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# Questions tagged [laplace-approximation]

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• 932
2 votes
2 answers
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### 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 ...
• 155
3 votes
2 answers
314 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 ...
• 954
8 votes
0 answers
382 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 ...
1 vote
0 answers
91 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 ...
• 954
19 votes
1 answer
3k 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 ...
• 327
1 vote
1 answer
666 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 ...
• 212
1 vote
0 answers
170 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 ...
• 1,714
1 vote
0 answers
285 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 ...
• 1,529
1 vote
0 answers
107 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.
• 140
11 votes
2 answers
1k views

### Bayesian inverse modeling with non-identifiable parameters?

If I have a physical model $$y = \frac{1}{\beta_0} (\beta_1 x_1 + \beta_2 x_2)$$ and want to estimate coefficients $\beta_0$, $\beta_1$, and $\beta_2$ from given data ...
• 869
3 votes
2 answers
391 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,...
• 613
3 votes
0 answers
262 views

• 535
0 votes
0 answers
249 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 ...
• 812
8 votes
2 answers
2k 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 ...
• 5,216
3 votes
0 answers
666 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 ...
• 131
4 votes
0 answers
960 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 ...
7 votes
1 answer
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 ...
• 71