Questions tagged [laplace-approximation]

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2answers
39 views

On the P-value of the variance of random intercept in glmer model

I'm using a logistic mixed-effect model with random intercept through glmer function from lme4 package. I want to test the ...
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0answers
13 views

Is roger-kenward not applicable to MCMCglmm as MCMCglmm is most accurate approximation to integral?

http://bbolker.github.io/mixedmodels-misc/glmmFAQ.html This mentions MCMCglmm is more accurate than anything else, i.e. Laplace approx, adaptive-quadrature (AGQ), PQL. Does this make Kenward-Roger ...
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0answers
13 views

Uncorrelated Samples from a non-conjugate (but well behaved) posterior

I'm trying to create a Dirichlet process mixture model with a kernel distribution similar to a product of gammas. (in fact, if I generate a latent random variable, it IS a product of (independent) ...
3
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2answers
85 views

Laplace approximation in high-dimensions

Obviously computing the inverse Hessian is hard when a probability distribution is fitted on high-dimensional datapoints. One idea to reduce computational cost would be to approximate the distribution ...
5
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0answers
67 views

Why does Quadratic (Normal/Laplace) Approximation fail on multilevel models?

In Statistical Rethinking, 2nd Edition, section 13.1, Richard McElreath says: Why doesn’t simple quadratic approximation, using for example quap, work with multilevel models? When a prior is itself a ...
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0answers
23 views

Laplace approximation for factorization models

How can we cheaply approximate the posteriors for a factorization model, with unobserved variables $\theta \in \mathbb{R}^{K \times U}, \beta \in \mathbb{R}^{K \times I}$ and log-likelihood of the ...
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0answers
56 views

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 ...
12
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1answer
814 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 ...
0
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1answer
307 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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0answers
97 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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0answers
132 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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0answers
102 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.
3
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2answers
593 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 ...
3
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2answers
223 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,...
3
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0answers
144 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, \...
5
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1answer
265 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\...
12
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1answer
1k 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 ...
3
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0answers
66 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) ...
3
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0answers
162 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 $\int p(D|\theta,M)p(\theta|M) d\,\theta$. ...
6
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1answer
313 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? ...
2
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2answers
515 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
349 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
199 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 ...
7
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2answers
1k 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 ...
3
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0answers
604 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 ...
4
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0answers
814 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
1k 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 ...