Reparameterization and Change of Variables
Stan supports a direct encoding of reparameterizations. Stan also supports changes of variables by directly incrementing the log probability accumulator with the log Jacobian of the transform.
Theoretical and practical background
A Bayesian posterior is technically a probability measure, which is a parameterization-invariant, abstract mathematical object.1
Stan’s modeling language, on the other hand, defines a probability density, which is a non-unique, parameterization-dependent function in \(\mathbb{R}^N \rightarrow \mathbb{R}^{+}\). In practice, this means a given model can be represented different ways in Stan, and different representations have different computational performances.
As pointed out by Gelman (2004) in a paper discussing the relation between parameterizations and Bayesian modeling, a change of parameterization often carries with it suggestions of how the model might change, because we tend to use certain natural classes of prior distributions. Thus, it’s not just that we have a fixed distribution that we want to sample from, with reparameterizations being computational aids. In addition, once we reparameterize and add prior information, the model itself typically changes, often in useful ways.
Reparameterizations
Reparameterizations may be implemented directly using the transformed parameters block or just in the model block.
Beta and Dirichlet priors
The beta and Dirichlet distributions may both be reparameterized from a vector of counts to use a mean and total count.
Beta distribution
For example, the Beta distribution is parameterized by two positive count parameters \(\alpha, \beta > 0\). The following example illustrates a hierarchical Stan model with a vector of parameters theta are drawn i.i.d. for a Beta distribution whose parameters are themselves drawn from a hyperprior distribution.
parameters {
real<lower=0> alpha;
real<lower=0> beta;
// ...
}
model {
alpha ~ ...
beta ~ ...
for (n in 1:N) {
theta[n] ~ beta(alpha, beta);
}
// ...
}It is often more natural to specify hyperpriors in terms of transformed parameters. In the case of the Beta, the obvious choice for reparameterization is in terms of a mean parameter \[ \phi = \alpha / (\alpha + \beta) \] and total count parameter \[ \lambda = \alpha + \beta. \] Following @[GelmanEtAl:2013, Chapter 5] the mean gets a uniform prior and the count parameter a Pareto prior with \(p(\lambda) \propto \lambda^{-2.5}\).
parameters {
real<lower=0, upper=1> phi;
real<lower=0.1> lambda;
// ...
}
transformed parameters {
real<lower=0> alpha = lambda * phi;
real<lower=0> beta = lambda * (1 - phi);
// ...
}
model {
phi ~ beta(1, 1); // uniform on phi, could drop
lambda ~ pareto(0.1, 1.5);
for (n in 1:N) {
theta[n] ~ beta(alpha, beta);
}
// ...
}The new parameters, phi and lambda, are declared in the parameters block and the parameters for the Beta distribution, alpha and beta, are declared and defined in the transformed parameters block. And If their values are not of interest, they could instead be defined as local variables in the model as follows.
model {
real alpha = lambda * phi
real beta = lambda * (1 - phi);
// ...
for (n in 1:N) {
theta[n] ~ beta(alpha, beta);
}
// ...
}With vectorization, this could be expressed more compactly and efficiently as follows.
model {
theta ~ beta(lambda * phi, lambda * (1 - phi));
// ...
}If the variables alpha and beta are of interest, they can be defined in the transformed parameter block and then used in the model.
Jacobians not necessary
Because the transformed parameters are being used, rather than given a distribution, there is no need to apply a Jacobian adjustment for the transform. For example, in the beta distribution example, alpha and beta have the correct posterior distribution.
Dirichlet priors
The same thing can be done with a Dirichlet, replacing the mean for the Beta, which is a probability value, with a simplex. Assume there are \(K > 0\) dimensions being considered (\(K=1\) is trivial and \(K=2\) reduces to the beta distribution case). The traditional prior is
parameters {
vector[K] alpha;
array[N] simplex[K] theta;
// ...
}
model {
alpha ~ // ...
for (n in 1:N) {
theta[n] ~ dirichlet(alpha);
}
}This provides essentially \(K\) degrees of freedom, one for each dimension of alpha, and it is not obvious how to specify a reasonable prior for alpha.
An alternative coding is to use the mean, which is a simplex, and a total count.
parameters {
simplex[K] phi;
real<lower=0> kappa;
array[N] simplex[K] theta;
// ...
