Questions tagged [laplace-approximation]

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Laplace approximation for small number of data

When a large number of data points is available, according to the central limit theorem, Laplace method can give an efficient, good approximation of posterior as a Gaussian distribution centered at ...
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1answer
210 views

How are PQL, REML, ML, Laplace, Gauss-Hermite related to each other?

While learning about the Generalized Linear Mixed Models, I often see the above terms. Sometimes it seems to me these are separate methods of estimation of (fixed? random? both?) effects, but when I ...
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1answer
75 views

What is the difference between approximate bayesian computation vs approximate bayesian inference?

What are the main differences between approximate bayesian computation vs approximate bayesian inference? Are they essentially the same? Do they refer to the same of different family of models? My ...
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23 views

Can Laplace Approximation be used when Gaussian Process is only a part of bigger model?

I felt in love with Gaussian Process models in Species Distribution Modelling, especially with the Laplace Approximation (LA), which is able to compute these models very fast. This way, I am able to ...
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57 views

Why does this improper prior = constant?

MacKay has an exercise on using Laplace's method for a Poisson model: $$ p(r \mid \lambda ) = \frac{e^{-\lambda} \lambda^r}{r!}, \qquad p(\lambda) = \frac{1}{\lambda} $$ And he asks the reader to ...
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65 views

Approximating the Kullback-Leibler Divergence with a Laplace approximation

Suppose I wish to compute the (asymptotic) Kullback-Leibler Divergence (KLD) between the exact Bayesian posterior $$q_{n}(\theta|x_{1:n}) \propto \pi(\theta)\prod_{i=1}^n p(x_i|\theta)$$ and the ...
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99 views

Equation (3.23) GP for ML book

This is the computation of the variance when we do Laplace Approximation for inference in binary classification. I do not understand why the variance is decomposed into these two terms.
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2answers
353 views

Bayesian inverse modeling with non-identifiable parameters?

If I have a physical model \begin{equation} y = \frac{1}{\beta_0} (\beta_1 x_1 + \beta_2 x_2) \end{equation} and want to estimate coefficients $\beta_0$, $\beta_1$, and $\beta_2$ from given data ...
2
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2answers
129 views

Lower-bound on covariance estimated via Laplace approximation?

I think when a posterior is approximated to be multivariate normal as in Laplace approximation, the covariance matrix is taken to be the negative inverse Hessian evaluated at the log-posterior maximum,...
2
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0answers
78 views

Which optimizer use for laplace approximation

I have been trying to estimate the marginal posterior for D variable using Laplace approximation: $p(\theta_i) \approx \left[\frac{\det{H}}{2\pi\det{H(\theta_i)}}\right]^{1/2} \exp\left[-L(\theta_i, \...
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1answer
249 views

Constant of Laplace approximation

I'm reading Example 3.16 of Robert & Casella's Monte Carlo Statistical Methods. It uses a Laplace approximation for approximating an integral related with the Gamma distribution namely $$\int_a^b\...
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719 views

Comparing Laplace Approximation and Variational Inference

Does anyone know of any references that look at the relationship between the Laplace approximation and variational inference (with normal approximating distributions)? Namely I'm looking for something ...
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64 views

Simple approximation of joint posterior

Consider the (hierarchical) Bayesian inference problem with two unknowns $(x,\theta)$ and data $y$. I'm using a very simple ("independence"?) approximation $$ p(x,\theta|y) \approx p(x|\theta_\star,y) ...
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110 views

Is BIC useful for a multimodal distribution?

Given a dataset $D$ and a model $M$ with parameters $\theta$, the Bayesian Information Criterion can be used to approximate the model's marginal likelihood $\int p(D|\theta,M)p(\theta|M) d\,\theta$. ...
5
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1answer
205 views

What's the relationship between Laplace approximation and Variational Bayes methods?

To be precise, I'm checking this presentation https://kaybrodersen.github.io/talks/Brodersen_2013_03_22.pdf, but I don't understand what is the connection between Laplace method and variational bayes? ...
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2answers
339 views

Modelling random effects as an autoregressive-autoregressive process

Has anyone ever come across an autoregressive-autogressive process. I am modelling positively correlated time-series of random effects in a fisheries model. They are integrated out via the Laplace ...
2
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1answer
320 views

Hessian for Laplace Approximation in Uncertainty Propagation

This is possibly a silly conceptual question, ... but anyway: Imagine I have a function: $f = F(\mathbf{x}) = F(x_1,x_2) = ax_1^2 + bx_2^3,$ where $x_1,x_2 \sim N(0,1)$ for example. For a naive ...
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0answers
168 views

Calculating variance using Laplace approximation for GP classification

I'm having some trouble implementing Algorithm 3.2 from Rasmussen and Williams. Namely, sometimes when I evaluate step 6, I obtain a negative variance, which I believe is impossible (and makes line ...
6
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2answers
992 views

Marginalization of GP regression hyperparameters with Laplace approximation

I am using Gaussian Processes (GP) for regression (via the gpml package for MATLAB). So far, I was optimizing the hyper-parameters by maximizing the log likelihood, but I would like to try a more ...
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0answers
548 views

Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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690 views

How to derive the Fisher information in the laplace approximation of a generalized linear mixed model?

I am currently using the Laplace approximation to fit some geostatistical models for binomial data. Regarding parameters estimation I do not have any problem. I can easily implement the Laplace ...
6
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1answer
878 views

Reference for generalized linear mixed models using Laplace approximation

I'm fitting a generalized linear mixed model in R using the Laplace approximation. I'm looking for a reference for the Laplace approximation used for that, or a reference regarding the comparison ...