# Chapter 16 Automatic Derivatives

As of version 1.0.0, NIMBLE can automatically provide numerically accurate derivatives of potentially arbitrary order for most calculations in models and/or nimbleFunctions. This feature enables methods such as Hamiltonian Monte Carlo (HMC, see package `nimbleHMC`

), Laplace approximation, and fast optimization with methods that use function gradients.

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 difference methods, used by R packages such as `numDeriv`

(Gilbert and Varadhan 2019) and `pracma`

(Borchers 2022), which approximate derivatives by calculating function values at extremely nearby points. Finite difference 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 in doing so (Kristensen et al. 2016). A general reference on AD is Griewank and Walther (2008).

Using a packaged AD algorithm should be as simple as setting `buildDerivs=TRUE`

in the model. On the other hand, writing new algorithms (as nimbleFunctions) that use AD requires understanding how the AD system works internally, including what can go wrong. Calls to `compileNimble`

that include AD features will result in slower C++ compilation.

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:

- how to turn on derivatives in a model and use them in Laplace approximation (for Hamiltonian Monte Carlo (HMC), see 7.11.2).
- how to modify user-defined functions and distributions to support derivatives.
- what functions are supported and not supported for derivatives.
- basics of obtaining derivatives in your own algorithms written as
`nimbleFunctions`

. - advanced methods of obtaining derivatives in
`nimbleFunctions`

, including*double-taping*. - how to get derivatives involving model calculations.
- automatic parameter transformations to give any model an unconstrained parameter space for algorithms to work in.
- an example showing use of nimble’s derivatives for maximum likelihood estimation.

First, make sure to set the option to enable derivative features. This should be `TRUE`

by default, but just in case:

`nimbleOptions(enableDerivs = TRUE)`

## 16.1 How to turn on derivatives in a model

To allow algorithms to use automatic derivatives for a model, include `buildDerivs=TRUE`

in the call to `nimbleModel`

. If you want derivatives to be set up for all models, you can run `nimbleOptions(buildModelDerivs = TRUE)`

and omit the `buildDerivs`

argument.

We’ll re-introduce the simple Poisson Generalized Linear Mixed Model (GLMM) example model from 7.11.2.1 and use Laplace approximation on it. 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:

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

We’ll simulate some values for `X`

.

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

Next, we build the model, including `buildDerivs=TRUE`

.

```
<- nimbleModel(model_code, constants = list(X = X), calculate = FALSE,
model buildDerivs = TRUE) # Here is the argument needed for AD.
```

### 16.1.1 Finish setting up the GLMM example

As preparation for the Laplace examples below, we need to finish setting up the GLMM. We could have provided data in the call to `nimbleModel`

, but instead we will simulate it using the model itself. Specifically, we will set parameter values, simulate data values, and then set those as the data to use.

```
$intercept <- 0
model$beta <- 0.2
model$sigma <- 0.5
model$calculate() # This will return NA because the model is not fully initialized. model
```

`## [1] NA`

```
$simulate(model$getDependencies('ran_eff'))
model$calculate() # Now the model is fully initialized: all nodes have valid values. model
```

`## [1] -80.74344`

`$setData('y') # Now the model has y marked as data, with values from simulation. model`

If you are not very familiar with using a `nimble`

model, that might have been confusing, but it was just for setting up the example.

Finally, we will make a compiled version of the model.

`<- compileNimble(model) Cmodel `

## 16.2 How to use Laplace approximation

Next we will show how to 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 (used for finding the MLE efficiently) uses third derivatves. These are described in detail by Skaug and Fournier (2006) and Fournier et al. (2012).

