| 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. |
parm |
This argument requires a vector of initial values equal in
length to the number of parameters. |
Data |
This required argument accepts a list of data. The list of
data must include |
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 |
Method |
This optional argument accepts a quoted string that
specifies the method used for Laplace Approximation. The default
method is |
Samples |
This argument indicates the number of posterior samples
to be taken with sampling importance resampling via the
|
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 |
This logical argument indicates whether or not Sampling
Importance Resampling (SIR) is conducted via the |
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 |
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 |
Type |
This argument specifies the type of parallel processing to
perform, accepting either |
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 |
Converged |
This is a logical indicator of whether or not
|
Covar |
This covariance matrix is estimated according to the
|
Deviance |
This is a vector of the iterative history of the
deviance in the |
History |
This is a matrix of the iterative history of the
parameters in the |
Initial.Values |
This is the vector of initial values that was
originally given to |
LML |
This is an approximation of the logarithm of the marginal
likelihood of the data (see the |
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
|
Monitor |
When |
Posterior |
When |
Step.Size.Final |
This is the final, scalar |
Step.Size.Initial |
This is the initial, scalar |
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
( |
Tolerance.Final |
This is the last |
Tolerance.Stop |
This is the |
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