Regression Models

Stan supports regression models from simple linear regressions to multilevel generalized linear models.

Linear regression

The simplest linear regression model is the following, with a single predictor and a slope and intercept coefficient, and normally distributed noise. This model can be written using standard regression notation as \[ y_n = \alpha + \beta x_n + \epsilon_n \quad\text{where}\quad \epsilon_n \sim \operatorname{normal}(0,\sigma). \]

This is equivalent to the following sampling involving the residual, \[ y_n - (\alpha + \beta X_n) \sim \operatorname{normal}(0,\sigma), \] and reducing still further, to \[ y_n \sim \operatorname{normal}(\alpha + \beta X_n, \, \sigma). \]

This latter form of the model is coded in Stan as follows.

data {
  int<lower=0> N;
  vector[N] x;
  vector[N] y;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(alpha + beta * x, sigma);
}

There are N observations and for each observation, \(n \in N\), we have predictor x[n] and outcome y[n]. The intercept and slope parameters are alpha and beta. The model assumes a normally distributed noise term with scale sigma. This model has improper priors for the two regression coefficients.

Matrix notation and vectorization

The sampling statement in the previous model is vectorized, with

y ~ normal(alpha + beta * x, sigma);

providing the same model as the unvectorized version,

for (n in 1:N) {
  y[n] ~ normal(alpha + beta * x[n], sigma);
}

In addition to being more concise, the vectorized form is much faster.1

In general, Stan allows the arguments to distributions such as normal to be vectors. If any of the other arguments are vectors or arrays, they have to be the same size. If any of the other arguments is a scalar, it is reused for each vector entry.

The other reason this works is that Stan’s arithmetic operators are overloaded to perform matrix arithmetic on matrices. In this case, because x is of type vector and beta of type real, the expression beta * x is of type vector. Because Stan supports vectorization, a regression model with more than one predictor can be written directly using matrix notation.

data {
  int<lower=0> N;   // number of data items
  int<lower=0> K;   // number of predictors
  matrix[N, K] x;   // predictor matrix
  vector[N] y;      // outcome vector
}
parameters {
  real alpha;           // intercept
  vector[K] beta;       // coefficients for predictors
  real<lower=0> sigma;  // error scale
}
model {
  y ~ normal(x * beta + alpha, sigma);  // likelihood
}

The constraint lower=0 in the declaration of sigma constrains the value to be greater than or equal to 0. With no prior in the model block, the effect is an improper prior on non-negative real numbers. Although a more informative prior may be added, improper priors are acceptable as long as they lead to proper posteriors.

In the model above, x is an \(N \times K\) matrix of predictors and beta a \(K\)-vector of coefficients, so x * beta is an \(N\)-vector of predictions, one for each of the \(N\) data items. These predictions line up with the outcomes in the \(N\)-vector y, so the entire model may be written using matrix arithmetic as shown. It would be possible to include a column of ones in the data matrix x to remove the alpha parameter.

The sampling statement in the model above is just a more efficient, vector-based approach to coding the model with a loop, as in the following statistically equivalent model.

model {
  for (n in 1:N) {
    y[n] ~ normal(x[n] * beta, sigma);
  }
}

With Stan’s matrix indexing scheme, x[n] picks out row n of the matrix x; because beta is a column vector, the product x[n] * beta is a scalar of type real.

Intercepts as inputs

In the model formulation

y ~ normal(x * beta, sigma);

there is no longer an intercept coefficient alpha. Instead, we have assumed that the first column of the input matrix x is a column of 1 values. This way, beta[1] plays the role of the intercept. If the intercept gets a different prior than the slope terms, then it would be clearer to break it out. It is also slightly more efficient in its explicit form with the intercept variable singled out because there’s one fewer multiplications; it should not make that much of a difference to speed, though, so the choice should be based on clarity.

The QR reparameterization

In the previous example, the linear predictor can be written as \(\eta = x \beta\), where \(\eta\) is a \(N\)-vector of predictions, \(x\) is a \(N \times K\) matrix, and \(\beta\) is a \(K\)-vector of coefficients. Presuming \(N \geq K\), we can exploit the fact that any design matrix \(x\) can be decomposed using the thin QR decomposition into an orthogonal matrix \(Q\) and an upper-triangular matrix \(R\), i.e. \(x = Q R\).

The functions qr_thin_Q and qr_thin_R implement the thin QR decomposition, which is to be preferred to the fat QR decomposition that would be obtained by using qr_Q and qr_R, as the latter would more easily run out of memory (see the Stan Functions Reference for more information on the qr_thin_Q and qr_thin_R functions). In practice, it is best to write \(x = Q^\ast R^\ast\) where \(Q^\ast = Q * \sqrt{n - 1}\) and \(R^\ast = \frac{1}{\sqrt{n - 1}} R\). Thus, we can equivalently write \(\eta = x \beta = Q R \beta = Q^\ast R^\ast \beta\). If we let \(\theta = R^\ast \beta\), then we have \(\eta = Q^\ast \theta\) and \(\beta = R^{\ast^{-1}} \theta\). In that case, the previous Stan program becomes

data {
  int<lower=0> N;   // number of data items
  int<lower=0> K;   // number of predictors
  matrix[N, K] x;   // predictor matrix
  vector[N] y;      // outcome vector
}
transformed data {
  matrix[N, K] Q_ast;
  matrix[K, K] R_ast;
  matrix[K, K] R_ast_inverse;
  // thin and scale the QR decomposition
  Q_ast = qr_thin_Q(x) * sqrt(N - 1);
  R_ast = qr_thin_R(x) / sqrt(N - 1);
  R_ast_inverse = inverse(R_ast);
}
parameters {
  real alpha;           // intercept
  vector[K] theta;      // coefficients on Q_ast
  real<lower=0> sigma;  // error scale
}
model {
  y ~ normal(Q_ast * theta + alpha, sigma);  // likelihood
}
generated quantities {
  vector[K] beta;
  beta = R_ast_inverse * theta; // coefficients on x
}

Since this Stan program generates equivalent predictions for \(y\) and the same posterior distribution for \(\alpha\), \(\beta\), and \(\sigma\) as the previous Stan program, many wonder why the version with this QR reparameterization performs so much better in practice, often both in terms of wall time and in terms of effective sample size. The reasoning is threefold:

  1. The columns of \(Q^\ast\) are orthogonal whereas the columns of \(x\) generally are not. Thus, it is easier for a Markov Chain to move around in \(\theta\)-space than in \(\beta\)-space.

  2. The columns of \(Q^\ast\) have the same scale whereas the columns of \(x\) generally do not. Thus, a Hamiltonian Monte Carlo algorithm can move around the parameter space with a smaller number of larger steps

  3. Since the covariance matrix for the columns of \(Q^\ast\) is an identity matrix, \(\theta\) typically has a reasonable scale if the units of \(y\) are also reasonable. This also helps HMC move efficiently without compromising numerical accuracy.

Consequently, this QR reparameterization is recommended for linear and generalized linear models in Stan whenever \(K > 1\) and you do not have an informative prior on the location of \(\beta\). It can also be worthwhile to subtract the mean from each column of \(x\) before obtaining the QR decomposition, which does not affect the posterior distribution of \(\theta\) or \(\beta\) but does affect \(\alpha\) and allows you to interpret \(\alpha\) as the expectation of \(y\) in a linear model.

Priors for coefficients and scales

See our general discussion of priors for tips on priors for parameters in regression models.

Later sections discuss univariate hierarchical priors and multivariate hierarchical priors, as well as priors used to identify models.

However, as described in QR-reparameterization section, if you do not have an informative prior on the location of the regression coefficients, then you are better off reparameterizing your model so that the regression coefficients are a generated quantity. In that case, it usually does not matter much what prior is used on on the reparameterized regression coefficients and almost any weakly informative prior that scales with the outcome will do.

Robust noise models

The standard approach to linear regression is to model the noise term \(\epsilon\) as having a normal distribution. From Stan’s perspective, there is nothing special about normally distributed noise. For instance, robust regression can be accommodated by giving the noise term a Student-\(t\) distribution. To code this in Stan, the sampling distribution is changed to the following.

data {
  // ...
  real<lower=0> nu;
}
// ...
model {
  y ~ student_t(nu, alpha + beta * x, sigma);
}

The degrees of freedom constant nu is specified as data.

