R: Laplace Approximation

LaplaceApproximation {LaplacesDemon} | R Documentation

## Laplace Approximation

### Description

The `LaplaceApproximation` function deterministically maximizes the logarithm of the unnormalized joint posterior density with one of several optimization algorithms. The goal of Laplace Approximation is to estimate the posterior mode and variance of each parameter. This function is useful for optimizing initial values and estimating a covariance matrix to be input into the `IterativeQuadrature`, `LaplacesDemon`, `PMC`, or `VariationalBayes` function, or sometimes for model estimation in its own right.

### Usage

    LaplaceApproximation(Model, parm, Data, Interval=1.0E-6,
         Iterations=100, Method="SPG", Samples=1000, CovEst="Hessian",
         sir=TRUE, Stop.Tolerance=1.0E-5, CPUs=1, Type="PSOCK")

### Arguments

`Model` | This required argument receives the model from a user-defined function. The user-defined function is where the model is specified. `LaplaceApproximation` passes two arguments to the model function, `parms` and `Data`. For more information, see the `LaplacesDemon` function and “LaplacesDemon Tutorial” vignette.
`parm` | This argument requires a vector of initial values equal in length to the number of parameters. `LaplaceApproximation` will attempt to optimize these initial values for the parameters, where the optimized values are the posterior modes, for later use with the `IterativeQuadrature`, `LaplacesDemon`, `PMC`, or the `VariationalBayes` function. The `GIV` function may be used to randomly generate initial values. Parameters must be continuous.
`Data` | This required argument accepts a list of data. The list of data must include `mon.names` which contains monitored variable names, and `parm.names` which contains parameter names. `LaplaceApproximation` must be able to determine the sample size of the data, and will look for a scalar sample size variable `n` or `N`. If not found, it will look for variable `y` or `Y`, and attempt to take its number of rows as sample size. `LaplaceApproximation` needs to determine sample size due to the asymptotic nature of this method. Sample size should be at least sqrt(J) with J exchangeable parameters.
`Interval` | This argument receives an interval for estimating approximate gradients. The logarithm of the unnormalized joint posterior density of the Bayesian model is evaluated at the current parameter value, and again at the current parameter value plus this interval.
`Iterations` | This argument accepts an integer that determines the number of iterations that `LaplaceApproximation` will attempt to maximize the logarithm of the unnormalized joint posterior density. `Iterations` defaults to 100. `LaplaceApproximation` will stop before this number of iterations if the tolerance is less than or equal to the `Stop.Tolerance` criterion. The required amount of computer memory increases with `Iterations`. If computer memory is exceeded, then all will be lost.
`Method` | This optional argument accepts a quoted string that specifies the method used for Laplace Approximation. The default method is `Method="SPG"`. Options include `"AGA"` for adaptive gradient ascent, `"BFGS"` for the Broyden-Fletcher-Goldfarb-Shanno algorithm, `"BHHH"` for the algorithm of Berndt et al., `"CG"` for conjugate gradient, `"DFP"` for the Davidon-Fletcher-Powell algorithm, `"HAR"` for adaptive hit-and-run, `"HJ"` for Hooke-Jeeves, `"LBFGS"` for limited-memory BFGS, `"LM"` for Levenberg-Marquardt, `"NM"` for Nelder-Mead, `"NR"` for Newton-Raphson, `"PSO"` for Particle Swarm Optimization, `"Rprop"` for resilient backpropagation, `"SGD"` for Stochastic Gradient Descent, `"SOMA"` for the Self-Organizing Migration Algorithm, `"SPG"` for Spectral Projected Gradient, `"SR1"` for Symmetric Rank-One, and `"TR"` for Trust Region.