To create a Laplace approximation specialized to the parameters of interest for this model, we use the nimbleFunction `buildLaplace`

. For many models, the setup code in `buildLaplace`

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`

. To see what default handling of nodes is being done for your model, use `setupMargNodes`

with the same node inputs as `buildLaplace`

.

```
<- buildLaplace(model, c('intercept','beta','sigma'))
glmm_laplace <- compileNimble(glmm_laplace, project = model) Cglmm_laplace
```

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.

```
# Get the Laplace approximation for one set of parameter values.
$calcLaplace(c(0, 0, 1)) Cglmm_laplace
```

`## [1] -65.57246`

`$gr_Laplace(c(0, 0, 1)) # Get the corresponding gradient. Cglmm_laplace`

`## [1] -1.866842 8.001648 -4.059556`

```
<- Cglmm_laplace$findMLE(c(0, 0, 1)) # Find the (approximate) MLE.
MLE $par # MLE parameter values MLE
```

`## [1] -0.1492313 0.1934101 0.5703648`

`$value # MLE log likelihood value MLE`

`## [1] -63.44875`

The final outputs show the MLE for `intercept`

, `beta`

, and `sigma`

, followed by the maximum (approximate) likelihood.

More information about the MLE can be obtained in two ways. The `summary`

method
can give estimated random effects and standard errors as well as the variance-covariance matrix
for the parameters and/or the random effects. The `summaryLaplace`

function
returns similar information but with names included in a more useful way. For example:

`$summary(MLE)$randomEffects$estimates Cglmm_laplace`

```
## [1] -0.33711223 -0.02963214 0.40581611 1.04780122 -0.36729920 0.26915416
## [7] -0.54949741 -0.11866631 0.10009139 -0.04408944
```

`summaryLaplace(Cglmm_laplace, MLE)$params`

```
## estimate se
## intercept -0.1492313 0.2465005
## beta 0.1934101 0.1467229
## sigma 0.5703648 0.2066583
```

`buildLaplace`

actually offers several choices in how computations are done, differing in how they use *double taping* for derivatives 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 `glmmTMB`

. In this case, nimble’s Laplace approximation is faster than `glmmTMB`

(on the machine used here), but that is not the point of this example. Here our interest is in checking that nimble’s Laplace approximation worked correctly in a case where we have an established tool such as `glmmTMB`

.

`library(glmmTMB)`

```
## Warning in checkDepPackageVersion(dep_pkg = "TMB"): Package version inconsistency detected.
## glmmTMB was built with TMB version 1.9.4
## Current TMB version is 1.9.7
## Please re-install glmmTMB from source or restore original 'TMB' package (see '?reinstalling' for more information)
```

```
<- as.numeric(model$y) # Re-arrage inputs for call to glmmTMB
y <- as.numeric(X)
X <- rep(1:10, 5)
group <- as.data.frame(cbind(X,y,group))
data <- glmmTMB(y ~ X + (1 | group), family = poisson, data = data)
tmb_fit summary(tmb_fit)
```

```
## Family: poisson ( log )
## Formula: y ~ X + (1 | group)
## Data: data
##
## AIC BIC logLik deviance df.resid
## 132.9 138.6 -63.4 126.9 47
##
## Random effects:
##
## Conditional model:
## Groups Name Variance Std.Dev.
## group (Intercept) 0.3253 0.5703
## Number of obs: 50, groups: group, 10
##
## Conditional model:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.1492 0.2465 -0.605 0.545
## X 0.1935 0.1467 1.319 0.187
```

`logLik(tmb_fit)`

`## 'log Lik.' -63.44875 (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 standard errors match.

## 16.3 How to support derivatives in user-defined functions and distributions

It is possible to activate derivative support for a user-defined function or distribution used in a model. Simply set `buildDerivs=TRUE`

in `nimbleFunction`

.

Here is an extremely toy example. Let’s say we want a model with one node that follows a user-defined distribution, which will happen to be the same as `dnorm`

(a normal distribution) for illustration.

The model code is:

```
<- nimbleCode({
toyCode ~ d_my_norm(mean = mu, sd = sigma)
x ~ dunif(-10, 10)
mu ~ dunif(0, 5)
sigma })
```

The user-defined distribution is:

```
<- nimbleFunction(
d_my_norm run = function(x = double(), mean = double(), sd = double(),
log = integer(0, default = 0)) {
<- -log(sqrt(2*pi*sd)) - 0.5*((x-mean)/sd)^2
ans if(log) return(ans)
return(exp(ans))
returnType(double())
},buildDerivs=TRUE
)
```

Now we can build the model with `buildDerivs=TRUE`

and compile it:

```
# Don't worry about the warnings from nimbleModel in this case.
<- nimbleModel(toyCode, inits = list(mu = 0, sigma = 1, x = 0),
toyModel buildDerivs = TRUE)
```

Now `toyModel`

can be used in algorithms such as HMC and Laplace approximation, as above, or new ones you might write, as below.