Logistic and probit regression

For binary outcomes, either of the closely related logistic or probit regression models may be used. These generalized linear models vary only in the link function they use to map linear predictions in \((-\infty,\infty)\) to probability values in \((0,1)\). Their respective link functions, the logistic function and the standard normal cumulative distribution function, are both sigmoid functions (i.e., they are both S-shaped).

A logistic regression model with one predictor and an intercept is coded as follows.

data {
  int<lower=0> N;
  vector[N] x;
  array[N] int<lower=0, upper=1> y;
}
parameters {
  real alpha;
  real beta;
}
model {
  y ~ bernoulli_logit(alpha + beta * x);
}

The noise parameter is built into the Bernoulli formulation here rather than specified directly.

Logistic regression is a kind of generalized linear model with binary outcomes and the log odds (logit) link function, defined by \[ \operatorname{logit}(v) = \log \left( \frac{v}{1-v} \right). \]

The inverse of the link function appears in the model: \[ \operatorname{logit}^{-1}(u) = \texttt{inv}\mathtt{\_}\texttt{logit}(u) = \frac{1}{1 + \exp(-u)}. \]

The model formulation above uses the logit-parameterized version of the Bernoulli distribution, which is defined by \[ \texttt{bernoulli}\mathtt{\_}\texttt{logit}\left(y \mid \alpha \right) = \texttt{bernoulli}\left(y \mid \operatorname{logit}^{-1}(\alpha)\right). \]

The formulation is also vectorized in the sense that alpha and beta are scalars and x is a vector, so that alpha + beta * x is a vector. The vectorized formulation is equivalent to the less efficient version

for (n in 1:N) {
  y[n] ~ bernoulli_logit(alpha + beta * x[n]);
}

Expanding out the Bernoulli logit, the model is equivalent to the more explicit, but less efficient and less arithmetically stable

for (n in 1:N) {
  y[n] ~ bernoulli(inv_logit(alpha + beta * x[n]));
}

Other link functions may be used in the same way. For example, probit regression uses the cumulative normal distribution function, which is typically written as

\[ \Phi(x) = \int_{-\infty}^x \textsf{normal}\left(y \mid 0,1 \right) \,\textrm{d}y. \]

The cumulative standard normal distribution function \(\Phi\) is implemented in Stan as the function Phi. The probit regression model may be coded in Stan by replacing the logistic model’s sampling statement with the following.

y[n] ~ bernoulli(Phi(alpha + beta * x[n]));

A fast approximation to the cumulative standard normal distribution function \(\Phi\) is implemented in Stan as the function Phi_approx.2 The approximate probit regression model may be coded with the following.

y[n] ~ bernoulli(Phi_approx(alpha + beta * x[n]));

Multi-logit regression

Multiple outcome forms of logistic regression can be coded directly in Stan. For instance, suppose there are \(K\) possible outcomes for each output variable \(y_n\). Also suppose that there is a \(D\)-dimensional vector \(x_n\) of predictors for \(y_n\). The multi-logit model with \(\textsf{normal}(0,5)\) priors on the coefficients is coded as follows.

data {
  int K;
  int N;
  int D;
  array[N] int y;
  matrix[N, D] x;
}
parameters {
  matrix[D, K] beta;
}
model {
  matrix[N, K] x_beta = x * beta;

  to_vector(beta) ~ normal(0, 5);

  for (n in 1:N) {
    y[n] ~ categorical_logit(x_beta[n]');

  }
}

where x_beta[n]' is the transpose of x_beta[n]. The prior on beta is coded in vectorized form. As of Stan 2.18, the categorical-logit distribution is not vectorized for parameter arguments, so the loop is required. The matrix multiplication is pulled out to define a local variable for all of the predictors for efficiency. Like the Bernoulli-logit, the categorical-logit distribution applies softmax internally to convert an arbitrary vector to a simplex, \[ \texttt{categorical}\mathtt{\_}\texttt{logit}\left(y \mid \alpha\right) = \texttt{categorical}\left(y \mid \texttt{softmax}(\alpha)\right), \] where \[ \texttt{softmax}(u) = \exp(u) / \operatorname{sum}\left(\exp(u)\right). \]

The categorical distribution with log-odds (logit) scaled parameters used above is equivalent to writing

y[n] ~ categorical(softmax(x[n] * beta));

Constraints on data declarations

The data block in the above model is defined without constraints on sizes K, N, and D or on the outcome array y. Constraints on data declarations provide error checking at the point data are read (or transformed data are defined), which is before sampling begins. Constraints on data declarations also make the model author’s intentions more explicit, which can help with readability. The above model’s declarations could be tightened to

int<lower=2> K;
int<lower=0> N;
int<lower=1> D;
array[N] int<lower=1, upper=K> y;

These constraints arise because the number of categories, K, must be at least two in order for a categorical model to be useful. The number of data items, N, can be zero, but not negative; unlike R, Stan’s for-loops always move forward, so that a loop extent of 1:N when N is equal to zero ensures the loop’s body will not be executed. The number of predictors, D, must be at least one in order for beta * x[n] to produce an appropriate argument for softmax(). The categorical outcomes y[n] must be between 1 and K in order for the discrete sampling to be well defined.

Constraints on data declarations are optional. Constraints on parameters declared in the parameters block, on the other hand, are not optional—they are required to ensure support for all parameter values satisfying their constraints. Constraints on transformed data, transformed parameters, and generated quantities are also optional.

Identifiability

Because softmax is invariant under adding a constant to each component of its input, the model is typically only identified if there is a suitable prior on the coefficients.

An alternative is to use \((K-1)\)-vectors by fixing one of them to be zero. The partially known parameters section discusses how to mix constants and parameters in a vector. In the multi-logit case, the parameter block would be redefined to use \((K - 1)\)-vectors

parameters {
  matrix[D, K - 1] beta_raw;
}

and then these are transformed to parameters to use in the model. First, a transformed data block is added before the parameters block to define a vector of zero values,

transformed data {
  vector[D] zeros = rep_vector(0, D);
}

which can then be appended to beta_raw to produce the coefficient matrix beta,

transformed parameters {
  matrix[D, K] beta = append_col(beta_raw, zeros);
}

The rep_vector(0, D) call creates a column vector of size D with all entries set to zero. The derived matrix beta is then defined to be the result of appending the vector zeros as a new column at the end of beta_raw; the vector zeros is defined as transformed data so that it doesn’t need to be constructed from scratch each time it is used.

This is not the same model as using \(K\)-vectors as parameters, because now the prior only applies to \((K-1)\)-vectors. In practice, this will cause the maximum likelihood solutions to be different and also the posteriors to be slightly different when taking priors centered around zero, as is typical for regression coefficients.

Parameterizing centered vectors

It is often convenient to define a parameter vector \(\beta\) that is centered in the sense of satisfying the sum-to-zero constraint, \[ \sum_{k=1}^K \beta_k = 0. \]

Such a parameter vector may be used to identify a multi-logit regression parameter vector (see the multi-logit section for details), or may be used for ability or difficulty parameters (but not both) in an IRT model (see the item-response model section for details).

\(K-1\) degrees of freedom

There is more than one way to enforce a sum-to-zero constraint on a parameter vector, the most efficient of which is to define the \(K\)-th element as the negation of the sum of the elements \(1\) through \(K-1\).

parameters {
  vector[K - 1] beta_raw;
  // ...
}
transformed parameters {
  vector[K] beta = append_row(beta_raw, -sum(beta_raw));
  // ...
}

Placing a prior on beta_raw in this parameterization leads to a subtly different posterior than that resulting from the same prior on beta in the original parameterization without the sum-to-zero constraint. Most notably, a simple prior on each component of beta_raw produces different results than putting the same prior on each component of an unconstrained \(K\)-vector beta. For example, providing a \(\textsf{normal}(0,5)\) prior on beta will produce a different posterior mode than placing the same prior on beta_raw.

Marginal distribution of sum-to-zero components

On the Stan forums, Aaron Goodman provided the following code to produce a prior with standard normal marginals on the components of beta,

model {
  beta ~ normal(0, inv(sqrt(1 - inv(K))));
  // ...
}

The components are not independent, as they must sum zero. No Jacobian is required because summation and negation are linear operations (and thus have constant Jacobians).