`Samples` | This argument indicates the number of posterior samples to be taken with sampling importance resampling via the `SIR` function, which occurs only when `sir=TRUE`. Note that the number of samples should increase with the number and intercorrelations of the parameters.
`CovEst` | This argument accepts a quoted string that indicates how the covariance matrix is estimated after the model finishes. This covariance matrix is used to obtain the standard deviation of each parameter, and may also be used for posterior sampling via Sampling Importance Resampling (SIR) (see the `sir` argument below), if converged. By default, the covariance matrix is approximated as the negative inverse of the `"Hessian"` matrix of second derivatives, estimated with Richardson extrapolation. Alternatives include `CovEst="Identity"`, `CovEst="OPG"`, or `CovEst="Sandwich"`. When `CovEst="Identity"`, the covariance matrix is not estimated, and is merely assigned an identity matrix. When `LaplaceApproximation` is performed internally by `LaplacesDemon`, an identity matrix is returned and scaled. When `CovEst="OPG"`, the covariance matrix is approximated with the inverse of the sum of the outer products of the gradient, which requires `X`, and either `y` or `Y` in the list of data. For OPG, a partial derivative is taken for each row in `X`, and each element in `y` or row in `Y`. Therefore, this requires N + NJ model evaluations for a data set with N records and J variables. The OPG method is an asymptotic approximation of the Hessian, and usually requires fewer calculations with a small data set, or more with large data sets. Both methods require a matrix inversion, which becomes costly as dimension grows. The Richardson-based Hessian method is more accurate, but requires more calculation in large dimensions. An alternative approach to obtaining covariance is to use the `BayesianBootstrap` on the data, or sample the posterior with iterative quadrature (`IterativeQuadrature`), MCMC (`LaplacesDemon`), or `VariationalBayes`.
`sir` | This logical argument indicates whether or not Sampling Importance Resampling (SIR) is conducted via the `SIR` function to draw independent posterior samples. This argument defaults to `TRUE`. Even when `TRUE`, posterior samples are drawn only when `LaplaceApproximation` has converged. Posterior samples are required for many other functions, including `plot.laplace` and `predict.laplace`. The only time that it is advantageous for `sir=FALSE` is when `LaplaceApproximation` is used to help the initial values for `IterativeQuadrature`, `LaplacesDemon`, `PMC`, or `VariationalBayes`, and it is unnecessary for time to be spent on sampling. Less time can be spent on sampling by increasing `CPUs`, which parallelizes the sampling.
`Stop.Tolerance` | This argument accepts any positive number and defaults to 1.0E-5. Tolerance is calculated each iteration, and the criteria varies by algorithm. The algorithm is considered to have converged to the user-specified `Stop.Tolerance` when the tolerance is less than or equal to the value of `Stop.Tolerance`, and the algorithm terminates at the end of the current iteration. Often, multiple criteria are used, in which case the maximum of all criteria becomes the tolerance. For example, when partial derivatives are taken, it is commonly required that the Euclidean norm of the partial derivatives is a criterion, and another common criterion is the Euclidean norm of the differences between the current and previous parameter values. Several algorithms have other, specific tolerances.
`CPUs` | This argument accepts an integer that specifies the number of central processing units (CPUs) of the multicore computer or computer cluster. This argument defaults to `CPUs=1`, in which parallel processing does not occur. Parallelization occurs only for sampling with `SIR` when `sir=TRUE`.
`Type` | This argument specifies the type of parallel processing to perform, accepting either `Type="PSOCK"` or `Type="MPI"`.

