◆ laplace_marginal_density_est()

template<typename LLFun , typename LLTupleArgs , typename CovarFun , typename ThetaVec , typename CovarArgs , require_t< is_all_arithmetic_scalar< ThetaVec, CovarArgs > > * = nullptr, require_eigen_vector_t< ThetaVec > * = nullptr>

auto stan::math::internal::laplace_marginal_density_est	(	LLFun &&	ll_fun,
		LLTupleArgs &&	ll_args,
		ThetaVec &&	theta_0,
		CovarFun &&	covariance_function,
		CovarArgs &&	covar_args,
		const laplace_options &	options,
		std::ostream *	msgs
	)

inline

For a latent Gaussian model with hyperparameters phi and latent variables theta, and observations y, this function computes an approximation of the log marginal density, p(y | phi).

This is done by marginalizing out theta, using a Laplace approxmation. The latter is obtained by finding the mode, via Newton's method, and computing the Hessian of the likelihood.

The convergence criterion for the Newton is a small change in log marginal density. The user controls the tolerance (i.e. threshold under which change is deemed small enough) and maximum number of steps.

A description of this algorithm can be found in:

(2023) Margossian, "General Adjoint-Differentiated Laplace approximation", https://arxiv.org/pdf/2306.14976. Additional references include:
(2020) Margossian et al, "HMC using an adjoint-differentiated Laplace...", NeurIPS, https://arxiv.org/abs/2004.12550.
(2006) Rasmussen and Williams, "Gaussian Processes for Machine Learning", second edition, MIT Press, algorithm 3.1.

Variables needed for the gradient or generating quantities are stored by reference.

Template Parameters

LLFun	Type with a valid `operator(ThetaVec, InnerLLTupleArgs)` where `InnerLLTupleArgs` are the elements of `LLTupleArgs`
LLTupleArgs	A tuple whose elements follow the types required for `LLFun`
ThetaVec	A type inheriting from `Eigen::EigenBase` with dynamic sized rows and 1 column.
CovarFun	A functor with an `operator()(CovarArgsElements..., {TrainTupleElements...\| PredTupleElements...})` method. The `operator()` method should accept as arguments the inner elements of `CovarArgs`. The return type of the `operator()` method should be a type inheriting from `Eigen::EigenBase` with dynamic sized rows and columns.
CovarArgs	A tuple of types to passed as the first arguments of `CovarFun::operator()`

Parameters

[in]	ll_fun	A log likelihood functor
[in]	ll_args	Tuple containing parameters for `LLFun`
[in]	theta_0	the initial guess for the Laplace approximation.
[in]	covariance_function	a function which returns the prior covariance.
[in]	covar_args	arguments for the covariance function.
[in]	options	A set of options for tuning the solver
[in,out]	msgs	stream for messages from likelihood and covariance

Returns

A struct containing

lmd the log marginal density, p(y | phi)
covariance the evaluated covariance function for the latent gaussian variable
theta a vector to store the mode
W_r A sparse matrix containing the square root of the negative hessian, if solver 1 or 2 are used.
L cholesky decomposition of stabilized inverse covariance
a element in the Newton step
l_grad the log density of the likelihood, evaluated at the mode

Definition at line 454 of file laplace_marginal_density.hpp.