To generate distributions with marginals other than standard normal, the resulting beta may be scaled by some factor sigma and translated to some new location mu.

QR decomposition

Aaron Goodman, on the Stan forums, also provided this approach, which calculates a QR decomposition in the transformed data block, then uses it to transform to a sum-to-zero parameter x,

transformed data{
  matrix[K, K] A = diag_matrix(rep_vector(1, K));
  matrix[K, K - 1] A_qr;
  for (i in 1:K - 1) A[K, i] = -1;
  A[K, K] = 0;
  A_qr = qr_Q(A)[ , 1:(K - 1)];
}
parameters {
  vector[K - 1] beta_raw;
}
transformed parameters{
   vector[K] beta =  A_qr * beta_raw;
}
model {
  beta_raw ~ normal(0, inv(sqrt(1 - inv(K))));
}

This produces a marginal standard normal distribution on the values of beta, which will sum to zero by construction of the QR decomposition.

Translated and scaled simplex

An alternative approach that’s less efficient, but amenable to a symmetric prior, is to offset and scale a simplex.

parameters {
  simplex[K] beta_raw;
  real beta_scale;
  // ...
}
transformed parameters {
  vector[K] beta;
  beta = beta_scale * (beta_raw - inv(K));
  // ...
}

Here inv(K) is just a short way to write 1.0 / K. Given that beta_raw sums to 1 because it is a simplex, the elementwise subtraction of inv(K) is guaranteed to sum to zero. Because the magnitude of the elements of the simplex is bounded, a scaling factor is required to provide beta with \(K\) degrees of freedom necessary to take on every possible value that sums to zero.

With this parameterization, a Dirichlet prior can be placed on beta_raw, perhaps uniform, and another prior put on beta_scale, typically for “shrinkage.”

Soft centering

Adding a prior such as \(\beta \sim \textsf{normal}(0,\sigma)\) will provide a kind of soft centering of a parameter vector \(\beta\) by preferring, all else being equal, that \(\sum_{k=1}^K \beta_k = 0\). This approach is only guaranteed to roughly center if \(\beta\) and the elementwise addition \(\beta + c\) for a scalar constant \(c\) produce the same likelihood (perhaps by another vector \(\alpha\) being transformed to \(\alpha - c\), as in the IRT models). This is another way of achieving a symmetric prior.

Ordered logistic and probit regression

Ordered regression for an outcome \(y_n \in \{ 1, \dotsc, k \}\) with predictors \(x_n \in \mathbb{R}^D\) is determined by a single coefficient vector \(\beta \in \mathbb{R}^D\) along with a sequence of cutpoints \(c \in \mathbb{R}^{K-1}\) sorted so that \(c_d < c_{d+1}\). The discrete output is \(k\) if the linear predictor \(x_n \beta\) falls between \(c_{k-1}\) and \(c_k\), assuming \(c_0 = -\infty\) and \(c_K = \infty\). The noise term is fixed by the form of regression, with examples for ordered logistic and ordered probit models.

Ordered logistic regression

The ordered logistic model can be coded in Stan using the ordered data type for the cutpoints and the built-in ordered_logistic distribution.

data {
  int<lower=2> K;
  int<lower=0> N;
  int<lower=1> D;
  array[N] int<lower=1, upper=K> y;
  array[N] row_vector[D] x;
}
parameters {
  vector[D] beta;
  ordered[K - 1] c;
}
model {
  for (n in 1:N) {
    y[n] ~ ordered_logistic(x[n] * beta, c);
  }
}

The vector of cutpoints c is declared as ordered[K - 1], which guarantees that c[k] is less than c[k + 1].

If the cutpoints were assigned independent priors, the constraint effectively truncates the joint prior to support over points that satisfy the ordering constraint. Luckily, Stan does not need to compute the effect of the constraint on the normalizing term because the probability is needed only up to a proportion.

Ordered probit

An ordered probit model could be coded in exactly the same way by swapping the cumulative logistic (inv_logit) for the cumulative normal (Phi).

data {
  int<lower=2> K;
  int<lower=0> N;
  int<lower=1> D;
  array[N] int<lower=1, upper=K> y;
  array[N] row_vector[D] x;
}
parameters {
  vector[D] beta;
  ordered[K - 1] c;
}
model {
  vector[K] theta;
  for (n in 1:N) {
    real eta;
    eta = x[n] * beta;
    theta[1] = 1 - Phi(eta - c[1]);
    for (k in 2:(K - 1)) {
      theta[k] = Phi(eta - c[k - 1]) - Phi(eta - c[k]);
    }
    theta[K] = Phi(eta - c[K - 1]);
    y[n] ~ categorical(theta);
  }
}

The logistic model could also be coded this way by replacing Phi with inv_logit, though the built-in encoding based on the softmax transform is more efficient and more numerically stable. A small efficiency gain could be achieved by computing the values Phi(eta - c[k]) once and storing them for re-use.

Hierarchical regression

The simplest multilevel model is a hierarchical model in which the data are grouped into \(L\) distinct categories (or levels). An extreme approach would be to completely pool all the data and estimate a common vector of regression coefficients \(\beta\). At the other extreme, an approach with no pooling assigns each level \(l\) its own coefficient vector \(\beta_l\) that is estimated separately from the other levels. A hierarchical model is an intermediate solution where the degree of pooling is determined by the data and a prior on the amount of pooling.

Suppose each binary outcome \(y_n \in \{ 0, 1 \}\) has an associated level, \(ll_n \in \{ 1, \dotsc, L \}\). Each outcome will also have an associated predictor vector \(x_n \in \mathbb{R}^D\). Each level \(l\) gets its own coefficient vector \(\beta_l \in \mathbb{R}^D\). The hierarchical structure involves drawing the coefficients \(\beta_{l,d} \in \mathbb{R}\) from a prior that is also estimated with the data. This hierarchically estimated prior determines the amount of pooling. If the data in each level are similar, strong pooling will be reflected in low hierarchical variance. If the data in the levels are dissimilar, weaker pooling will be reflected in higher hierarchical variance.

The following model encodes a hierarchical logistic regression model with a hierarchical prior on the regression coefficients.

data {
  int<lower=1> D;
  int<lower=0> N;
  int<lower=1> L;
  array[N] int<lower=0, upper=1> y;
  array[N] int<lower=1, upper=L> ll;
  array[N] row_vector[D] x;
}
parameters {
  array[D] real mu;
  array[D] real<lower=0> sigma;
  array[L] vector[D] beta;
}
model {
  for (d in 1:D) {
    mu[d] ~ normal(0, 100);
    for (l in 1:L) {
      beta[l, d] ~ normal(mu[d], sigma[d]);
    }
  }
  for (n in 1:N) {
    y[n] ~ bernoulli(inv_logit(x[n] * beta[ll[n]]));
  }
}

The standard deviation parameter sigma gets an implicit uniform prior on \((0,\infty)\) because of its declaration with a lower-bound constraint of zero. Stan allows improper priors as long as the posterior is proper. Nevertheless, it is usually helpful to have informative or at least weakly informative priors for all parameters; see the regression priors section for recommendations on priors for regression coefficients and scales.

Optimizing the model

Where possible, vectorizing sampling statements leads to faster log probability and derivative evaluations. The speed boost is not because loops are eliminated, but because vectorization allows sharing subcomputations in the log probability and gradient calculations and because it reduces the size of the expression tree required for gradient calculations.

The first optimization vectorizes the for-loop over D as

mu ~ normal(0, 100);
for (l in 1:L) {
  beta[l] ~ normal(mu, sigma);
}

The declaration of beta as an array of vectors means that the expression beta[l] denotes a vector. Although beta could have been declared as a matrix, an array of vectors (or a two-dimensional array) is more efficient for accessing rows; see the indexing efficiency section for more information on the efficiency tradeoffs among arrays, vectors, and matrices.

