Make basic INLA interface and simple marginalisation routine

pymc_extras/inference/fit.py

@@ -36,7 +36,17 @@ def fit(method: str, **kwargs) -> az.InferenceData:
 
         return fit_pathfinder(**kwargs)
 
-    if method == "laplace":
+    elif method == "laplace":
         from pymc_extras.inference.laplace import fit_laplace
 
         return fit_laplace(**kwargs)
+
+    elif method == "INLA":
+        from pymc_extras.inference.laplace import fit_INLA
+
+        return fit_INLA(**kwargs)
+
+    else:
+        raise ValueError(
+            f"method '{method}' not supported. Use one of 'pathfinder', 'laplace' or 'INLA'."
+        )

pymc_extras/inference/inla.py

@@ -0,0 +1,164 @@
+import arviz as az
+import numpy as np
+import pymc as pm
+import pytensor
+import pytensor.tensor as pt
+
+from better_optimize.constants import minimize_method
+from numpy.typing import ArrayLike
+from pytensor.tensor import TensorVariable
+from pytensor.tensor.optimize import minimize
+
+
+def get_conditional_gaussian_approximation(
+    x: TensorVariable,
+    Q: TensorVariable | ArrayLike,
+    mu: TensorVariable | ArrayLike,
+    model: pm.Model | None = None,
+    method: minimize_method = "BFGS",
+    use_jac: bool = True,
+    use_hess: bool = False,
+    optimizer_kwargs: dict | None = None,
+) -> list[TensorVariable]:
+    """
+    Returns an estimate the a posteriori probability of a latent Gaussian field x and its mode x0 using the Laplace approximation.
+
+    That is:
+    y | x, sigma ~ N(Ax, sigma^2 W)
+    x | params ~ N(mu, Q(params)^-1)
+
+    We seek to estimate p(x | y, params) with a Gaussian:
+
+    log(p(x | y, params)) = log(p(y | x, params)) + log(p(x | params)) + const
+
+    Let f(x) = log(p(y | x, params)). From the definition of our model above, we have log(p(x | params)) = -0.5*(x - mu).T Q (x - mu) + 0.5*logdet(Q).
+
+    This gives log(p(x | y, params)) = f(x) - 0.5*(x - mu).T Q (x - mu) + 0.5*logdet(Q). We will estimate this using the Laplace approximation by Taylor expanding f(x) about the mode.
+
+    Thus:
+
+    1. Maximize log(p(x | y, params)) = f(x) - 0.5*(x - mu).T Q (x - mu) wrt x (note that logdet(Q) does not depend on x) to find the mode x0.
+
+    2. Use the Laplace approximation expanded about the mode: p(x | y, params) ~= N(mu=x0, tau=Q - f''(x0)).
+
+    Parameters
+    ----------
+    x: TensorVariable
+        The parameter with which to maximize wrt (that is, find the mode in x). In INLA, this is the latent Gaussian field x~N(mu,Q^-1).
+    Q: TensorVariable | ArrayLike
+        The precision matrix of the latent field x.
+    mu: TensorVariable | ArrayLike
+        The mean of the latent field x.
+    model: Model
+        PyMC model to use.
+    method: minimize_method
+        Which minimization algorithm to use.
+    use_jac: bool
+        If true, the minimizer will compute the gradient of log(p(x | y, params)).
+    use_hess: bool
+        If true, the minimizer will compute the Hessian log(p(x | y, params)).
+    optimizer_kwargs: dict
+        Kwargs to pass to scipy.optimize.minimize.
+
+    Returns
+    -------
+    x0, p(x | y, params): list[TensorVariable]
+        Mode and Laplace approximation for posterior.
+    """
+    model = pm.modelcontext(model)
+
+    # f = log(p(y | x, params))
+    f_x = model.logp()
+