## 16.4 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 options to control how linear algebra is handled.

Features that are not supported for AD include:

- cumulative (“p”) and quantile (“q”) functions for distributions;
- truncated distributions (because they use cumulative distribution functions), for any derivatives;
- discrete distributions, for derivatives with respect to the random variable;
- random number generation;
- some specific functions including
`step`

,`%%`

(mod),`eigen`

,`svd`

, and`besselK`

; and - most model operations (other than
`model$calculate`

, which is supported) including`model$calculateDiff`

,`model$simulate`

,`model$getBound`

, and`model$getParam`

.

Note that functions and distributions that are not supported can still be used in a model, as long as no algorithm tries to take derivatives with respect to node(s) that is (are) not supported. For example, `x ~ dcat(p[1:3])`

declares `x`

to be a discrete random variable following a categorical distribution. An algorithm can use derivatives with respect to `p[1:3]`

(or with respect to parent nodes of `p[1:3]`

) because the calculation of the log probability of `x`

given `p[1:3]`

is continuous with respect to `p`

. However, derivatives with respect to `x`

will not work. In the case of truncated distributions, neither derivatives with respect to parameters nor to random variable are supported.

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`

. Note that if`b`

is coded as a (possibly negative) integer,`pow_int`

will be used. For example`pow(tau, -2)`

will be converted to`pow_int(tau, -2)`

.- 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 jargon) “atomics”. 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 system simply uses all the scalar operations that compose the linear algebra operation.

Another important feature of AD is that sometimes values get “baked in” to AD calculations, meaning they are used *and permanently retained* from the first set of calculations and then can’t have their value changed later (unless an algorithm does a “reset”, described below). For people writing user-defined distributions and functions, a brief summary of what can get baked in includes:

- the extent (iteration values) of any for-loops.
- values of any arguments that are not of type ‘double’, including e.g. the ‘log’ argument in
`d_my_norm`

. (‘d’ functions called from models always have`log = TRUE`

, so in that case it is not a problem.) - the evaluation path followed by any if-then-else calls.
- values of any integer indices.

See below for more thorough explanations.

## 16.5 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. From here on, this part of the chapter is oriented towards algorithm developers. 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.

```
<- nimbleFunction(
derivs_demo 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)) {
<- 1:(1 + length(x)) # total length of d and x
wrt 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). Some nimbleFunctions don’t have`setup`

code, but`setup`

code is required when there will be a call to`derivs`

. If, as here, you don’t need the`setup`

to do anything, you can simply use`setup=TRUE`

, which is equivalent to`setup=function(){}`

. - 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`

is a vector of derivative orders to return, which can include any of`0`

(the function value, not a derivative!),`1`

(1st order), or`2`

(2nd order). Higher-order derivatives can be obtained by double-taping, described below.- Basics of
`nimbleFunction`

programming are covered in Chapters 11-15. These include type declarations (`double()`

for scalar,`double(1)`

for vector), the distinction between`setup`

code and`run`

code, and how to write more methods (of which`run`

is simply a default name when there is only one method). `derivs`

can alternatively be called`nimDerivs`

. In fact the former will be converted to the latter internally.

Let’s see this work.

```
<- derivs_demo()
my_derivs_demo <- compileNimble(my_derivs_demo)
C_my_derivs_demo <- 1.2
d <- c(2.1, 2.2)
x $derivsRun(d, x) C_my_derivs_demo
```

```
## 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 index. 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.