This model can be further sped up and at the same time made more arithmetically stable by replacing the application of inverse-logit inside the Bernoulli distribution with the logit-parameterized Bernoulli,3

for (n in 1:N) {
  y[n] ~ bernoulli_logit(x[n] * beta[ll[n]]);
}

Unlike in R or BUGS, loops, array access and assignments are fast in Stan because they are translated directly to C++. In most cases, the cost of allocating and assigning to a container is more than made up for by the increased efficiency due to vectorizing the log probability and gradient calculations. Thus the following version is faster than the original formulation as a loop over a sampling statement.

{
  vector[N] x_beta_ll;
  for (n in 1:N) {
    x_beta_ll[n] = x[n] * beta[ll[n]];
  }
  y ~ bernoulli_logit(x_beta_ll);
}

The brackets introduce a new scope for the local variable x_beta_ll; alternatively, the variable may be declared at the top of the model block.

In some cases, such as the above, the local variable assignment leads to models that are less readable. The recommended practice in such cases is to first develop and debug the more transparent version of the model and only work on optimizations when the simpler formulation has been debugged.

Hierarchical priors

Priors on priors, also known as “hyperpriors,” should be treated the same way as priors on lower-level parameters in that as much prior information as is available should be brought to bear. Because hyperpriors often apply to only a handful of lower-level parameters, care must be taken to ensure the posterior is both proper and not overly sensitive either statistically or computationally to wide tails in the priors.

Boundary-avoiding priors for MLE in hierarchical models

The fundamental problem with maximum likelihood estimation (MLE) in the hierarchical model setting is that as the hierarchical variance drops and the values cluster around the hierarchical mean, the overall density grows without bound. As an illustration, consider a simple hierarchical linear regression (with fixed prior mean) of \(y_n \in \mathbb{R}\) on \(x_n \in \mathbb{R}^K\), formulated as \[\begin{align*} y_n & \sim \textsf{normal}(x_n \beta, \sigma) \\ \beta_k & \sim \textsf{normal}(0,\tau) \\ \tau & \sim \textsf{Cauchy}(0,2.5) \\ \end{align*}\]

In this case, as \(\tau \rightarrow 0\) and \(\beta_k \rightarrow 0\), the posterior density \[ p(\beta,\tau,\sigma|y,x) \propto p(y|x,\beta,\tau,\sigma) \] grows without bound. See the plot of Neal’s funnel density, which has similar behavior.

There is obviously no MLE estimate for \(\beta,\tau,\sigma\) in such a case, and therefore the model must be modified if posterior modes are to be used for inference. The approach recommended by Chung et al. (2013) is to use a gamma distribution as a prior, such as \[ \sigma \sim \textsf{Gamma}(2, 1/A), \] for a reasonably large value of \(A\), such as \(A = 10\).

Item-response theory models

Item-response theory (IRT) models the situation in which a number of students each answer one or more of a group of test questions. The model is based on parameters for the ability of the students, the difficulty of the questions, and in more articulated models, the discriminativeness of the questions and the probability of guessing correctly; see Gelman and Hill (2007, pps. 314–320) for a textbook introduction to hierarchical IRT models and Curtis (2010) for encodings of a range of IRT models in BUGS.

Data declaration with missingness

The data provided for an IRT model may be declared as follows to account for the fact that not every student is required to answer every question.

data {
  int<lower=1> J;                     // number of students
  int<lower=1> K;                     // number of questions
  int<lower=1> N;                     // number of observations
  array[N] int<lower=1, upper=J> jj;  // student for observation n
  array[N] int<lower=1, upper=K> kk;  // question for observation n
  array[N] int<lower=0, upper=1> y;   // correctness for observation n
}

This declares a total of N student-question pairs in the data set, where each n in 1:N indexes a binary observation y[n] of the correctness of the answer of student jj[n] on question kk[n].

The prior hyperparameters will be hard coded in the rest of this section for simplicity, though they could be coded as data in Stan for more flexibility.

1PL (Rasch) model

The 1PL item-response model, also known as the Rasch model, has one parameter (1P) for questions and uses the logistic link function (L).

The model parameters are declared as follows.

parameters {
  real delta;            // mean student ability
  array[J] real alpha;   // ability of student j - mean ability
  array[K] real beta;    // difficulty of question k
}

The parameter alpha[J] is the ability coefficient for student j and beta[k] is the difficulty coefficient for question k. The non-standard parameterization used here also includes an intercept term delta, which represents the average student’s response to the average question.4

The model itself is as follows.

model {
  alpha ~ std_normal();         // informative true prior
  beta ~ std_normal();          // informative true prior
  delta ~ normal(0.75, 1);      // informative true prior
  for (n in 1:N) {
    y[n] ~ bernoulli_logit(alpha[jj[n]] - beta[kk[n]] + delta);
  }
}

This model uses the logit-parameterized Bernoulli distribution, where \[ \texttt{bernoulli}\mathtt{\_}\texttt{logit}\left(y \mid \alpha\right) = \texttt{bernoulli}\left(y \mid \operatorname{logit}^{-1}(\alpha)\right). \]

The key to understanding it is the term inside the bernoulli_logit distribution, from which it follows that \[ \Pr[y_n = 1] = \operatorname{logit}^{-1}\left(\alpha_{jj[n]} - \beta_{kk[n]} + \delta\right). \]

The model suffers from additive identifiability issues without the priors. For example, adding a term \(\xi\) to each \(\alpha_j\) and \(\beta_k\) results in the same predictions. The use of priors for \(\alpha\) and \(\beta\) located at 0 identifies the parameters; see Gelman and Hill (2007) for a discussion of identifiability issues and alternative approaches to identification.

For testing purposes, the IRT 1PL model distributed with Stan uses informative priors that match the actual data generation process used to simulate the data in R (the simulation code is supplied in the same directory as the models). This is unrealistic for most practical applications, but allows Stan’s inferences to be validated. A simple sensitivity analysis with fatter priors shows that the posterior is fairly sensitive to the prior even with 400 students and 100 questions and only 25% missingness at random. For real applications, the priors should be fit hierarchically along with the other parameters, as described in the next section.

Multilevel 2PL model

The simple 1PL model described in the previous section is generalized in this section with the addition of a discrimination parameter to model how noisy a question is and by adding multilevel priors for the question difficulty and discrimination parameters. The model parameters are declared as follows.

parameters {
  real mu_beta;                // mean question difficulty
  vector[J] alpha;             // ability for j - mean
  vector[K] beta;              // difficulty for k
  vector<lower=0>[K] gamma;    // discrimination of k
  real<lower=0> sigma_beta;    // scale of difficulties
  real<lower=0> sigma_gamma;   // scale of log discrimination
}

The parameters should be clearer after the model definition.

model {
  alpha ~ std_normal();
  beta ~ normal(0, sigma_beta);
  gamma ~ lognormal(0, sigma_gamma);
  mu_beta ~ cauchy(0, 5);
  sigma_beta ~ cauchy(0, 5);
  sigma_gamma ~ cauchy(0, 5);
  y ~ bernoulli_logit(gamma[kk] .* (alpha[jj] - (beta[kk] + mu_beta)));
}

The std_normal function is used here, defined by \[ \texttt{std}\mathtt{\_}\texttt{normal}(y) = \textsf{normal}\left(y \mid 0, 1\right). \]

The sampling statement is also vectorized using elementwise multiplication; it is equivalent to

for (n in 1:N) {
  y[n] ~ bernoulli_logit(gamma[kk[n]]
                         * (alpha[jj[n]] - (beta[kk[n]] + mu_beta));
}

The 2PL model is similar to the 1PL model, with the additional parameter gamma[k] modeling how discriminative question k is. If gamma[k] is greater than 1, responses are more attenuated with less chance of getting a question right at random. The parameter gamma[k] is constrained to be positive, which prohibits there being questions that are easier for students of lesser ability; such questions are not unheard of, but they tend to be eliminated from most testing situations where an IRT model would be applied.

The model is parameterized here with student abilities alpha being given a standard normal prior. This is to identify both the scale and the location of the parameters, both of which would be unidentified otherwise; see the problematic posteriors chapter for further discussion of identifiability. The difficulty and discrimination parameters beta and gamma then have varying scales given hierarchically in this model. They could also be given weakly informative non-hierarchical priors, such as

beta ~ normal(0, 5);
gamma ~ lognormal(0, 2);

The point is that the alpha determines the scale and location and beta and gamma are allowed to float.

