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 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:
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)Next, we build the model, including buildDerivs=TRUE.
model <- nimbleModel(model_code, constants = list(X = X), calculate = FALSE,
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.
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] NAmodel$simulate(model$getDependencies('ran_eff'))
model$calculate() # Now the model is fully initialized: all nodes have valid values.## [1] -80.74344model$setData('y') # Now the model has y marked as data, with values from simulation.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.
Cmodel <- compileNimble(model)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.
glmm_laplace <- buildLaplace(model, c('intercept','beta','sigma'))
Cglmm_laplace <- compileNimble(glmm_laplace, project = model)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.
Cglmm_laplace$calcLaplace(c(0, 0, 1)) ## [1] -65.57246Cglmm_laplace$gr_Laplace(c(0, 0, 1)) # Get the corresponding gradient.## [1] -1.866842 8.001648 -4.059556MLE <- Cglmm_laplace$findMLE(c(0, 0, 1)) # Find the (approximate) MLE.
MLE$par # MLE parameter values## [1] -0.1492313 0.1934101 0.5703648MLE$value # MLE log likelihood value## [1] -63.44875The final outputs show the MLE for intercept, beta, and sigma, followed by the maximum (approximate) likelihood.
More information about the MLE can be obtained in two ways. The summary method
can give estimated and standard errors as well as the variance-covariance matrix
for the parameters and/or the random effects. The summaryLaplace function
returns similar information but with names included in a more useful way. For example:
Cglmm_laplace$summary(MLE)$randomEffects$estimates## [1] -0.33711223 -0.02963214 0.40581611 1.04780122 -0.36729920 0.26915416
## [7] -0.54949741 -0.11866631 0.10009139 -0.04408944summaryLaplace(Cglmm_laplace, MLE)$params## estimate se
## intercept -0.1492313 0.2465005
## beta 0.1934101 0.1467229
## sigma 0.5703648 0.2066583buildLaplace actually offers several choices in how computations are done, differing in how they use double taping for derivatives or not. In some cases one or another choice might be more efficient. See help(buildLaplace) if you want to explore it further.
Finally, let’s confirm that it worked by comparing to results from package glmmTMB. In this case, nimble’s Laplace approximation is faster than glmmTMB, but it is not the point of this example. Here our interest is in checking that nimble’s Laplace approximation worked correctly in a case where we have an established tool such as glmmTMB.
library(glmmTMB)
y <- as.numeric(model$y) # Re-arrage inputs for call to glmmTMB
X <- as.numeric(X)
group <- rep(1:10, 5)
data <- as.data.frame(cbind(X,y,group))
tmb_fit <- glmmTMB(y ~ X + (1 | group), family = poisson, data = data)
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 0.187logLik(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 maximum likelihood values 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:
toyCode <- nimbleCode({
x ~ d_my_norm(mean = mu, sd = sigma)
mu ~ dunif(-10, 10)
sigma ~ dunif(0, 5)
})The user-defined distribution is:
d_my_norm <- nimbleFunction(
run = function(x = double(), mean = double(), sd = double(),
log = integer(0, default = 0)) {
ans <- -log(sqrt(2*pi*sd)) - 0.5*((x-mean)/sd)^2
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.
toyModel <- nimbleModel(toyCode, inits = list(mu = 0, sigma = 1, x = 0),
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 operations are handled.
Features that are not supported for AD include:
- stochastic indexing (i.e. the index itself in a latent state) 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, anddinterval. - some specific functions including
step,%%(mod),eigen,svd, andbessel_k. - specific model operations including
model$calculateDiff,model$simulate,model$getBound, andmodel$getParam.
Some details on what is supported include:
round,floor,ceil, andtruncare all supported with derivatives defined to be zero everywhere.pow(a, b)requires positiveaandb.- A new function
pow_int(a, b)returnspow(a, round(b))and thus sets all derivatives with respect tobto zero. This allows valid derivatives with respect toaeven 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”. 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 backsolveWhen 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 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 havelog = TRUE, so in that case it is not a problem.) - the evaluation path followed by any if-then-else calls.
- values of any 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 manual 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.
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
setupcode allows the nimbleFunction to have multiple methods (i.e. to behave like a class definition in object-oriented programming). Some nimbleFunctions don’t havesetupcode, butsetupcode is required when there will be a call toderivs. If, as here, you don’t need thesetupto do anything, you can simply usesetup=TRUE, which is equivalent tosetup=function(){}. - Any arguments to
runthat are real numbers (i.e. regular double precision numbers, but not integers or logicals) will have derivatives tracked when called throughderivs. - The “with-respect-to” (
wrt) argument gives indices of the arguments for which you want derivatives. In this case, we’re including all elements ofdandX. The indices form one sequence over all arguments. orderis a vector of derivative orders to return, which can include any of0(the function value, not a derivative!),1(1st order), or2(2nd order). Higher-order derivatives can be obtained by double-taping, described below.- Basics of
nimbleFunctionprogramming are covered in Chapters 11-15. These include type declarations (double()for scalar,double(1)for vector), the distinction betweensetupcode andruncode, and how to write more methods (of whichrunis simply a default name when there is only one method). derivscan alternatively be callednimDerivs. 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)
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 0.1027602We 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:
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 0.1027602We 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. 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)
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 -0.08563352One 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()).