+    # log(p(x | y, params)) only including terms that depend on x for the minimization step (logdet(Q) ignored as it is a constant wrt x)
+    log_x_posterior = f_x - 0.5 * (x - mu).T @ Q @ (x - mu)
+
+    # Maximize log(p(x | y, params)) wrt x to find mode x0
+    x0, _ = minimize(
+        objective=-log_x_posterior,
+        x=x,
+        method=method,
+        jac=use_jac,
+        hess=use_hess,
+        optimizer_kwargs=optimizer_kwargs,
+    )
+
+    # require f''(x0) for Laplace approx
+    hess = pytensor.gradient.hessian(f_x, x)
+    hess = pytensor.graph.replace.graph_replace(hess, {x: x0})
+
+    # Could be made more efficient with adding diagonals only
+    tau = Q - hess
+
+    # Currently x is passed both as the query point for f(x, args) = logp(x | y, params) AND as an initial guess for x0. This may cause issues if the query point is
+    # far from the mode x0 or in a neighbourhood which results in poor convergence.
+    return x0, pm.MvNormal(f"{x.name}_laplace_approx", mu=x0, tau=tau)
+
+
+def get_log_marginal_likelihood(
+    x: TensorVariable,
+    Q: TensorVariable | ArrayLike,
+    mu: TensorVariable | ArrayLike,
+    model: pm.Model | None = None,
+    method: minimize_method = "BFGS",
+    use_jac: bool = True,
+    use_hess: bool = False,
+    optimizer_kwargs: dict | None = None,
+) -> TensorVariable:
+    model = pm.modelcontext(model)
+
+    x0, laplace_approx = get_conditional_gaussian_approximation(
+        x, Q, mu, model, method, use_jac, use_hess, optimizer_kwargs
+    )
+    log_laplace_approx = pm.logp(laplace_approx, model.rvs_to_values[x])
+
+    _, logdetQ = pt.nlinalg.slogdet(Q)
+    log_x_likelihood = (
+        -0.5 * (x - mu).T @ Q @ (x - mu) + 0.5 * logdetQ - 0.5 * x.shape[0] * np.log(2 * np.pi)
+    )
+
+    log_likelihood = (  # logp(y | params) =
+        model.logp()  # logp(y | x, params)
+        + log_x_likelihood  # * logp(x | params)
+        - log_laplace_approx  # / logp(x | y, params)
+    )
+
+    return log_likelihood
+
+
+def fit_INLA(
+    x: TensorVariable,
+    Q: TensorVariable | ArrayLike,
+    mu: TensorVariable | ArrayLike,
+    model: pm.Model | None = None,
+    method: minimize_method = "BFGS",
+    use_jac: bool = True,
+    use_hess: bool = False,
+    optimizer_kwargs: dict | None = None,
+) -> az.InferenceData:
+    model = pm.modelcontext(model)
+
+    # logp(y | params)
+    log_likelihood = get_log_marginal_likelihood(
+        x, Q, mu, model, method, use_jac, use_hess, optimizer_kwargs
+    )
+
+    # TODO How to obtain prior? It can parametrise Q, mu, y, etc. Not sure if we could extract from model.logp somehow. Otherwise simply specify as a user input
+    prior = None
+    params = None
+    log_prior = pm.logp(prior, model.rvs_to_values[params])
+
+    # logp(params | y) = logp(y | params) + logp(params) + const
+    log_posterior = log_likelihood + log_prior
+
+    # TODO log_marginal_x_likelihood is almost the same as log_likelihood, but need to do some sampling?
+    log_marginal_x_likelihood = None
+    log_marginal_x_posterior = log_marginal_x_likelihood + log_prior
+
+    # TODO can we sample over log likelihoods?
+    # Marginalize params
+    idata_params = log_posterior.sample()  # TODO something like NUTS, QMC, etc.?
+    idata_x = log_marginal_x_posterior.sample()
+
+    # Bundle up idatas somehow
+    return idata_params, idata_x