The beta parameter is here given a non-centered parameterization, with parameter mu_beta serving as the mean beta location. An alternative would’ve been to take:

beta ~ normal(mu_beta, sigma_beta);

and

y[n] ~ bernoulli_logit(gamma[kk[n]] * (alpha[jj[n]] - beta[kk[n]]));

Non-centered parameterizations tend to be more efficient in hierarchical models; see the reparameterization section for more information on non-centered reparameterizations.

The intercept term mu_beta can’t itself be modeled hierarchically, so it is given a weakly informative \(\textsf{Cauchy}(0,5)\) prior. Similarly, the scale terms, sigma_beta, and sigma_gamma, are given half-Cauchy priors. As mentioned earlier, the scale and location for alpha are fixed to ensure identifiability. The truncation in the half-Cauchy prior is implicit; explicit truncation is not necessary because the log probability need only be calculated up to a proportion and the scale variables are constrained to \((0,\infty)\) by their declarations.

Priors for identifiability

Location and scale invariance

One application of (hierarchical) priors is to identify the scale and/or location of a group of parameters. For example, in the IRT models discussed in the previous section, there is both a location and scale non-identifiability. With uniform priors, the posteriors will float in terms of both scale and location. See the collinearity section for a simple example of the problems this poses for estimation.

The non-identifiability is resolved by providing a standard normal (i.e., \(\textsf{normal}(0,1)\)) prior on one group of coefficients, such as the student abilities. With a standard normal prior on the student abilities, the IRT model is identified in that the posterior will produce a group of estimates for student ability parameters that have a sample mean of close to zero and a sample variance of close to one. The difficulty and discrimination parameters for the questions should then be given a diffuse, or ideally a hierarchical prior, which will identify these parameters by scaling and locating relative to the student ability parameters.

Collinearity

Another case in which priors can help provide identifiability is in the case of collinearity in a linear regression. In linear regression, if two predictors are collinear (i.e, one is a linear function of the other), then their coefficients will have a correlation of 1 (or -1) in the posterior. This leads to non-identifiability. By placing normal priors on the coefficients, the maximum likelihood solution of two duplicated predictors (trivially collinear) will be half the value than would be obtained by only including one.

Separability

In a logistic regression, if a predictor is positive in cases of 1 outcomes and negative in cases of 0 outcomes, then the maximum likelihood estimate for the coefficient for that predictor diverges to infinity. This divergence can be controlled by providing a prior for the coefficient, which will “shrink” the estimate back toward zero and thus identify the model in the posterior.

Similar problems arise for sampling with improper flat priors. The sampler will try to draw large values. By providing a prior, the posterior will be concentrated around finite values, leading to well-behaved sampling.

Multivariate priors for hierarchical models

In hierarchical regression models (and other situations), several individual-level variables may be assigned hierarchical priors. For example, a model with multiple varying intercepts and slopes within might assign them a multivariate prior.

As an example, the individuals might be people and the outcome income, with predictors such as education level and age, and the groups might be states or other geographic divisions. The effect of education level and age as well as an intercept might be allowed to vary by state. Furthermore, there might be state-level predictors, such as average state income and unemployment level.

Multivariate regression example

Gelman and Hill (2007, chap. 13, Chapter 17) provide a discussion of a hierarchical model with \(N\) individuals organized into \(J\) groups. Each individual has a predictor row vector \(x_n\) of size \(K\); to unify the notation, they assume that \(x_{n,1} = 1\) is a fixed “intercept” predictor. To encode group membership, they assume individual \(n\) belongs to group \(jj[n] \in \{ 1, \dotsc, J \}\). Each individual \(n\) also has an observed outcome \(y_n\) taking on real values.

Likelihood

The model is a linear regression with slope and intercept coefficients varying by group, so that \(\beta_j\) is the coefficient \(K\)-vector for group \(j\). The likelihood function for individual \(n\) is then just \[ y_n \sim \textsf{normal}(x_n \, \beta_{jj[n]}, \, \sigma) \quad\text{for}\quad n \in \{ 1, \dotsc, N \}. \]

Coefficient prior

Gelman and Hill model the coefficient vectors \(\beta_j\) as being drawn from a multivariate distribution with mean vector \(\mu\) and covariance matrix \(\Sigma\), \[ \beta_j \sim \textsf{multivariate normal}(\mu_j, \, \Sigma) \quad\text{for}\quad j \in \{ 1, \dotsc, J \}. \]

Below, we discuss the full model of Gelman and Hill, which uses group-level predictors to model \(\mu\); for now, we assume \(\mu\) is a simple vector parameter.

Hyperpriors

For hierarchical modeling, the group-level mean vector \(\mu\) and covariance matrix \(\Sigma\) must themselves be given priors. The group-level mean vector can be given a reasonable weakly-informative prior for independent coefficients, such as \[ \mu_j \sim \textsf{normal}(0,5). \] If more is known about the expected coefficient values \(\beta_{j, k}\), this information can be incorporated into the prior for \(\mu_j\).

For the prior on the covariance matrix, Gelman and Hill suggest using a scaled inverse Wishart. That choice was motivated primarily by convenience as it is conjugate to the multivariate likelihood function and thus simplifies Gibbs sampling

In Stan, there is no restriction to conjugacy for multivariate priors, and we in fact recommend a slightly different approach. Like Gelman and Hill, we decompose our prior into a scale and a matrix, but are able to do so in a more natural way based on the actual variable scales and a correlation matrix. Specifically, we define \[ \Sigma = \texttt{diag}\mathtt{\_}\texttt{matrix}(\tau) \times \Omega \times \texttt{diag}\mathtt{\_}\texttt{matrix}(\tau), \] where \(\Omega\) is a correlation matrix and \(\tau\) is the vector of coefficient scales. This mapping from scale vector \(\tau\) and correlation matrix \(\Omega\) can be inverted, using \[ \tau_k = \sqrt{\Sigma_{k,k}} \quad\textsf{and}\quad \Omega_{i, j} = \frac{\Sigma_{i, j}}{\tau_i \, \tau_j}. \]

The components of the scale vector \(\tau\) can be given any reasonable prior for scales, but we recommend something weakly informative like a half-Cauchy distribution with a small scale, such as \[ \tau_k \sim \textsf{Cauchy}(0, 2.5) \quad\text{for}\quad k \in \{ 1, \dotsc, K \} \quad\text{constrained\ by}\quad \tau_k > 0. \] As for the prior means, if there is information about the scale of variation of coefficients across groups, it should be incorporated into the prior for \(\tau\). For large numbers of exchangeable coefficients, the components of \(\tau\) itself (perhaps excluding the intercept) may themselves be given a hierarchical prior.

Our final recommendation is to give the correlation matrix \(\Omega\) an LKJ prior with shape \(\eta \geq 1\),5

\[ \Omega \sim \textsf{LKJCorr}(\eta). \]

The LKJ correlation distribution is defined by \[ \textsf{LKJCorr}\left(\Sigma \mid \eta\right) \propto \operatorname{det}\left(\Sigma\right)^{\eta - 1}. \]

The basic behavior of the LKJ correlation distribution is similar to that of a beta distribution. For \(\eta = 1\), the result is a uniform distribution. Despite being the identity over correlation matrices, the marginal distribution over the entries in that matrix (i.e., the correlations) is not uniform between -1 and 1. Rather, it concentrates around zero as the dimensionality increases due to the complex constraints.

For \(\eta > 1\), the density increasingly concentrates mass around the unit matrix, i.e., favoring less correlation. For \(\eta < 1\), it increasingly concentrates mass in the other direction, i.e., favoring more correlation.

The LKJ prior may thus be used to control the expected amount of correlation among the parameters \(\beta_j\). For a discussion of decomposing a covariance prior into a prior on correlation matrices and an independent prior on scales, see Barnard, McCulloch, and Meng (2000).

Group-level predictors for prior mean

To complete Gelman and Hill’s model, suppose each group \(j \in \{ 1, \dotsc, J \}\) is supplied with an \(L\)-dimensional row-vector of group-level predictors \(u_j\). The prior mean for the \(\beta_j\) can then itself be modeled as a regression, using an \(L\)-dimensional coefficient vector \(\gamma\). The prior for the group-level coefficients then becomes \[ \beta_j \sim \textsf{multivariate normal}(u_j \, \gamma, \Sigma) \]

The group-level coefficients \(\gamma\) may themselves be given independent weakly informative priors, such as \[ \gamma_l \sim \textsf{normal}(0,5). \] As usual, information about the group-level means should be incorporated into this prior.

Coding the model in Stan

The Stan code for the full hierarchical model with multivariate priors on the group-level coefficients and group-level prior means follows its definition.

data {
  int<lower=0> N;              // num individuals
  int<lower=1> K;              // num ind predictors
  int<lower=1> J;              // num groups
  int<lower=1> L;              // num group predictors
  array[N] int<lower=1, upper=J> jj;  // group for individual
  matrix[N, K] x;              // individual predictors
  array[J] row_vector[L] u;    // group predictors
  vector[N] y;                 // outcomes
}
parameters {
  corr_matrix[K] Omega;        // prior correlation
  vector<lower=0>[K] tau;      // prior scale
  matrix[L, K] gamma;          // group coeffs
  array[J] vector[K] beta;     // indiv coeffs by group
  real<lower=0> sigma;         // prediction error scale
}
model {
  tau ~ cauchy(0, 2.5);
  Omega ~ lkj_corr(2);
  to_vector(gamma) ~ normal(0, 5);
  {
    array[J] row_vector[K] u_gamma;
    for (j in 1:J) {
      u_gamma[j] = u[j] * gamma;
    }
    beta ~ multi_normal(u_gamma, quad_form_diag(Omega, tau));
  }
  for (n in 1:N) {
    y[n] ~ normal(x[n] * beta[jj[n]], sigma);
  }
}

The hyperprior covariance matrix is defined implicitly through the quadratic form in the code because the correlation matrix Omega and scale vector tau are more natural to inspect in the output; to output Sigma, define it as a transformed parameter. The function quad_form_diag is defined so that quad_form_diag(Sigma, tau) is equivalent to diag_matrix(tau) * Sigma * diag_matrix(tau), where diag_matrix(tau) returns the matrix with tau on the diagonal and zeroes off diagonal; the version using quad_form_diag should be faster. For details on these and other matrix arithmetic operators and functions, see the function reference manual.

Optimization through vectorization

The code in the Stan program above can be sped up dramatically by replacing:

for (n in 1:N) {
  y[n] ~ normal(x[n] * beta[jj[n]], sigma);
}

with the vectorized form:

{
  vector[N] x_beta_jj;
  for (n in 1:N) {
    x_beta_jj[n] = x[n] * beta[jj[n]];
  }
  y ~ normal(x_beta_jj, sigma);
}

The outer brackets create a local scope in which to define the variable x_beta_jj, which is then filled in a loop and used to define a vectorized sampling statement. The reason this is such a big win is that it allows us to take the log of sigma only once and it greatly reduces the size of the resulting expression graph by packing all of the work into a single density function.

Although it is tempting to redeclare beta and include a revised model block sampling statement,

parameters {
  matrix[J, K] beta;
// ...
}
model {
  y ~ normal(rows_dot_product(x, beta[jj]), sigma);
  // ...
}

this fails because it breaks the vectorization of sampling for beta,6

beta ~ multi_normal(...);

which requires beta to be an array of vectors. Both vectorizations are important, so the best solution is to just use the loop above, because rows_dot_product cannot do much optimization in and of itself because there are no shared computations.

The code in the Stan program above also builds up an array of vectors for the outcomes and for the multivariate normal, which provides a major speedup by reducing the number of linear systems that need to be solved and differentiated.

{
  matrix[K, K] Sigma_beta;
  Sigma_beta = quad_form_diag(Omega, tau);
  for (j in 1:J) {
    beta[j] ~ multi_normal((u[j] * gamma)', Sigma_beta);
  }
}

In this example, the covariance matrix Sigma_beta is defined as a local variable so as not to have to repeat the quadratic form computation \(J\) times. This vectorization can be combined with the Cholesky-factor optimization in the next section.

Optimization through Cholesky factorization

The multivariate normal density and LKJ prior on correlation matrices both require their matrix parameters to be factored. Vectorizing, as in the previous section, ensures this is only done once for each density. An even better solution, both in terms of efficiency and numerical stability, is to parameterize the model directly in terms of Cholesky factors of correlation matrices using the multivariate version of the non-centered parameterization. For the model in the previous section, the program fragment to replace the full matrix prior with an equivalent Cholesky factorized prior is as follows.

data {
  matrix[L, J] u;              // group predictors transposed
  // ...
}
parameters {
  matrix[K, J] z;
  cholesky_factor_corr[K] L_Omega;
  matrix[K, L] gamma;
  // ...
}
transformed parameters {
  matrix[K, J] beta;
  beta = gamma * u + diag_pre_multiply(tau, L_Omega) * z;
}
model {
  to_vector(z) ~ std_normal();
  L_Omega ~ lkj_corr_cholesky(2);
  // ...
}

The data variable u was originally an array of vectors, which is efficient for access; here it is redeclared as a matrix in order to use it in matrix arithmetic. Moreover, it is transposed, along with gamma and beta, to minimize the number of transposition operations. The new parameter L_Omega is the Cholesky factor of the original correlation matrix Omega, so that

Omega = L_Omega * L_Omega'

The prior scale vector tau is unchanged, and furthermore, pre-multiplying the Cholesky factor by the scale produces the Cholesky factor of the final covariance matrix,

Sigma_beta
  = quad_form_diag(Omega, tau)
  = diag_pre_multiply(tau, L_Omega) * diag_pre_multiply(tau, L_Omega)'

where the diagonal pre-multiply compound operation is defined by

diag_pre_multiply(a, b) = diag_matrix(a) * b

The new variable z is declared as a matrix, the entries of which are given independent standard normal priors; the to_vector operation turns the matrix into a vector so that it can be used as a vectorized argument to the univariate normal density. This results in every column of z being a \(K\)-variate normal random vector with the identity as covariance matrix. Therefore, multiplying z by the Cholesky factor of the covariance matrix and adding the mean (u * gamma)' produces a beta distributed as in the original model, where the variance is, letting \(L = \mathrm{diag}(\tau)\,\Omega_L\),

\[ \begin{aligned} \mathbb{V}[\beta] &= \mathbb{E}\big((L \, z)(L \, z)^\top) \\ &= \mathbb{E}\big((L \, z \, z^\top \, L^\top) \\ &= L \, \mathbb{E}(z \, z^\top) \, L^\top \\ &= L \, L^\top =(\mathrm{diag}(\tau)\,\Omega_L)\,(\mathrm{diag}(\tau)\,\Omega_L)^\top \\ &= \mathrm{diag}(\tau)\,\Omega\,\mathrm{diag}(\tau) \\ &= \Sigma. \end{aligned} \] Where we have used the linearity of expectations (line 2 to 3), the definition of \(\Omega = \Omega_L \, \Omega_L^\top\), and the fact that \(\mathbb{E}(z \, z^\top) = I\) since \(z \sim \mathcal{N}(0, I)\).

Omitting the remaining data declarations, which are the same as before with the exception of u, the optimized model is as follows.

parameters {
  matrix[K, J] z;
  cholesky_factor_corr[K] L_Omega;
  vector<lower=0, upper=pi() / 2>[K] tau_unif;  // prior scale
  matrix[K, L] gamma;                        // group coeffs
  real<lower=0> sigma;                       // prediction error scale
}
transformed parameters {
  vector<lower=0>[K] tau = 2.5 * tan(tau_unif);
  matrix[K, J] beta = gamma * u + diag_pre_multiply(tau, L_Omega) * z;
}
model {
  vector[N] mu;
  for(n in 1:N) {
    mu[n] = x[n, ] * beta[, jj[n]];
  }
  to_vector(z) ~ std_normal();
  L_Omega ~ lkj_corr_cholesky(2);
  to_vector(gamma) ~ normal(0, 5);
  y ~ normal(mu, sigma);
}

This model also reparameterizes the prior scale tau to avoid potential problems with the heavy tails of the Cauchy distribution. The statement tau_unif ~ uniform(0, pi() / 2) can be omitted from the model block because Stan increments the log posterior for parameters with uniform priors without it.