16.5.3 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))).
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)
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 -0.1602812Note 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$derivsRun with new arguments of the same size(s):
C_my_derivs_demo$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 1.230497However, if we call C_my_derivs_demo$derivsRun with length(x) different from 2, the result will be garbage. If we need to change the size of the arguments, we need to re-record the tape. This is done with the reset argument.
Here is a slightly more general version of the derivs_demo allowing a user to reset the tape. It also takes order and wrt as arguments instead of hard-coding them.
derivs_demo4 <- nimbleFunction(
setup = function() {},
run = function(d = double(), x = double(1)) {
return(exp(-d*x))
returnType(double(1))
},
methods = list(
derivsRun = function(d = double(), x = double(1),
wrt = integer(1), order = integer(1),
reset = logical(0, default=FALSE)) {
return(derivs(run(d, x), wrt = wrt, order = order, reset = reset))
returnType(ADNimbleList())
}
),
buildDerivs = 'run'
)Now we will illustrate the use of reset. To make shorter output, we’ll request only the Jacobian.
my_derivs_demo4 <- derivs_demo4()
C_my_derivs_demo4 <- compileNimble(my_derivs_demo4)
d <- 1.2
x <- c(2.1, 2.2)
# On the first call, reset is ignored because the tape must be recorded.
C_my_derivs_demo4$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>
## 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 ofstartandendare in effect baked into a tape. - the outcomes of any if-then-else conditions in a nimbleFunction are changed.
- values of any
integerorlogicalinputs change. - any other member data used in calculations changes. It is possible to create a variable in the
setupfunction and use it in methods such asrun. 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.
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)
d <- 1.2
x <- c(2.1, 2.2)
C_my_derivs_demo5$derivsJacobian(d, x, wrt = 1:3, order = 0:2)## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -0.1689652 -0.1569948
## Field "jacobian":
## [,1] [,2] [,3]
## [1,] 0.3548269 0.1222986 0.0000000
## [2,] 0.3453885 0.0000000 0.1170325
## Field "hessian":
## , , 1
##
## [,1] [,2] [,3]
## [1,] -0.74513642 -0.08786189 0
## [2,] -0.08786189 -0.05020679 0
## [3,] 0.00000000 0.00000000 0
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] -0.7598548 0 -0.10047667
## [2,] 0.0000000 0 0.00000000
## [3,] -0.1004767 0 -0.05480546We can compare these results to those shown above from the same values of d and x. The value here is the same as jacobian[1:2, 1] above. The jacobian here is the same as t(hessian[1, 1:3, 1:2]) or (by symmetry of second derivatives) t(hessian[1:3, 1, 1:2]) above. And the hessian[i,j,k] here contains the third derivative of the k-th output of run(d, x) with respect to d, the i-th input, and the j-th input.
It is also possible to call derivs on (i.e. tape) a function containing a derivs call from which some 0-th and/or 2nd order derivatives are extracted. It is even possible to triple tape by calling derivs on a function calling derivs on a function calling derivs, and so on.
16.7 Derivatives involving model calculations
Obtaining derivatives involving model calculations takes some special considerations, and there are two ways to do it.
We will use the Poisson GLMM above as an example model for which we want derivatives. We can obtain derivatives with respect to any variables for all or any subset of model calculations.
16.7.1 Method 1: nimDerivs of model$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$calculate as the function taped by derivs (here shown by its alternative name nimDerivs for illustration).
derivs_nf <- nimbleFunction(
setup = function(model, with_respect_to_nodes, calc_nodes) {},
run = function(order = integer(1),
reset = logical(0, default=FALSE)) {
ans <- nimDerivs(model$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, andcalc_nodesassetuparguments makes them available as member data forrunor other methods. modelwill be a model object returned fromnimbleModel.with_respect_to_nodeswill be the names of nodes we want derivatives with respect to.calc_nodeswill be the nodes to be calculated, in the order given.ordercan contain any of0(value),1(1st order), or2(2nd order) derivatives requested, as above.resetshould be TRUE if the AD tape should be reset (re-recorded), as above. There are additional situations when a tape formodel$calculateshould be reset, discussed below.
Thus, nimDerivs(model$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$calculate(calc_nodes). The internal handling of this case is distinct from other calls to nimDerivs.
Now we can make an instance of derivs_nf, compile the model and nimbleFunction instance, and look at the results. We will assume the model was built as above in the Laplace example.
wrt_nodes <- c('intercept','beta', 'sigma')
calc_nodes <- model$getDependencies(wrt_nodes)
derivs_all <- derivs_nf(model, wrt_nodes, calc_nodes)
cModel <- compileNimble(model)
cDerivs_all <- compileNimble(derivs_all, project = model)
derivs_result <- cDerivs_all$run(order = 0:2)
derivs_result## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -80.74344
## Field "jacobian":
## [,1] [,2] [,3]
## [1,] -9.358104 -0.3637619 -5.81414
## Field "hessian":
## , , 1
##
## [,1] [,2] [,3]
## [1,] -62.35820 -16.21705 0.00000
## [2,] -16.21705 -69.47074 0.00000
## [3,] 0.00000 0.00000 -45.11516As above, using order = 0:2 results in the the value (0th order), Jacobian (1st order), and Hessian (2nd order). The function model$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$expandNodeNames("m", returnScalarElements=TRUE).
The actual values used as inputs are the current values in the compiled model object. These would typically be set by code like values(model, with_respect_to_nodes) <<- my_values before the call to nimDerivs in the above example.
As above, derivatives can also be calculated in uncompiled execution, but they will be much slower and less accurate. Uncompiled execution is mostly useful for checking that compiled derivatives are working correctly. This will use values in the uncompiled model object.
derivs_all$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 -45.11517We 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.
derivs_nf2 <- nimbleFunction(
setup = function(model, wrt_nodes, calc_nodes) {
derivsInfo <- makeModelDerivsInfo(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)
returnType(ADNimbleList())
}
),
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$derivsRun(params, order = 0:2)## nimbleList object of type NIMBLE_ADCLASS
## Field "value":
## [1] -80.74344
## Field "jacobian":
## [,1] [,2] [,3]
## [1,] -9.358104 -0.3637619 -2.90707
## Field "hessian":
## , , 1
##
## [,1] [,2] [,3]
## [1,] -62.35820 -16.21705 0.00000
## [2,] -16.21705 -69.47074 0.00000
## [3,] 0.00000 0.00000 -14.18586Notice that these results are the same as we saw above except for third elements, which represent derivatives with respect to sigma above and to log(sigma) here.
There are several important new points here:
The only valid way to get values into the model that will be recorded on the AD tape is with
values(model, nodes) <<- some_valuesas shown. Other ways such asnimCopyormodel[[node]] <<- some_valueare not currently supported to work with AD.The call to
nimDerivsofrunmust be told some information about the model calculations that will be done inside ofrun.modelis of course the model that will be used.constantNodesis a vector of node names whose values are needed for the calculations but are not expected to change until you usereset = TRUE. Values of these nodes will be baked into the AD tape untilreset = TRUE.updateNodesis a vector of node names whose values are needed for the calculations, might change between calls even withreset = FALSE, and are neither part ofwrt_nodesnor a deterministic part ofcalc_nodes. The functionmakeModelDerivsInfo, as shown in thesetupcode, determines what is usually needed forconstantNodesandupdateNodes.In Method 1 above, the NIMBLE compiler automatically determines
constantNodesandupdateNodesusingmakeModelDerivsInfobased on the inputs tonimDerivs(model$calculate(...),...).One can use double-taping, but if so both calls to
nimDerivsneed themodel,updateNodes, andconstantNodesarguments, 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:
updateNodesandconstantNodesare vectors of nodes names that together comprise any nodes that are necessary formodel$calculate(calc_nodes)but are:- not in the
wrtargument tonimDerivs(only relevant for Method 1), and - not assigned into the model using
values(model, nodes) <<- some_valuesprior tomodel$calculate(only relevant for Method 2), and - not a deterministic node in
calc_nodes.
- not in the
updateNodesincludes node names satisfying those conditions whose values might change between uses of the tape regardless of theresetargument.constantNodesincludes node names satisfying those conditions whose values will be baked into the tape (will not change) until the next call withreset=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]"(The “lifted” nodes here are for the exp(intercept + beta*X[i,j] + ran_eff[i]), i.e. the inputs to dpois. See chapters @ref(#cha-lightning-intro) and @ref(#cha-using-models) 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:
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 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:
model$calculate(model$getDependencies('precision', determOnly=TRUE))In Method 1 above, NIMBLE automatically uses makeModelDerivsInfo based on the code nimDerivs(model$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 ~ dweibullin a model means \(x \in (0, \infty)\). - If scalar \(x \in (0, 1)\), then \(g(x) = \mbox{logit}(x)\). For example,
x ~ dbetain 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] ~ dwishartorx[1:n, 1:n] ~ dinvwishartin 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] ~ ddirchin 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,xfollows 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.