### Details

The Laplace Approximation or Laplace Method is a family of asymptotic techniques used to approximate integrals. Laplace's method accurately approximates unimodal posterior moments and marginal posterior distributions in many cases. Since it is not applicable in all cases, it is recommended here that Laplace Approximation is used cautiously in its own right, or preferably, it is used before MCMC.

After introducing the Laplace Approximation (Laplace, 1774, p. 366–367), a proof was published later (Laplace, 1814) as part of a mathematical system of inductive reasoning based on probability. Laplace used this method to approximate posterior moments.

Since its introduction, the Laplace Approximation has been applied successfully in many disciplines. In the 1980s, the Laplace Approximation experienced renewed interest, especially in statistics, and some improvements in its implementation were introduced (Tierney et al., 1986; Tierney et al., 1989). Only since the 1980s has the Laplace Approximation been seriously considered by statisticians in practical applications.

There are many variations of Laplace Approximation, with an effort toward replacing Markov chain Monte Carlo (MCMC) algorithms as the dominant form of numerical approximation in Bayesian inference. The run-time of Laplace Approximation is a little longer than Maximum Likelihood Estimation (MLE), usually shorter than variational Bayes, and much shorter than MCMC (Azevedo and Shachter, 1994).

The speed of Laplace Approximation depends on the optimization algorithm selected, and typically involves many evaluations of the objective function per iteration (where an MCMC algorithm with a multivariate proposal usually evaluates once per iteration), making many MCMC algorithms faster per iteration. The attractiveness of Laplace Approximation is that it typically improves the objective function better than iterative quadrature, MCMC, and PMC when the parameters are in low-probability regions. Laplace Approximation is also typically faster than MCMC and PMC because it is seeking point-estimates, rather than attempting to represent the target distribution with enough simulation draws. Laplace Approximation extends MLE, but shares similar limitations, such as its asymptotic nature with respect to sample size and that marginal posterior distributions are Gaussian. Bernardo and Smith (2000) note that Laplace Approximation is an attractive family of numerical approximation algorithms, and will continue to develop.

`LaplaceApproximation` seeks a global maximum of the logarithm of the unnormalized joint posterior density. The approach differs by `Method`. The `LaplacesDemon` function uses the `LaplaceApproximation` algorithm to optimize initial values and save time for the user.

Most optimization algorithms assume that the logarithm of the unnormalized joint posterior density is defined and differentiable. Some methods calculate an approximate gradient for each initial value as the difference in the logarithm of the unnormalized joint posterior density due to a slight increase in the parameter.

When `Method="AGA"`, the direction and distance for each parameter is proposed based on an approximate truncated gradient and an adaptive step size. The step size parameter, which is often plural and called rate parameters in other literature, is adapted each iteration with the univariate version of the Robbins-Monro stochastic approximation in Garthwaite (2010). The step size shrinks when a proposal is rejected and expands when a proposal is accepted.

Gradient ascent is criticized for sometimes being relatively slow when close to the maximum, and its asymptotic rate of convergence is inferior to other methods. However, compared to other popular optimization algorithms such as Newton-Raphson, an advantage of the gradient ascent is that it works in infinite dimensions, requiring only sufficient computer memory. Although Newton-Raphson converges in fewer iterations, calculating the inverse of the negative Hessian matrix of second-derivatives is more computationally expensive and subject to singularities. Therefore, gradient ascent takes longer to converge, but is more generalizable.

When `Method="BFGS"`, the BFGS algorithm is used, which was proposed by Broyden (1970), Fletcher (1970), Goldfarb (1970), and Shanno (1970), independently. BFGS may be the most efficient and popular quasi-Newton optimiziation algorithm. As a quasi-Newton algorithm, the Hessian matrix is approximated using rank-one updates specified by (approximate) gradient evaluations. Since BFGS is very popular, there are many variations of it. This is a version by Nash that has been adapted from the Rvmmin package, and is used in the `optim` function of base R. The approximate Hessian is not guaranteed to converge to the Hessian. When BFGS is used, the approximate Hessian is not used to calculate the final covariance matrix.

When `Method="BHHH"`, the algorithm of Berndt et al. (1974) is used, which is commonly pronounced B-triple H. The BHHH algorithm is a quasi-Newton method that includes a step-size parameter, partial derivatives, and an approximation of a covariance matrix that is calculated as the inverse of the sum of the outer product of the gradient (OPG), calculated from each record. The OPG method becomes more costly with data sets with more records. Since partial derivatives must be calculated per record of data, the list of data has special requirements with this method, and must include design matrix `X`, and dependent variable `y` or `Y`. Records must be row-wise. An advantage of BHHH over NR (see below) is that the covariance matrix is necessarily positive definite, and gauranteed to provide an increase in LP each iteration (given a small enough step-size), even in convex areas. The covariance matrix is better approximated with larger data sample sizes, and when closer to the maximum of LP. Disadvantages of BHHH include that it can give small increases in LP, especially when far from the maximum or when LP is highly non-quadratic.

When `Method="CG"`, a nonlinear conjugate gradient algorithm is used. CG uses partial derivatives, but does not use the Hessian matrix or any approximation of it. CG usually requires more iterations to reach convergence than other algorithms that use the Hessian or an approximation. However, since the Hessian becomes computationally expensive as the dimension of the model grows, CG is applicable to large dimensional models when `CovEst="Hessian"` is avoided. CG was originally developed by Hestenes and Stiefel (1952), though this version is adapted from the `Rcgminu` function in package Rcgmin.

When `Method="DFP"`, the Davidon-Fletcher-Powell algorithm is used. DFP was the first popular, multidimensional, quasi-Newton optimization algorithm. The DFP update of an approximate Hessian matrix maintains symmetry and positive-definiteness. The approximate Hessian is not guaranteed to converge to the Hessian. When DFP is used, the approximate Hessian is not used to calculate the final covariance matrix. Although DFP is very effective, it was superseded by the BFGS algorithm.

When `Method="HAR"`, a hit-and-run algorithm with a multivariate proposal and adaptive length is used. The length parameter is adapted each iteration with the univariate version of the Robbins-Monro stochastic approximation in Garthwaite (2010). The length shrinks when a proposal is rejected and expands when a proposal is accepted. This is the same algorithm as the HARM or Hit-And-Run Metropolis MCMC algorithm with adaptive length, except that a Metropolis step is not used.

When `Method="HJ"`, the Hooke-Jeeves (1961) algorithm is used. This was adapted from the `HJK` algorithm in package dfoptim. Hooke-Jeeves is a derivative-free, direct search method. Each iteration involves two steps: an exploratory move and a pattern move. The exploratory move explores local behavior, and the pattern move takes advantage of pattern direction. It is sometimes described as a hill-climbing algorithm. If the solution improves, it accepts the move, and otherwise rejects it. Step size decreases with each iteration. The decreasing step size can trap it in local maxima, where it gets stuck and convergences erroneously. Users are encouraged to attempt again after what seems to be convergence, starting from the latest point. Although getting stuck at local maxima can be problematic, the Hooke-Jeeves algorithm is also attractive because it is simple, fast, does not depend on derivatives, and is otherwise relatively robust.

When `Method="LBFGS"`, the limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm is called in `optim`, once per iteration.

When `Method="LM"`, the Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963) is used. Also known as the Levenberg-Marquardt Algorithm (LMA) or the Damped Least-Squares (DLS) method, LM is a trust region (not to be confused with TR below) quasi-Newton optimization algorithm that provides minimizes nonlinear least squares, and has been adapted here to maximize LP. LM uses partial derivatives and approximates the Hessian with outer-products. It is suitable for nonlinear optimization up to a few hundred parameters, but loses its efficiency in larger problems due to matrix inversion. LM is considered between the Gauss-Newton algorithm and gradient descent. When far from the solution, LM moves slowly like gradient descent, but is guaranteed to converge. When LM is close to the solution, LM becomes a damped Gauss-Newton method. This was adapted from the `lsqnonlin` algorithm in package pracma.

When `Method="NM"`, the Nelder-Mead (1965) algorithm is used. This was adapted from the `nelder_mead` function in package pracma. Nelder-Mead is a derivative-free, direct search method that is known to become inefficient in large-dimensional problems. As the dimension increases, the search direction becomes increasingly orthogonal to the steepest ascent (usually descent) direction. However, in smaller dimensions, it is a popular algorithm. At each iteration, three steps are taken to improve a simplex: reflection, extension, and contraction.

When `Method="NR"`, the Newton-Raphson optimization algorithm, also known as Newton's Method, is used. Newton-Raphson uses derivatives and a Hessian matrix. The algorithm is included for its historical significance, but is known to be problematic when starting values are far from the targets, and calculating and inverting the Hessian matrix can be computationally expensive. As programmed here, when the Hessian is problematic, it tries to use only the derivatives, and when that fails, a jitter is applied. Newton-Raphson should not be the first choice of the user, and BFGS should always be preferred.

When `Method="PSO"`, the Standard Particle Swarm Optimization 2007 algorithm is used. A swarm of particles is moved according to velocity, neighborhood, and the best previous solution. The neighborhood for each particle is a set of informing particles. PSO is derivative-free. PSO has been adapted from the `psoptim` function in package pso.

When `Method="Rprop"`, the approximate gradient is taken for each parameter in each iteration, and its sign is compared to the approximate gradient in the previous iteration. A weight element in a weight vector is associated with each approximate gradient. A weight element is multiplied by 1.2 when the sign does not change, or by 0.5 if the sign changes. The weight vector is the step size, and is constrained to the interval [0.001, 50], and initial weights are 0.0125. This is the resilient backpropagation algorithm, which is often denoted as the “Rprop-” algorithm of Riedmiller (1994).

When `Method="SGD"`, a stochastic gradient descent algorithm is used that is designed only for big data, and gained popularity after successful use in the NetFlix competition. This algorithm has special requirements for the `Model` specification function and the `Data` list. See the “LaplacesDemon Tutorial” vignette for more information.