Prediction, forecasting, and backcasting

Stan models can be used for “predicting” the values of arbitrary model unknowns. When predictions are about the future, they’re called “forecasts;” when they are predictions about the past, as in climate reconstruction or cosmology, they are sometimes called “backcasts” (or “aftcasts” or “hindcasts” or “antecasts,” depending on the author’s feelings about the opposite of “fore”).

Programming predictions

As a simple example, the following linear regression provides the same setup for estimating the coefficients beta as in our very first example, using y for the N observations and x for the N predictor vectors. The model parameters and model for observations are exactly the same as before.

To make predictions, we need to be given the number of predictions, N_new, and their predictor matrix, x_new. The predictions themselves are modeled as a parameter y_new. The model statement for the predictions is exactly the same as for the observations, with the new outcome vector y_new and prediction matrix x_new.

data {
  int<lower=1> K;
  int<lower=0> N;
  matrix[N, K] x;
  vector[N] y;

  int<lower=0> N_new;
  matrix[N_new, K] x_new;
}
parameters {
  vector[K] beta;
  real<lower=0> sigma;

  vector[N_new] y_new;                  // predictions
}
model {
  y ~ normal(x * beta, sigma);          // observed model

  y_new ~ normal(x_new * beta, sigma);  // prediction model
}

Predictions as generated quantities

Where possible, the most efficient way to generate predictions is to use the generated quantities block. This provides proper Monte Carlo (not Markov chain Monte Carlo) inference, which can have a much higher effective sample size per iteration.

// ...data as above...

parameters {
  vector[K] beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(x * beta, sigma);
}
generated quantities {
  vector[N_new] y_new;
  for (n in 1:N_new) {
    y_new[n] = normal_rng(x_new[n] * beta, sigma);
  }
}

Now the data are just as before, but the parameter y_new is now declared as a generated quantity, and the prediction model is removed from the model and replaced by a pseudo-random draw from a normal distribution.

Overflow in generated quantities

It is possible for values to overflow or underflow in generated quantities. The problem is that if the result is NaN, then any constraints placed on the variables will be violated. It is possible to check a value assigned by an RNG and reject it if it overflows, but this is both inefficient and leads to biased posterior estimates. Instead, the conditions causing overflow, such as trying to generate a negative binomial random variate with a mean of \(2^{31}\), must be intercepted and dealt with. This is typically done by reparameterizing or reimplementing the random number generator using real values rather than integers, which are upper-bounded by \(2^{31} - 1\) in Stan.

Multivariate outcomes

Most regressions are set up to model univariate observations (be they scalar, boolean, categorical, ordinal, or count). Even multinomial regressions are just repeated categorical regressions. In contrast, this section discusses regression when each observed value is multivariate. To relate multiple outcomes in a regression setting, their error terms are provided with covariance structure.

This section considers two cases, seemingly unrelated regressions for continuous multivariate quantities and multivariate probit regression for boolean multivariate quantities.

Seemingly unrelated regressions

The first model considered is the “seemingly unrelated” regressions (SUR) of econometrics where several linear regressions share predictors and use a covariance error structure rather than independent errors (Zellner 1962; Greene 2011).

The model is easy to write down as a regression, \[\begin{align*} y_n &= x_n \, \beta + \epsilon_n \\ \epsilon_n &\sim \textsf{multivariate normal}(0, \Sigma) \end{align*}\]

where \(x_n\) is a \(J\)-row-vector of predictors (\(x\) is an \((N \times J)\) matrix), \(y_n\) is a \(K\)-vector of observations, \(\beta\) is a \((K \times J)\) matrix of regression coefficients (vector \(\beta_k\) holds coefficients for outcome \(k\)), and \(\Sigma\) is covariance matrix governing the error. As usual, the intercept can be rolled into \(x\) as a column of ones.

The basic Stan code is straightforward (though see below for more optimized code for use with LKJ priors on correlation).

data {
  int<lower=1> K;
  int<lower=1> J;
  int<lower=0> N;
  array[N] vector[J] x;
  array[N] vector[K] y;
}
parameters {
  matrix[K, J] beta;
  cov_matrix[K] Sigma;
}
model {
  array[N] vector[K] mu;
  for (n in 1:N) {
    mu[n] = beta * x[n];
  }
  y ~ multi_normal(mu, Sigma);
}

For efficiency, the multivariate normal is vectorized by precomputing the array of mean vectors and sharing the same covariance matrix.

Following the advice in the multivariate hierarchical priors section, we will place a weakly informative normal prior on the regression coefficients, an LKJ prior on the correlations and a half-Cauchy prior on standard deviations. The covariance structure is parameterized in terms of Cholesky factors for efficiency and arithmetic stability.

// ...
parameters {
  matrix[K, J] beta;
  cholesky_factor_corr[K] L_Omega;
  vector<lower=0>[K] L_sigma;
}
model {
  array[N] vector[K] mu;
  matrix[K, K] L_Sigma;

  for (n in 1:N) {
    mu[n] = beta * x[n];

  }

  L_Sigma = diag_pre_multiply(L_sigma, L_Omega);

  to_vector(beta) ~ normal(0, 5);
  L_Omega ~ lkj_corr_cholesky(4);
  L_sigma ~ cauchy(0, 2.5);

  y ~ multi_normal_cholesky(mu, L_Sigma);
}

The Cholesky factor of the covariance matrix is then reconstructed as a local variable and used in the model by scaling the Cholesky factor of the correlation matrices. The regression coefficients get a prior all at once by converting the matrix beta to a vector.

If required, the full correlation or covariance matrices may be reconstructed from their Cholesky factors in the generated quantities block.

Multivariate probit regression

The multivariate probit model generates sequences of boolean variables by applying a step function to the output of a seemingly unrelated regression.

The observations \(y_n\) are \(D\)-vectors of boolean values (coded 0 for false, 1 for true). The values for the observations \(y_n\) are based on latent values \(z_n\) drawn from a seemingly unrelated regression model (see the previous section), \[\begin{align*} z_n &= x_n \, \beta + \epsilon_n \\ \epsilon_n &\sim \textsf{multivariate normal}(0, \Sigma) \end{align*}\]

These are then put through the step function to produce a \(K\)-vector \(z_n\) of boolean values with elements defined by \[ y_{n, k} = \operatorname{I}\left(z_{n, k} > 0\right), \] where \(\operatorname{I}()\) is the indicator function taking the value 1 if its argument is true and 0 otherwise.

Unlike in the seemingly unrelated regressions case, here the covariance matrix \(\Sigma\) has unit standard deviations (i.e., it is a correlation matrix). As with ordinary probit and logistic regressions, letting the scale vary causes the model (which is defined only by a cutpoint at 0, not a scale) to be unidentified (see Greene (2011)).

Multivariate probit regression can be coded in Stan using the trick introduced by Albert and Chib (1993), where the underlying continuous value vectors \(y_n\) are coded as truncated parameters. The key to coding the model in Stan is declaring the latent vector \(z\) in two parts, based on whether the corresponding value of \(y\) is 0 or 1. Otherwise, the model is identical to the seemingly unrelated regression model in the previous section.

First, we introduce a sum function for two-dimensional arrays of integers; this is going to help us calculate how many total 1 values there are in \(y\).

functions {
  int sum2d(array[,] int a) {
    int s = 0;
    for (i in 1:size(a)) {
      s += sum(a[i]);
    }
    return s;
  }
}

The function is trivial, but it’s not a built-in for Stan and it’s easier to understand the rest of the model if it’s pulled into its own function so as not to create a distraction.

The data declaration block is much like for the seemingly unrelated regressions, but the observations y are now integers constrained to be 0 or 1.

data {
  int<lower=1> K;
  int<lower=1> D;
  int<lower=0> N;
  array[N, D] int<lower=0, upper=1> y;
  array[N] vector[K] x;
}

After declaring the data, there is a rather involved transformed data block whose sole purpose is to sort the data array y into positive and negative components, keeping track of indexes so that z can be easily reassembled in the transformed parameters block.

transformed data {
  int<lower=0> N_pos;
  array[sum2d(y)] int<lower=1, upper=N> n_pos;
  array[size(n_pos)] int<lower=1, upper=D> d_pos;
  int<lower=0> N_neg;
  array[(N * D) - size(n_pos)] int<lower=1, upper=N> n_neg;
  array[size(n_neg)] int<lower=1, upper=D> d_neg;

  N_pos = size(n_pos);
  N_neg = size(n_neg);
  {
    int i;
    int j;
    i = 1;
    j = 1;
    for (n in 1:N) {
      for (d in 1:D) {
        if (y[n, d] == 1) {
          n_pos[i] = n;
          d_pos[i] = d;
          i += 1;
        } else {
          n_neg[j] = n;
          d_neg[j] = d;
          j += 1;
        }
      }
    }
  }
}

The variables N_pos and N_neg are set to the number of true (1) and number of false (0) observations in y. The loop then fills in the sequence of indexes for the positive and negative values in four arrays.