### 16.5.1 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:

`$derivsRun(d, x) my_derivs_demo`

```
## 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.

### 16.5.2 Holding some local variables out of derivative tracking

Sometimes one wants to omit tracking of certain variables in derivative calculations. There are three ways to do this: naming variables in an `ignore`

set; ensuring a variable’s type is `integer`

(not simply `numeric`

with an integer value) or `logical`

; or using the `ADbreak`

function. The `ignore`

method is coarse, holding a variable such as a for-loop counter entirely out of derivative tracking. The `ADbreak`

method provides more fine-grained control and is essential when function arguments might be passed from models. Here is an example of `ignore`

, while `ADbreak`

is described below.

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:

```
<- nimbleFunction(
derivs_demo2 setup = function() {},
run = function(d = double(), x = double(1)) {
<- numeric(length = length(x))
ans for(i in 1:length(x))
<- exp(-d*x[i])
ans[i] return(ans)
returnType(double(1))
},methods = list(
derivsRun = function(d = double(), x = double(1)) {
<- 1:(1 + length(x)) # total length of d and x
wrt 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.

```
<- derivs_demo2()
my_derivs_demo2 <- compileNimble(my_derivs_demo2)
C_my_derivs_demo2 <- 1.2
d <- c(2.1, 2.2)
x $derivsRun(d, x)$jacobian C_my_derivs_demo2
```

```
## [,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)`

.)

Because derivatives are not tracked for integer or logical variables, arguments to `run`

that have `integer`

or `logical`

types will not have derivatives tracked.

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())`

.

#### 16.5.2.1 Holding function arguments (including variables passed from a model) out of derivative tracking

Derivatives can be tracked through function calls, including from a model to a user-defined distribution or function (see 16.3 and 16.5.3). However, if one function is tracking derivatives for `x`

and passes `x`

as an argument to another function that does not track its derivatives, there will be an error (most likely from `compileNimble`

). The problem is that the C++ variable types used for tracking derivatives or not are different. Importantly, variables passed from models always have derivatives tracked.

For this situation, you can use `ADbreak`

to stop derivative tracking. For example, if one has a nimbleFunction argument `x`

, one would do `x_noDeriv <- ADbreak(x)`

and then use `x_noDeriv`

in subsequent calculations. This “breaks” derivative tracking in the sense that the derivative of `x_noDeriv`

with respect to `x`

is 0.

Including `"x"`

in the `ignore`

set won’t work because, if `x`

is a function argument, its type will then be changed to be a non-derivative type. If a derivative-tracking type is then passed as the value for `x`

there will be a mismatch in types. Since variables passed from models always allow derivative-tracking, one should not `ignore`

the corresponding function arguments.

The `ADbreak`

method can be useful in other situations as well, not only for function arguments whose derivatives you don’t want to track.

Note that `ADbreak`

only works for scalars. If you need to break AD tracking for a non-scalar, you have to write a for-loop to apply `ADbreak`

for each element. For example:

```
<- nimNumeric(length(x), init=FALSE)
x_noDeriv <- 1L
i for(i in 1:length(x)) x_noDeriv[i] <- ADbreak(x[i])
```

### 16.5.3 Using AD with multiple nimbleFunctions

Derivatives will be tracked through 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)))`

.

```
<- nimbleFunction(
nf_sqrt run = function(x = double(1)) {
return(sqrt(x))
returnType(double(1))
},buildDerivs = TRUE
)
<- nimbleFunction(
derivs_demo3 setup = function() {},
run = function(d = double(), x = double(1)) {
<- exp(-d*x)
ans <- nf_sqrt(ans)
ans return(ans)
returnType(double(1))
}, methods = list(
derivsRun = function(d = double(), x = double(1)) {
<- 1:(1 + length(x)) # total length of d and x
wrt return(derivs(run(d, x), wrt = wrt, order = 0:2))
returnType(ADNimbleList())
}
),buildDerivs = 'run'
)
```

And then let’s see it work:

```
<- derivs_demo3()
my_derivs_demo3 <- compileNimble(my_derivs_demo3)
C_my_derivs_demo3 <- 1.2
d <- c(2.1, 2.2)
x $derivsRun(d, x)$jacobian C_my_derivs_demo3
```

```
## [,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())`

. One can’t take derivatives of `nf_sqrt`

on its own, but it can be called by a method of a nimbleFunction that can have its derivatives taken (e.g. the `run`

method of a `derivs_demo3`

object).

### 16.5.4 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 \(dx/dt\).) In CppAD, the basic numeric object type (double) is replaced with a special type and corresponding versions of all the basic math operations so that those operations can track not just the values but also derivative information.