When `Method="SOMA"`, a population of ten particles or individuals moves in the direction of the best particle, the leader. The leader does not move in each iteration, and a line-search is used for each non-leader, up to three times the difference in parameter values between each non-leader and leader. This algorithm is derivative-free and often considered in the family of evolution algorithms. Numerous model evaluations are performed per non-leader per iteration. This algorithm was adapted from package soma.

When `Method="SPG"`, a Spectral Projected Gradient algorithm is used. SPG is a non-monotone algorithm that is suitable for high-dimensional models. The approximate gradient is used, but the Hessian matrix is not. When used with large models, `CovEst="Hessian"` should be avoided. SPG has been adapted from the `spg` function in package BB.

When `Method="SR1"`, the Symmetric Rank-One (SR1) algorithm is used. SR1 is a quasi-Newton algorithm, and the Hessian matrix is approximated, often without being positive-definite. At the posterior modes, the true Hessian is usually positive-definite, but this is often not the case during optimization when the parameters have not yet reached the posterior modes. Other restrictions, including constraints, often result in the true Hessian being indefinite at the solution. For these reasons, SR1 often outperforms BFGS. The approximate Hessian is not guaranteed to converge to the Hessian. When SR1 is used, the approximate Hessian is not used to calculate the final covariance matrix.

When `Method="TR"`, the Trust Region algorithm of Nocedal and Wright (1999) is used. The TR algorithm attempts to reach its objective in the fewest number of iterations, is therefore very efficient, as well as safe. The efficiency of TR is attractive when model evaluations are expensive. The Hessian is approximated each iteration, making TR best suited to models with small to medium dimensions, say up to a few hundred parameters. TR has been adapted from the `trust` function in package trust.

### Value

`LaplaceApproximation` returns an object of class `laplace` that is a list with the following components:

`Call` | This is the matched call of `LaplaceApproximation`.
`Converged` | This is a logical indicator of whether or not `LaplaceApproximation` converged within the specified `Iterations` according to the supplied `Stop.Tolerance` criterion. Convergence does not indicate that the global maximum has been found, but only that the tolerance was less than or equal to the `Stop.Tolerance` criterion.
`Covar` | This covariance matrix is estimated according to the `CovEst` argument. The `Covar` matrix may be scaled and input into the `Covar` argument of the `LaplacesDemon` or `PMC` function for further estimation, or the diagonal of this matrix may be used to represent the posterior variance of the parameters, provided the algorithm converged and matrix inversion was successful. To scale this matrix for use with Laplace's Demon or PMC, multiply it by 2.38^2/d, where d is the number of initial values.
`Deviance` | This is a vector of the iterative history of the deviance in the `LaplaceApproximation` function, as it sought convergence.
`History` | This is a matrix of the iterative history of the parameters in the `LaplaceApproximation` function, as it sought convergence.
`Initial.Values` | This is the vector of initial values that was originally given to `LaplaceApproximation` in the `parm` argument.
`LML` | This is an approximation of the logarithm of the marginal likelihood of the data (see the `LML` function for more information). When the model has converged and `sir=TRUE`, the NSIS method is used. When the model has converged and `sir=FALSE`, the LME method is used. This is the logarithmic form of equation 4 in Lewis and Raftery (1997). As a rough estimate of Kass and Raftery (1995), the LME-based LML is worrisome when the sample size of the data is less than five times the number of parameters, and `LML` should be adequate in most problems when the sample size of the data exceeds twenty times the number of parameters (p. 778). The LME is inappropriate with hierarchical models. However `LML` is estimated, it is useful for comparing multiple models with the `BayesFactor` function.
`LP.Final` | This reports the final scalar value for the logarithm of the unnormalized joint posterior density.
`LP.Initial` | This reports the initial scalar value for the logarithm of the unnormalized joint posterior density.
`Minutes` | This is the number of minutes that `LaplaceApproximation` was running, and this includes the initial checks as well as drawing posterior samples and creating summaries.
`Monitor` | When `sir=TRUE`, a number of independent posterior samples equal to `Samples` is taken, and the draws are stored here as a matrix. The rows of the matrix are the samples, and the columns are the monitored variables.
`Posterior` | When `sir=TRUE`, a number of independent posterior samples equal to `Samples` is taken, and the draws are stored here as a matrix. The rows of the matrix are the samples, and the columns are the parameters.
`Step.Size.Final` | This is the final, scalar `Step.Size` value at the end of the `LaplaceApproximation` algorithm.
`Step.Size.Initial` | This is the initial, scalar `Step.Size`.
`Summary1` | This is a summary matrix that summarizes the point-estimated posterior modes. Uncertainty around the posterior modes is estimated from the covariance matrix. Rows are parameters. The following columns are included: Mode, SD (Standard Deviation), LB (Lower Bound), and UB (Upper Bound). The bounds constitute a 95% probability interval.
`Summary2` | This is a summary matrix that summarizes the posterior samples drawn with sampling importance resampling (`SIR`) when `sir=TRUE`, given the point-estimated posterior modes and the covariance matrix. Rows are parameters. The following columns are included: Mode, SD (Standard Deviation), LB (Lower Bound), and UB (Upper Bound). The bounds constitute a 95% probability interval.
`Tolerance.Final` | This is the last `Tolerance` of the `LaplaceApproximation` algorithm.
`Tolerance.Stop` | This is the `Stop.Tolerance` criterion.