pymc_extras/inference/laplace.py

@@ -15,7 +15,6 @@
 
 import logging
 
-from collections.abc import Callable
 from functools import reduce
 from importlib.util import find_spec
 from itertools import product
@@ -30,7 +29,6 @@
 
 from arviz import dict_to_dataset
 from better_optimize.constants import minimize_method
-from numpy.typing import ArrayLike
 from pymc import DictToArrayBijection
 from pymc.backends.arviz import (
     coords_and_dims_for_inferencedata,
@@ -41,8 +39,6 @@
 from pymc.model.transform.conditioning import remove_value_transforms
 from pymc.model.transform.optimization import freeze_dims_and_data
 from pymc.util import get_default_varnames
-from pytensor.tensor import TensorVariable
-from pytensor.tensor.optimize import minimize
 from scipy import stats
 
 from pymc_extras.inference.find_map import (
@@ -56,102 +52,6 @@
 _log = logging.getLogger(__name__)
 
 
-def get_conditional_gaussian_approximation(
-    x: TensorVariable,
-    Q: TensorVariable | ArrayLike,
-    mu: TensorVariable | ArrayLike,
-    args: list[TensorVariable] | None = None,
-    model: pm.Model | None = None,
-    method: minimize_method = "BFGS",
-    use_jac: bool = True,
-    use_hess: bool = False,
-    optimizer_kwargs: dict | None = None,
-) -> Callable:
-    """
-    Returns a function to estimate the a posteriori log probability of a latent Gaussian field x and its mode x0 using the Laplace approximation.
-
-    That is:
-    y | x, sigma ~ N(Ax, sigma^2 W)
-    x | params ~ N(mu, Q(params)^-1)
-
-    We seek to estimate log(p(x | y, params)):
-
-    log(p(x | y, params)) = log(p(y | x, params)) + log(p(x | params)) + const
-
-    Let f(x) = log(p(y | x, params)). From the definition of our model above, we have log(p(x | params)) = -0.5*(x - mu).T Q (x - mu) + 0.5*logdet(Q).
-
-    This gives log(p(x | y, params)) = f(x) - 0.5*(x - mu).T Q (x - mu) + 0.5*logdet(Q). We will estimate this using the Laplace approximation by Taylor expanding f(x) about the mode.
-
-    Thus:
-
-    1. Maximize log(p(x | y, params)) = f(x) - 0.5*(x - mu).T Q (x - mu) wrt x (note that logdet(Q) does not depend on x) to find the mode x0.
-
-    2. Substitute x0 into the Laplace approximation expanded about the mode: log(p(x | y, params)) ~= -0.5*x.T (-f''(x0) + Q) x + x.T (Q.mu + f'(x0) - f''(x0).x0) + 0.5*logdet(Q).
-
-    Parameters
-    ----------
-    x: TensorVariable
-        The parameter with which to maximize wrt (that is, find the mode in x). In INLA, this is the latent field x~N(mu,Q^-1).
-    Q: TensorVariable | ArrayLike
-        The precision matrix of the latent field x.
-    mu: TensorVariable | ArrayLike
-        The mean of the latent field x.
-    args: list[TensorVariable]
-        Args to supply to the compiled function. That is, (x0, logp) = f(x, *args). If set to None, assumes the model RVs are args.
-    model: Model
-        PyMC model to use.
-    method: minimize_method
-        Which minimization algorithm to use.
-    use_jac: bool
-        If true, the minimizer will compute the gradient of log(p(x | y, params)).
-    use_hess: bool
-        If true, the minimizer will compute the Hessian log(p(x | y, params)).
-    optimizer_kwargs: dict
-        Kwargs to pass to scipy.optimize.minimize.
-
-    Returns
-    -------
-    f: Callable
-        A function which accepts a value of x and args and returns [x0, log(p(x | y, params))], where x0 is the mode. x is currently both the point at which to evaluate logp and the initial guess for the minimizer.
-    """
-    model = pm.modelcontext(model)
-
-    if args is None:
-        args = model.continuous_value_vars + model.discrete_value_vars
-
-    # f = log(p(y | x, params))
-    f_x = model.logp()
-    jac = pytensor.gradient.grad(f_x, x)
-    hess = pytensor.gradient.jacobian(jac.flatten(), x)
-
-    # log(p(x | y, params)) only including terms that depend on x for the minimization step (logdet(Q) ignored as it is a constant wrt x)
-    log_x_posterior = f_x - 0.5 * (x - mu).T @ Q @ (x - mu)
-
-    # Maximize log(p(x | y, params)) wrt x to find mode x0
-    x0, _ = minimize(
-        objective=-log_x_posterior,
-        x=x,
-        method=method,
-        jac=use_jac,
-        hess=use_hess,
-        optimizer_kwargs=optimizer_kwargs,
-    )
-
-    # require f'(x0) and f''(x0) for Laplace approx
-    jac = pytensor.graph.replace.graph_replace(jac, {x: x0})
-    hess = pytensor.graph.replace.graph_replace(hess, {x: x0})
-
-    # Full log(p(x | y, params)) using the Laplace approximation (up to a constant)
-    _, logdetQ = pt.nlinalg.slogdet(Q)
-    conditional_gaussian_approx = (
-        -0.5 * x.T @ (-hess + Q) @ x + x.T @ (Q @ mu + jac - hess @ x0) + 0.5 * logdetQ
-    )
-
-    # Currently x is passed both as the query point for f(x, args) = logp(x | y, params) AND as an initial guess for x0. This may cause issues if the query point is
-    # far from the mode x0 or in a neighbourhood which results in poor convergence.
-    return pytensor.function(args, [x0, conditional_gaussian_approx])
-
-
 def laplace_draws_to_inferencedata(
     posterior_draws: list[np.ndarray[float | int]], model: pm.Model | None = None
 ) -> az.InferenceData: