Questions tagged [laplace-approximation]

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0answers
29 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 ...
2
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2answers
281 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
84 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,...
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0answers
59 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
232 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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452 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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0answers
60 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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0answers
88 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
160 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
258 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 ...