}
transformed parameters {
vector[K] alpha = kappa * phi;
// ...
}
model {
phi ~ // ...
kappa ~ // ...
for (n in 1:N) {
theta[n] ~ dirichlet(alpha);
}
// ...
}Now it is much easier to formulate priors, because phi is the expected value of theta and kappa (minus K) is the strength of the prior mean measured in number of prior observations.
Transforming unconstrained priors: probit and logit
If the variable \(u\) has a \(\textsf{uniform}(0, 1)\) distribution, then \(\operatorname{logit}(u)\) is distributed as \(\textsf{logistic}(0, 1)\). This is because inverse logit is the cumulative distribution function (cdf) for the logistic distribution, so that the logit function itself is the inverse CDF and thus maps a uniform draw in \((0, 1)\) to a logistically-distributed quantity.
Things work the same way for the probit case: if \(u\) has a \(\textsf{uniform}(0, 1)\) distribution, then \(\Phi^{-1}(u)\) has a \(\textsf{normal}(0, 1)\) distribution. The other way around, if \(v\) has a \(\textsf{normal}(0, 1)\) distribution, then \(\Phi(v)\) has a \(\textsf{uniform}(0, 1)\) distribution.
In order to use the probit and logistic as priors on variables constrained to \((0, 1)\), create an unconstrained variable and transform it appropriately. For comparison, the following Stan program fragment declares a \((0, 1)\)-constrained parameter theta and gives it a beta prior, then uses it as a parameter in a distribution (here using foo as a placeholder).
parameters {
real<lower=0, upper=1> theta;
// ...
}
model {
theta ~ beta(a, b);
// ...
y ~ foo(theta);
// ...
}If the variables a and b are one, then this imposes a uniform distribution theta. If a and b are both less than one, then the density on theta has a U shape, whereas if they are both greater than one, the density of theta has an inverted-U or more bell-like shape.
Roughly the same result can be achieved with unbounded parameters that are probit or inverse-logit-transformed. For example,
parameters {
real theta_raw;
// ...
}
transformed parameters {
real<lower=0, upper=1> theta = inv_logit(theta_raw);
// ...
}
model {
theta_raw ~ logistic(mu, sigma);
// ...
y ~ foo(theta);
// ...
}In this model, an unconstrained parameter theta_raw gets a logistic prior, and then the transformed parameter theta is defined to be the inverse logit of theta_raw. In this parameterization, inv_logit(mu) is the mean of the implied prior on theta. The prior distribution on theta will be flat if sigma is one and mu is zero, and will be U-shaped if sigma is larger than one and bell shaped if sigma is less than one.
When moving from a variable in \((0, 1)\) to a simplex, the same trick may be performed using the softmax function, which is a multinomial generalization of the inverse logit function. First, consider a simplex parameter with a Dirichlet prior.
parameters {
simplex[K] theta;
// ...
}
model {
theta ~ dirichlet(a);
// ...
y ~ foo(theta);
}Now a is a vector with K rows, but it has the same shape properties as the pair a and b for a beta; the beta distribution is just the distribution of the first component of a Dirichlet with parameter vector \([a b]^{\top}\). To formulate an unconstrained prior, the exact same strategy works as for the beta.
parameters {
vector[K] theta_raw;
// ...
}
transformed parameters {
simplex[K] theta = softmax(theta_raw);
// ...
}
model {
theta_raw ~ multi_normal_cholesky(mu, L_Sigma);
}The multivariate normal is used for convenience and efficiency with its Cholesky-factor parameterization. Now the mean is controlled by softmax(mu), but we have additional control of covariance through L_Sigma at the expense of having on the order of \(K^2\) parameters in the prior rather than order \(K\). If no covariance is desired, the number of parameters can be reduced back to \(K\) using a vectorized normal distribution as follows.
theta_raw ~ normal(mu, sigma);where either or both of mu and sigma can be vectors.
Changes of variables
Changes of variables are applied when the transformation of a parameter is characterized by a distribution. The standard textbook example is the lognormal distribution, which is the distribution of a variable \(y > 0\) whose logarithm \(\log y\) has a normal distribution. The distribution is being assigned to \(\log y\).