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 \(x^2\), which is 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, which are calculated by stepping through the calculation sequence a *second* time

What was just described is called *forward* mode in 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 \((1/dx) = (x^2) / dz\). Again, values are calculated first, followed by derivatives in a second pass through the calculation sequence, this time in reverse order.

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 choices 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 and it can be used to get third and higher order derivatives.

### 16.5.5 Resetting a `nimDerivs`

call

The first time a call to `nimDerivs`

runs, it records a tape of the operations in its first argument (`run(d, x)`

in the examples here). 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):`$derivsRun(-0.4, c(3.2, 5.1)) C_my_derivs_demo`

```
## 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.

```
<- nimbleFunction(
derivs_demo4 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))
returnType(ADNimbleList())
}
),buildDerivs = 'run'
)
```

Now we will illustrate the use of `reset`

. To make shorter output, we’ll request only the Jacobian.

```
<- derivs_demo4()
my_derivs_demo4 <- compileNimble(my_derivs_demo4)
C_my_derivs_demo4 <- 1.2
d <- c(2.1, 2.2)
x # On the first call, reset is ignored because the tape must be recorded.
$derivsRun(d, x, 1:3, 1) C_my_derivs_demo4
```

```
## 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.
$derivsRun(-0.4, c(3.2, 5.1), 1:3, 1) C_my_derivs_demo4
```

```
## 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
$derivsRun(1.2, c(2.1, 2.2, 2.3), 1:4, 1, reset=TRUE) C_my_derivs_demo4
```

```
## 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
$derivsRun(-0.4, c(3.2, 5.1, 4.5), 1:4, 1) C_my_derivs_demo4
```

```
## 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>
##
```

You could also make multiple instances of `derivs_demo4`

and 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.

#### 16.5.5.1 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.
- values of any
`integer`

or`logical`

inputs change. - 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.

## 16.6 Advanced uses: double taping

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 (i.e. from `derivs`

) 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.

```
<- nimbleFunction(
derivs_demo5 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)) {
<- derivs(run(d, x), wrt = wrt, order = 1, reset = reset)
ans <- ans$jacobian[,1] # derivatives wrt 'd' only
jac return(jac)
returnType(double(1))
},derivsJacobian = function(d = double(), x = double(1),
wrt = integer(1),
order = integer(1),
reset = logical(0, default = FALSE)) {
<- nimInteger(value = 1, length = 1)
innerWrt <- nimDerivs(jacobian_run_wrt_d(d, x, wrt = innerWrt, reset = reset),
ans 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.

```
<- derivs_demo5()
my_derivs_demo5 <- compileNimble(my_derivs_demo5)
C_my_derivs_demo5 <- 1.2
d <- c(2.1, 2.2)
x $derivsJacobian(d, x, wrt = 1:3, order = 0:2) C_my_derivs_demo5
```

```
## 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 can 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.

## 16.7 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.

### 16.7.1 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 its alternative name `nimDerivs`

for illustration).

```
<- nimbleFunction(
derivs_nf setup = function(model, with_respect_to_nodes, calc_nodes) {},
run = function(order = integer(1),
reset = logical(0, default=FALSE)) {
<- nimDerivs(model$calculate(calc_nodes), wrt = with_respect_to_nodes,
ans 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 returned 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 Laplace example.

```
<- c('intercept','beta', 'sigma')
wrt_nodes <- model$getDependencies(wrt_nodes)
calc_nodes <- derivs_nf(model, wrt_nodes, calc_nodes)
derivs_all <- compileNimble(model)
cModel <- compileNimble(derivs_all, project = model)
cDerivs_all <- cDerivs_all$run(order = 0:2)
derivs_result 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 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]`

. This will usually be the same ordering as the names returned by `model$expandNodeNames("m", returnScalarElements=TRUE)`

.

The actual values used as inputs are the current values in the compiled model object. These would typically be set by code like `values(model, with_respect_to_nodes) <<- my_values`

before the call to `nimDerivs`

in the above example.

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.

`$run(order = 0:2) derivs_all`

```
## 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.

### 16.7.2 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 to take derivatives of a 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. 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).