### Author(s)

Statisticat, LLC software@bayesian-inference.com

### References

Azevedo-Filho, A. and Shachter, R. (1994). "Laplace's Method Approximations for Probabilistic Inference in Belief Networks with Continuous Variables". In "Uncertainty in Artificial Intelligence", Mantaras, R. and Poole, D., Morgan Kauffman, San Francisco, CA, p. 28–36.

Bernardo, J.M. and Smith, A.F.M. (2000). "Bayesian Theory". John Wiley \& Sons: West Sussex, England.

Berndt, E., Hall, B., Hall, R., and Hausman, J. (1974), "Estimation and Inference in Nonlinear Structural Models". Annals of Economic and Social Measurement, 3, p. 653–665.

Broyden, C.G. (1970). "The Convergence of a Class of Double Rank Minimization Algorithms: 2. The New Algorithm". Journal of the Institute of Mathematics and its Applications, 6, p.76–90.

Fletcher, R. (1970). "A New Approach to Variable Metric Algorithms". Computer Journal, 13(3), p. 317–322.

Garthwaite, P., Fan, Y., and Sisson, S. (2010). "Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process."

Goldfarb, D. (1970). "A Family of Variable Metric Methods Derived by Variational Means". Mathematics of Computation, 24(109), p. 23–26.

Hestenes, M.R. and Stiefel, E. (1952). "Methods of Conjugate Gradients for Solving Linear Systems". Journal of Research of the National Bureau of Standards, 49(6), p. 409–436.

Hooke, R. and Jeeves, T.A. (1961). "'Direct Search' Solution of Numerical and Statistical Problems". Journal of the Association for Computing Machinery, 8(2), p. 212–229.

Kass, R.E. and Raftery, A.E. (1995). "Bayes Factors". Journal of the American Statistical Association, 90(430), p. 773–795.

Laplace, P. (1774). "Memoire sur la Probabilite des Causes par les Evenements." l'Academie Royale des Sciences, 6, 621–656. English translation by S.M. Stigler in 1986 as "Memoir on the Probability of the Causes of Events" in Statistical Science, 1(3), 359–378.

Laplace, P. (1814). "Essai Philosophique sur les Probabilites." English translation in Truscott, F.W. and Emory, F.L. (2007) from (1902) as "A Philosophical Essay on Probabilities". ISBN 1602063281, translated from the French 6th ed. (1840).

Levenberg, K. (1944). "A Method for the Solution of Certain Non-Linear Problems in Least Squares". Quarterly of Applied Mathematics, 2, p. 164–168.

Lewis, S.M. and Raftery, A.E. (1997). "Estimating Bayes Factors via Posterior Simulation with the Laplace-Metropolis Estimator". Journal of the American Statistical Association, 92, p. 648–655.

Marquardt, D. (1963). "An Algorithm for Least-Squares Estimation of Nonlinear Parameters". SIAM Journal on Applied Mathematics, 11(2), p. 431–441.

Nelder, J.A. and Mead, R. (1965). "A Simplex Method for Function Minimization". The Computer Journal, 7(4), p. 308–313.

Nocedal, J. and Wright, S.J. (1999). "Numerical Optimization". Springer-Verlag.

Riedmiller, M. (1994). "Advanced Supervised Learning in Multi-Layer Perceptrons - From Backpropagation to Adaptive Learning Algorithms". Computer Standards and Interfaces, 16, p. 265–278.

Shanno, D.F. (1970). "Conditioning of quasi-Newton Methods for Function Minimization". Mathematics of Computation, 24(111), p. 647–650.

Tierney, L. and Kadane, J.B. (1986). "Accurate Approximations for Posterior Moments and Marginal Densities". Journal of the American Statistical Association, 81(393), p. 82–86.

Tierney, L., Kass. R., and Kadane, J.B. (1989). "Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions". Journal of the American Statistical Association, 84(407), p. 710–716.

Zelinka, I. (2004). "SOMA - Self Organizing Migrating Algorithm". In: Onwubolu G.C. and Babu, B.V., editors. "New Optimization Techniques in Engineering". Springer: Berlin, Germany.

### See Also

`BayesFactor`, `BayesianBootstrap`, `IterativeQuadrature`, `LaplacesDemon`, `GIV`, `LML`, `optim`, `PMC`, `SIR`, and `VariationalBayes`.

### Examples

    # The accompanying Examples vignette is a compendium of examples.
    ####################  Load the LaplacesDemon Library  #####################
    library(LaplacesDemon)

    ##############################  Demon Data  ###############################
    data(demonsnacks)
    y <- log(demonsnacks$Calories)
    X <- cbind(1, as.matrix(log(demonsnacks[,10]+1)))
    J <- ncol(X)
    for (j in 2:J) X[,j] <- CenterScale(X[,j])

    #########################  Data List Preparation  #########################
    mon.names <- "mu[1]"
    parm.names <- as.parm.names(list(beta=rep(0,J), sigma=0))
    pos.beta <- grep("beta", parm.names)
    pos.sigma <- grep("sigma", parm.names)
    PGF <- function(Data) {
         beta <- rnorm(Data$J)
         sigma <- runif(1)
         return(c(beta, sigma))
         }
    MyData <- list(J=J, PGF=PGF, X=X, mon.names=mon.names,
         parm.names=parm.names, pos.beta=pos.beta, pos.sigma=pos.sigma, y=y)

    ##########################  Model Specification  ##########################
    Model <- function(parm, Data)
         {
         ### Parameters
         beta <- parm[Data$pos.beta]
         sigma <- interval(parm[Data$pos.sigma], 1e-100, Inf)
         parm[Data$pos.sigma] <- sigma
         ### Log-Prior
         beta.prior <- sum(dnormv(beta, 0, 1000, log=TRUE))
         sigma.prior <- dhalfcauchy(sigma, 25, log=TRUE)
         ### Log-Likelihood
         mu <- tcrossprod(Data$X, t(beta))
         LL <- sum(dnorm(Data$y, mu, sigma, log=TRUE))
         ### Log-Posterior
         LP <- LL + beta.prior + sigma.prior
         Modelout <- list(LP=LP, Dev=-2*LL, Monitor=mu[1],
              yhat=rnorm(length(mu), mu, sigma), parm=parm)
         return(Modelout)
         }

    ############################  Initial Values  #############################
    #Initial.Values <- GIV(Model, MyData, PGF=TRUE)
    Initial.Values <- rep(0,J+1)

    Fit <- LaplaceApproximation(Model, Initial.Values, Data=MyData,
         Iterations=100, Method="NM", CPUs=1)
    Fit
    print(Fit)
    #PosteriorChecks(Fit)
    #caterpillar.plot(Fit, Parms="beta")
    #plot(Fit, MyData, PDF=FALSE)
    #Pred <- predict(Fit, Model, MyData, CPUs=1)
    #summary(Pred, Discrep="Chi-Square")
    #plot(Pred, Style="Covariates", Data=MyData)
    #plot(Pred, Style="Density", Rows=1:9)
    #plot(Pred, Style="Fitted")
    #plot(Pred, Style="Jarque-Bera")
    #plot(Pred, Style="Predictive Quantiles")
    #plot(Pred, Style="Residual Density")
    #plot(Pred, Style="Residuals")
    #Levene.Test(Pred)
    #Importance(Fit, Model, MyData, Discrep="Chi-Square")

    #Fit$Covar is scaled (2.38^2/d) and submitted to LaplacesDemon as Covar.
    #Fit$Summary[,1] is submitted to LaplacesDemon as Initial.Values.
    #End

[Package LaplacesDemon version 16.1.1 Index]