The change of variables requires an adjustment to the probability to account for the distortion caused by the transform. For this to work, univariate changes of variables must be monotonic and differentiable everywhere in their support. Multivariate changes of variables must be injective and differentiable everywhere in their support, and they must map \(\mathbb{R}^N \rightarrow \mathbb{R}^N\).
The probability must be scaled by a Jacobian adjustment equal to the absolute determinant of the Jacobian of the transform. In the univariate case, the Jacobian adjustment is simply the absolute derivative of the transform.
In the case of log normals, if \(y\)’s logarithm is normal with mean \(\mu\) and deviation \(\sigma\), then the distribution of \(y\) is given by \[ p(y) = \textsf{normal}(\log y \mid \mu, \sigma) \, \left| \frac{d}{dy} \log y \right| = \textsf{normal}(\log y \mid \mu, \sigma) \, \frac{1}{y}. \] Stan works on the log scale to prevent underflow, where \[ \log p(y) = \log \textsf{normal}(\log y \mid \mu, \sigma) - \log y. \]
In Stan, the change of variables can be applied in the sampling statement. To adjust for the curvature, the log probability accumulator is incremented with the log absolute derivative of the transform. The lognormal distribution can thus be implemented directly in Stan as follows.2
parameters {
real<lower=0> y;
// ...
}
model {
log(y) ~ normal(mu, sigma);
target += -log(y);
// ...
}It is important, as always, to declare appropriate constraints on parameters; here y is constrained to be positive.
It would be slightly more efficient to define a local variable for the logarithm, as follows.
model {
real log_y;
log_y = log(y);
log_y ~ normal(mu, sigma);
target += -log_y;
// ...
}If y were declared as data instead of as a parameter, then the adjustment can be ignored because the data will be constant and Stan only requires the log probability up to a constant.
Change of variables vs. transformations
This section illustrates the difference between a change of variables and a simple variable transformation. A transformation samples a parameter, then transforms it, whereas a change of variables transforms a parameter, then samples it. Only the latter requires a Jacobian adjustment.
It does not matter whether the probability function is expressed using a sampling statement, such as
log(y) ~ normal(mu, sigma);or as an increment to the log probability function, as in
target += normal_lpdf(log(y) | mu, sigma);Gamma and inverse gamma distribution
Like the log normal, the inverse gamma distribution is a distribution of variables whose inverse has a gamma distribution. This section contrasts two approaches, first with a transform, then with a change of variables.
The transform based approach to sampling y_inv with an inverse gamma distribution can be coded as follows.
parameters {
real<lower=0> y;
}
transformed parameters {
real<lower=0> y_inv;
y_inv = 1 / y;
}
model {
y ~ gamma(2,4);
}The change-of-variables approach to sampling y_inv with an inverse gamma distribution can be coded as follows.
parameters {
real<lower=0> y_inv;
}
transformed parameters {
real<lower=0> y;
y = 1 / y_inv; // change variables
}
model {
y ~ gamma(2,4);
target += -2 * log(y_inv); // Jacobian adjustment;
}The Jacobian adjustment is the log of the absolute derivative of the transform, which in this case is
\[ \log \left| \frac{d}{du} \left( \frac{1}{u} \right) \right| = \log \left| - u^{-2} \right| = \log u^{-2} = -2 \log u. \]
Multivariate changes of variables
In the case of a multivariate transform, the log of the absolute determinant of the Jacobian of the transform must be added to the log probability accumulator. In Stan, this can be coded as follows in the general case where the Jacobian is not a full matrix.
parameters {
vector[K] u; // multivariate parameter
// ...
}
transformed parameters {
vector[K] v; // transformed parameter
matrix[K, K] J; // Jacobian matrix of transform
// ... compute v as a function of u ...
// ... compute J[m, n] = d.v[m] / d.u[n] ...
target += log(abs(determinant(J)));
// ...
}
model {
v ~ // ...
// ...
}If the determinant of the Jacobian is known analytically, it will be more efficient to apply it directly than to call the determinant function, which is neither efficient nor particularly stable numerically.