(Using `log(sigma)`

instead of `sigma`

is useful so an algorithm such as optimization or HMC can use an unconstrained parameter space. A constraint such as `sigma > 0`

can be difficult for many algorithms. See below for NIMBLE’s automated parameter transformation system if you need to do this systematically.)

Here is a nimbleFunction to do that.

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

Let’s see how this can be used.

```
<- derivs_nf2(model, wrt_nodes, calc_nodes)
derivs_all2 <- compileNimble(derivs_all2, project = model)
cDerivs_all2 <- values(model, wrt_nodes)
params 3] <- log(params[3])
params[$derivsRun(params, order = 0:2) cDerivs_all2
```

```
## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -80.74344
## Field "jacobian":
## [,1] [,2] [,3]
## [1,] -9.358104 -0.3637619 -2.90707
## 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 -14.18586
```

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`makeModelDerivsInfo`

, 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`makeModelDerivsInfo`

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.

#### 16.7.2.1 Advanced topic: more about `constantNodes`

and `updateNodes`

In most cases, you can obtain `constantNodes`

and `updateNodes`

from `makeModelDerivsInfo`

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) <<- some_values`

prior to`model$calculate`

(only relevant for Method 2), and - not a deterministic node in
`calc_nodes`

.

- not in the
`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:

`$getDependencies('ran_eff[1]') model`

```
## [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]"
```

(The “lifted” nodes here are for the `exp(intercept + beta*X[i,j] + ran_eff[i])`

, i.e. the inputs to `dpois`

. See chapters 2 and 13 to learn about lifted nodes.)

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 line `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 `makeModelDerivsInfo`

inspects the model and determines the usual needs for `updateNodes`

and `constantNodes`

. By default, data nodes are put in `constantNodes`

. Use `dataAsConstantNodes = FALSE`

in `makeModelDerivsInfo`

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:

```
~ dgamma(0.01, 0.01)
precision # <other lines>
~ dnorm(0, tau = precision) ran_eff[i]
```

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 `makeModelDerivsInfo`

). 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. These considerations are not unique to AD but rather are part of model-generic programming (see chapter 15).

An example of model-generic programming to update any lifted nodes that depend on `precision`

would be:

`$calculate(model$getDependencies('precision', determOnly=TRUE)) model`

In Method 1 above, NIMBLE automatically uses `makeModelDerivsInfo`

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. Hence, the two functions (`run`

and `derivsRun`

) must be written in coordination.

## 16.8 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 the inverse transformation back to the original parameters. Denoting `x`

as an original parameter and `g(x)`

as the transformed parameter, the cases handled include:

- If scalar \(x \in (0, \infty)\), then \(g(x) = \log(x)\). For example,
`x ~ dweibull`

in a model means \(x \in (0, \infty)\). - If scalar \(x \in (0, 1)\), then \(g(x) = \mbox{logit}(x)\). For example,
`x ~ dbeta`

in a model means \(x \in (0, 1)\) - If scalar \(x \in (a, \infty)\), then \(g(x) = \log(x - a)\).
- If scalar \(x \in (-\infty, b)\), then \(g(x) = -\log(b - x)\).
- If scalar \(x \in (a, b)\), then \(g(x) = \mbox{logit}((x-a)/(b-a))\). For example
`x ~ dunif(a, b)`

in a model means \(x \in (a, b)\). - If matrix
`x[1:n, 1:n] ~ dwishart`

or`x[1:n, 1:n] ~ dinvwishart`

in a model, then \(g(x) = \mbox{chol}(x)\) 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 precision or covariance matrix. That means it must be positive definite and thus have positive diagonal elements of \(\mbox{chol}(x)\). - If vector
`x[1:n] ~ ddirch`

in a model, then \(g(x[1]) = \mbox{logit}(x[1])\) and \(g(x[i]) = \mbox{logit}(x[i] / (1-\sum_{k=1}^{i-1}x[i]))\) for \(i > 1\). In this case,`x`

follows a Dirichlet distribution and thus has the simplex constraint that \(\sum_{i=1}^n x[i] = 1\). - If vector
`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.

The worked example next will illustrate use of the parameter transformation system.