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Fisher information in Laplace approximation

Let $X$ and $Y$ be continuous random variables with Probability Density Function (PDF) $f_X$ and $f_Y$, respectively. Upon observing $Y=y$, the log-posterior PDF is given by Bayes' rule in log form: $$...
W. Zhu's user avatar
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Time Efficient Method of Asymmetric Laplace MLE of Location Parameter

Reviewing "Tests of fit for the asymmetric Laplace distribution" under "Maximum Likelihood Estimation", the proposed method of determining the $\hat{\theta}$ seems problematic from ...
JakeTheSnake's user avatar
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Laplace approximation, MAP vs MLE and wiki's notations

I was trying to understand Laplace approximation in statistics and so I was going through the wikipedia article. I don't know much about statistics and I am already getting a bit confused by the ...
roi_saumon's user avatar
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Fitting GLMM using lme4 package, fitting algorithm

I'm a little confused here. So I'm using glmm model to fit user/item interaction. If user #12 liked movie #15 it's 1 otherwise it's 0. Here is my model:  ...
Efecan Bahcivanoglu's user avatar
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What is the meaning of graph singal for graph constructed from correlation matrix?

In the highly cited paper "The Emerging Field of Signal Processing on Graphs", the authors defined graph singal for a graph of N vertices as a vector of length N, with each element of the ...
Patrick's user avatar
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3 votes
2 answers
206 views

Is it bad to have large random effect variance when fixed effect estimates are small?

I'm running a generalized linear mixed models (GLMM) and am comparing the Laplace approximation to the Gauss-Hermite Quadrature. I have three predictor variables (fixed effects) and each is a number ...
burphound's user avatar
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Laplace approximation from a log-posterior in R

I would like to perform a Laplace approximation of a log-posterior. The evolution of a cancer cell at given time $t_j$, $j = 1,\cdots,n$ for an experiment $i$ follows the following Poisson ...
Mathieu Rousseau's user avatar
2 votes
0 answers
178 views

Gaussian Processes: multi-class Laplace approximation

In Chapter 3.5 of Gaussian Process for Machine Learning book by Rasmussen and Williams (R&W 2006), authors present a Laplace approximation for a multi-class Gaussian Process (GP) classifier. ...
Metod Jazbec's user avatar
2 votes
2 answers
534 views

Inference: How is the Laplace approximation actually useful to us compared with MLE and MAP?

I was reading a few different sources (including the "Machine Learning and Pattern Recognition" book by Bishop) about the Laplace integral approximation method for inference. However, I am ...
Rocky the Owl's user avatar
1 vote
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336 views

Random Fourier Features approximating a kernel inverse?

There is a method I have been studying called Spectral Normalised Neural Gaussian Processes which leaves me with a question I cannot answer. In this method, they utilize Random Fourier Features but ...
Joff's user avatar
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Why is the Hessian in the Laplace approximation negative

The Laplace approximation builds from the Taylor expansion of the MAP estimate, where the first derivative is 0. The second order Taylor series goes... $$ f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}...
Joff's user avatar
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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 ...
user1988's user avatar
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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 ...
Dion's user avatar
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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 ...
January Board's user avatar
1 vote
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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 ...
Dion's user avatar
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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 ...
humbleasker's user avatar
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 ...
Kirk Walla's user avatar
1 vote
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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 ...
jds's user avatar
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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 ...
Jeremias K's user avatar
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1 vote
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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.
Skullgreymon's user avatar
11 votes
2 answers
1k 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 ...
hatmatrix's user avatar
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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,...
CBowman's user avatar
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3 votes
0 answers
262 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, \...
Charlotte's user avatar
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5 votes
1 answer
321 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\...
ZHU's user avatar
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16 votes
2 answers
3k 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 ...
aleshing's user avatar
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3 votes
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76 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) ...
Patrick's user avatar
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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 $\int p(D|\theta,M)p(\theta|M) d\,\theta$. ...
ostrichgroomer's user avatar
6 votes
1 answer
540 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? ...
Renzo Mauricio Guzmán Anaya's user avatar
2 votes
2 answers
982 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 ...
dave fournier's user avatar
2 votes
1 answer
460 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 ...
tisPrimeTime's user avatar
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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 ...
Kevin Yang's user avatar
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 ...
lacerbi's user avatar
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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 ...
Hector's user avatar
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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 ...
Emanuele  Giorgi's user avatar
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 ...
Tatiana's user avatar
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