The parameters are declared as follows.

parameters {
  matrix[D, K] beta;
  cholesky_factor_corr[D] L_Omega;
  vector<lower=0>[N_pos] z_pos;
  vector<upper=0>[N_neg] z_neg;
}

These include the regression coefficients beta and the Cholesky factor of the correlation matrix, L_Omega. This time there is no scaling because the covariance matrix has unit scale (i.e., it is a correlation matrix; see above).

The critical part of the parameter declaration is that the latent real value \(z\) is broken into positive-constrained and negative-constrained components, whose size was conveniently calculated in the transformed data block. The transformed data block’s real work was to allow the transformed parameter block to reconstruct \(z\).

transformed parameters {
  array[N] vector[D] z;
  for (n in 1:N_pos) {
    z[n_pos[n], d_pos[n]] = z_pos[n];
  }
  for (n in 1:N_neg) {
    z[n_neg[n], d_neg[n]] = z_neg[n];
  }
}

At this point, the model is simple, pretty much recreating the seemingly unrelated regression.

model {
  L_Omega ~ lkj_corr_cholesky(4);
  to_vector(beta) ~ normal(0, 5);
  {
    array[N] vector[D] beta_x;
    for (n in 1:N) {
      beta_x[n] = beta * x[n];
    }
    z ~ multi_normal_cholesky(beta_x, L_Omega);
  }
}

This simple form of model is made possible by the Albert and Chib-style constraints on z.

Finally, the correlation matrix itself can be put back together in the generated quantities block if desired.

generated quantities {
  corr_matrix[D] Omega;
  Omega = multiply_lower_tri_self_transpose(L_Omega);
}

The same could be done for the seemingly unrelated regressions in the previous section.

Applications of pseudorandom number generation

The main application of pseudorandom number generator (PRNGs) is for posterior inference, including prediction and posterior predictive checks. They can also be used for pure data simulation, which is like a posterior predictive check with no conditioning. See the function reference manual for a complete description of the syntax and usage of pseudorandom number generators.

Prediction

Consider predicting unobserved outcomes using linear regression. Given predictors \(x_1, \dotsc, x_N\) and observed outcomes \(y_1, \dotsc, y_N\), and assuming a standard linear regression with intercept \(\alpha\), slope \(\beta\), and error scale \(\sigma\), along with improper uniform priors, the posterior over the parameters given \(x\) and \(y\) is \[ p\left(\alpha, \beta, \sigma \mid x, y \right) \propto \prod_{n=1}^N \textsf{normal}\left(y_n \mid \alpha + \beta x_n, \sigma\right). \]

For this model, the posterior predictive inference for a new outcome \(\tilde{y}_m\) given a predictor \(\tilde{x}_m\), conditioned on the observed data \(x\) and \(y\), is \[ p\left(\tilde{y}_n \mid \tilde{x}_n, x, y\right) = \int_{(\alpha,\beta,\sigma)} \textsf{normal}\left(\tilde{y}_n \mid \alpha + \beta \tilde{x}_n, \sigma\right) \times p\left(\alpha, \beta, \sigma \mid x, y\right) \,\textrm{d}(\alpha,\beta,\sigma). \]

To code the posterior predictive inference in Stan, a standard linear regression is combined with a random number in the generated quantities block.

data {
  int<lower=0> N;
  vector[N] y;
  vector[N] x;
  int<lower=0> N_tilde;
  vector[N_tilde] x_tilde;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(alpha + beta * x, sigma);
}
generated quantities {
  vector[N_tilde] y_tilde;
  for (n in 1:N_tilde) {
    y_tilde[n] = normal_rng(alpha + beta * x_tilde[n], sigma);
  }
}

Given observed predictors \(x\) and outcomes \(y\), y_tilde will be drawn according to \(p\left(\tilde{y} \mid \tilde{x}, y, x\right)\). This means that, for example, the posterior mean for y_tilde is the estimate of the outcome that minimizes expected square error (conditioned on the data and model).

Posterior predictive checks

A good way to investigate the fit of a model to the data, a critical step in Bayesian data analysis, is to generate simulated data according to the parameters of the model. This is carried out with exactly the same procedure as before, only the observed data predictors \(x\) are used in place of new predictors \(\tilde{x}\) for unobserved outcomes. If the model fits the data well, the predictions for \(\tilde{y}\) based on \(x\) should match the observed data \(y\).

To code posterior predictive checks in Stan requires only a slight modification of the prediction code to use \(x\) and \(N\) in place of \(\tilde{x}\) and \(\tilde{N}\),

generated quantities {
  vector[N] y_tilde;
  for (n in 1:N) {
    y_tilde[n] = normal_rng(alpha + beta * x[n], sigma);
  }
}

Gelman et al. (2013) recommend choosing several posterior draws \(\tilde{y}^{(1)}, \dotsc, \tilde{y}^{(M)}\) and plotting each of them alongside the data \(y\) that was actually observed. If the model fits well, the simulated \(\tilde{y}\) will look like the actual data \(y\).

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References

Albert, J. H., and S. Chib. 1993. “Bayesian Analysis of Binary and Polychotomous Response Data.” Journal of the American Statistical Association 88: 669–79.
Barnard, John, Robert McCulloch, and Xiao-Li Meng. 2000. “Modeling Covariance Matrices in Terms of Standard Deviations and Correlations, with Application to Shrinkage.” Statistica Sinica, 1281–311.
Chung, Yeojin, Sophia Rabe-Hesketh, Vincent Dorie, Andrew Gelman, and Jingchen Liu. 2013. “A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models.” Psychometrika 78 (4): 685–709.
Curtis, S. McKay. 2010. “BUGS Code for Item Response Theory.” Journal of Statistical Software 36 (1): 1–34.
Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. Third Edition. London: Chapman & Hall / CRC Press.
Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel-Hierarchical Models. Cambridge, United Kingdom: Cambridge University Press.
Greene, William H. 2011. Econometric Analysis. 7th ed. Prentice-Hall.
Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. “Generating Random Correlation Matrices Based on Vines and Extended Onion Method.” Journal of Multivariate Analysis 100: 1989–2001.
Zellner, Arnold. 1962. “An Efficient Method of Estimating Seemingly Unrelated Regression Equations and Tests for Aggregation Bias.” Journal of the American Statistical Association 57: 348–68.

Footnotes

  1. Unlike in Python and R, which are interpreted, Stan is translated to C++ and compiled, so loops and assignment statements are fast. Vectorized code is faster in Stan because (a) the expression tree used to compute derivatives can be simplified, leading to fewer virtual function calls, and (b) computations that would be repeated in the looping version, such as log(sigma) in the above model, will be computed once and reused.↩︎

  2. The Phi_approx function is a rescaled version of the inverse logit function, so while the scale is roughly the same \(\Phi\), the tails do not match.↩︎

  3. The Bernoulli-logit distribution builds in the log link function, taking \[\texttt{bernoulli}\mathtt{\_}\texttt{logit}\left(y \mid \alpha\right) = \texttt{bernoulli}\left(y \mid \operatorname{logit}^{-1}(\alpha)\right).\]↩︎

  4. Gelman and Hill (2007) treat the \(\delta\) term equivalently as the location parameter in the distribution of student abilities.↩︎

  5. The prior is named for Lewandowski, Kurowicka, and Joe, as it was derived by inverting the random correlation matrix generation strategy of Lewandowski, Kurowicka, and Joe (2009).↩︎

  6. Thanks to Mike Lawrence for pointing this out in the GitHub issue for the manual.↩︎