www.coin-or.org/CppAD.
Borchers, Hans W. 2022. Pracma: Practical Numerical Math Functions. https://CRAN.R-project.org/package=pracma.
Fournier, David A., Hans J. Skaug, Johnoel Ancheta, James Ianelli, Arni Magnusson, Mark N. Maunder, Anders Nielsen, and John Sibert. 2012. AD Model Builder: Using Automatic Differentiation for Statistical Inference of Highly Parameterized Complex Nonlinear Models.” Optimization Methods and Software 27 (2): 233–49. https://doi.org/10.1080/10556788.2011.597854.
Gilbert, Paul, and Ravi Varadhan. 2019. numDeriv: Accurate Numerical Derivatives. https://CRAN.R-project.org/package=numDeriv.
Griewank, Andreas, and Andrea Walther. 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. Second edition. Philadelphia, PA: Society for Industrial; Applied Mathematic.
Kristensen, Kasper, Anders Nielsen, Casper W. Berg, Hans Skaug, and Bradley M. Bell. 2016. TMB: Automatic Differentiation and Laplace Approximation.” Journal of Statistical Software 70 (5): 1–21. https://doi.org/10.18637/jss.v070.i05.
Neal, Radford M. 2011. MCMC Using Hamiltonian Dynamics.” In Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC Handbooks of Modern Statistical Methods. Chapman; Hall/CRC. http://www.crcnetbase.com/doi/abs/10.1201/b10905-6.
Skaug, Hans J., and David A. Fournier. 2006. “Automatic Approximation of the Marginal Likelihood in Non-Gaussian Hierarchical Models.” Computational Statistics & Data Analysis 51 (2): 699–709. https://doi.org/10.1016/j.csda.2006.03.005.