In many cases, the Jacobian matrix will be triangular, so that only the diagonal elements will be required for the determinant calculation. Triangular Jacobians arise when each element v[k] of the transformed parameter vector only depends on elements u[1], …, u[k] of the parameter vector. For triangular matrices, the determinant is the product of the diagonal elements, so the transformed parameters block of the above model can be simplified and made more efficient by recoding as follows.
transformed parameters {
// ...
vector[K] J_diag; // diagonals of Jacobian matrix
// ...
// ... compute J[k, k] = d.v[k] / d.u[k] ...
target += sum(log(J_diag));
// ...
}Vectors with varying bounds
Stan allows scalar and non-scalar upper and lower bounds to be declared in the constraints for a container data type. The transforms are calculated and their log Jacobians added to the log density accumulator; the Jacobian calculations are described in detail in the reference manual chapter on constrained parameter transforms.
Varying lower bounds
For example, suppose there is a vector parameter \(\alpha\) with a vector \(L\) of lower bounds. The simplest way to deal with this if \(L\) is a constant is to shift a lower-bounded parameter.
data {
int N;
vector[N] L; // lower bounds
// ...
}
parameters {
vector<lower=L>[N] alpha_raw;
// ...
}The above is equivalent to manually calculating the vector bounds by the following.
data {
int N;
vector[N] L; // lower bounds
// ...
}
parameters {
vector<lower=0>[N] alpha_raw;
// ...
}
transformed parameters {
vector[N] alpha = L + alpha_raw;
// ...
}The Jacobian for adding a constant is one, so its log drops out of the log density.
Even if the lower bound is a parameter rather than data, there is no Jacobian required, because the transform from \((L, \alpha_{\textrm{raw}})\) to \((L + \alpha_{\textrm{raw}}, \alpha_{\textrm{raw}})\) produces a Jacobian derivative matrix with a unit determinant.
It’s also possible to implement the transform using an array or vector of parameters as bounds (with the requirement that the type of the variable must match the bound type) in the following.
data {
int N;
vector[N] L; // lower bounds
// ...
}
parameters {
vector<lower=0>[N] alpha_raw;
vector<lower=L + alpha_raw>[N] alpha;
// ...
}This is equivalent to directly transforming an unconstrained parameter and accounting for the Jacobian.
data {
int N;
vector[N] L; // lower bounds
// ...
}
parameters {
vector[N] alpha_raw;
// ...
}
transformed parameters {
vector[N] alpha = L + exp(alpha_raw);
// ...
}
model {
target += sum(alpha_raw); // log Jacobian
// ...
}The adjustment in the log Jacobian determinant of the transform mapping \(\alpha_{\textrm{raw}}\) to \(\alpha = L + \exp(\alpha_{\textrm{raw}})\). The details are simple in this case because the Jacobian is diagonal; see the reference manual chapter on constrained parameter transforms for full details. Here \(L\) can even be a vector containing parameters that don’t depend on \(\alpha_{\textrm{raw}}\); if the bounds do depend on \(\alpha_{\textrm{raw}}\) then a revised Jacobian needs to be calculated taking into account the dependencies.
Varying upper and lower bounds
Suppose there are lower and upper bounds that vary by parameter. These can be applied to shift and rescale a parameter constrained to \((0, 1)\). This is easily accomplished as the following.
data {
int N;
vector[N] L; // lower bounds
vector[N] U; // upper bounds
// ...
}
parameters {
vector<lower=L, upper=U>[N] alpha;
// ...
}The same may be accomplished by manually constructing the transform as follows.
data {
int N;
vector[N] L; // lower bounds
vector[N] U; // upper bounds
// ...
}
parameters {
vector<lower=0, upper=1>[N] alpha_raw;
// ...
}
transformed parameters {
vector[N] alpha = L + (U - L) .* alpha_raw;
}The expression U - L is multiplied by alpha_raw elementwise to produce a vector of variables in \((0, U-L)\), then adding \(L\) results in a variable ranging between \((L, U)\).
In this case, it is important that \(L\) and \(U\) are constants, otherwise a Jacobian would be required when multiplying by \(U - L\).
References
Footnotes
This is in contrast to (penalized) maximum likelihood estimates, which are not parameterization invariant.↩︎
This example is for illustrative purposes only; the recommended way to implement the lognormal distribution in Stan is with the built-in
lognormalprobability function; see the functions reference manual for details.↩︎