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Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Bayesian Q-learning

Suppose that, for every state $s$, there is a set of actions $\mathcal{A}(s)$ that can be chosen in that state. Let $Q(s, a)$ denote the expected utility of choosing action $a \in \mathcal{A}(s)$ in ...
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Getting started with bayesian structural models using MCMC

I'm trying to learn bayesian structural time series analysis. For a variety of reasons I need to use Python (mostly pymc3) not R so please do not suggest the ...
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What is the correct way to write the elastic net?

I am confused about the correct way to write the elastic net. After reading some research papers there seems to be three forms 1) $\exp\{-\lambda_1|\beta_k|-\lambda_2\beta_k^2\}$ 2) $\exp\{-\frac{(\...
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Help me understand the Bayesian kernel density estimation (Sibisi and Skilling, 1996)

Sibisi and Skilling (1996, also mentioned in the 1997 paper) define Bayesian kernel density as $$ f(x) = \int dx' \,\phi(x')\, K(x, x') \tag{2} $$ Here the kernel $K$ is an assigned smooth ...
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170 views

Is probability fundamentally about reference classes (real or imagined)?

Question: It seems that frequentism and Bayesianism may not really be different as far as the the ultimate basis for what a probability is (relative frequency within a reference class) - it's just ...
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450 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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250 views

Dealing with dependent data in a Bayesian model

Background: Consider a series of dependent data points, $$ y_1,y_2,y_3,\cdots,y_N. $$ In cases where the dependence is well described by an exponentially decaying auto-correlation function, it is ...
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Horseshoe priors and random slope/intercept regressions

I'm interested in using the horseshoe prior (or the related hierarchical-shrinkage family of priors) for regression coefficients of a traditional multilevel regression (e.g., random slopes/intercepts)....
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Robust Gamma Regression

I am modeling some spectroscopic data where the response of the instrument to the size of the input is strictly positive and non-linear. Gamma regression seems like a good choice to explain the data, ...
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437 views

Density estimation/approximation from MCMC samples

I'm looking to accurately describe the density function of a multivariate posterior probability distribution based on samples from MCMC. As far as I know, in most cases this is done either with a ...
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151 views

Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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130 views

Bayesian inference via approximate data likelihood

Suppose that we have a very large i.i.d. sample $x_1,...,x_n$ and a data likelihood defined by $$p(x | \theta,\beta) = \prod_ip(x_i | \theta,\beta)$$. Further suppose that $\theta$ is the parameter ...
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Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
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105 views

why use diagonal $\Sigma$ when working with Bayes decision theory?

My prof. said in the class that for Bayes decision rule, the likelihood is Gaussian and in practice, we will almost always work with a diagonal $\Sigma$. Why is that? I know that a diagonal $\Sigma$ ...
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How to construct “reference priors”?

I have been reading about noninformative priors. Two of the most popular priors of this kind seem to be the Jeffreys prior and the reference prior. The Jeffreys prior has a clear construction, being ...
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728 views

Is my OpenBUGS / WinBUGS model well specified?

I've just started trying to use OpenBUGS for Bayesian analysis of stochastic volatility models. In particular, I'm trying to calculate stochastic covariance, similar to the DC-MSV model specified by ...
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140 views

Is this problem Bayesian? And can I use variational approximation?

Suppose there are $N$ samples of observations $\mathbf X(n)$ ($n=1,\cdots,N$), which are given by probability distribution $p(\mathbf X(n)|\mathbf Z(n))$ with their conditions are given by hidden ...
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83 views

Rationale behind Good–Turing frequency estimation?

Good–Turing frequency estimation is a smoothing estimator for estimating a multinomial distribution. It seems very convoluted. From mathematical statistics point of view, what is the rationale ...
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Trouble replicating simple example of Bayesian inference

On pages 20-21 of John Kruschke's Doing Bayesian Data Analysis book (2nd ed.), there is an introductory illustration of Bayesian inference. We know that balls can have four sizes: 1, 2, 3 and 4, but ...
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Do shrinkage estimators solve the Neyman-Scott paradox?

I read the following SE question: What problem do shrinkage methods solve? And I wondered if shrinkage estimators provide a consistent estimator of the sample variance in a "mixed-effects" model using ...
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219 views

Why can't the complete class theorem be easily generalized to all locally-compact spaces?

So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, ...
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332 views

Bayesian linear regression - posterior distribution

This is about bayesian linear regression. In this link http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node2.html there's a derivation for = The part that I don't understand is how it is ...
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How to fix this implementation of Bayesian regularization for ANNs?

I have implemented the Levenberg-Marquard algorithm (from Hagan's "Artifical Neural Network Design" -- 2014) for a two layer network with 20 neurons in the hidden layer. This network can beautifully ...
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Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
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446 views

Can you use a gaussian process to model the smoothness of residuals?

I see a lot of use of Gaussian Processes for regression - fitting a GP model to data points, with a prior specifying the smoothness of the function, and using it to predict new values. However, I'm ...
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Noninformative prior for variance, understanding and coding

I have three questions regarding the understanding behind and implementation of a noninformative prior for variance. I'm attempting to build a Metropolis sampler and I'm trying to sample from a ...
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374 views

AR(1) model - which prior to use?

I want to use the following univariate model: $y_t = \mu_t + \epsilon_t, \ \epsilon_t \sim N(0,1)$ $\mu_t = \phi \mu_{t-1} + \omega_t, \ \omega_t \sim N(0,\sigma_\omega^2)$ That is, $\mu_t$ follows ...
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Making sense of standard deviation after sampling using Cholesky

I have an inverse problem with over 65,000 degrees of freedom. I am using Bayesian formulation to solve this problem. After using the optimization algorithm to obtain MAP solution, I want to calculate ...
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240 views

Bayesian estimates for Deming regression coinciding with least-squares estimates

Consider the following Deming model with independent replicates : $$x_{i,j} \mid \theta_{i} \sim {\cal N}(\theta_{i}, \gamma_X^2), \quad y_{i,j} \mid \theta_{i} \sim {\cal N}(\alpha+\beta\theta_{i}, \...
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3k views

How to write up and report a Bayesian analysis?

Bayesian Estimation Supersedes the t-Test for John K. Kruschke is one of the most important papers that I had read explaining how to run the Bayesian analysis and how to make the plots. But the ...
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Question 10.9 from Bayesian Data Analysis, what does accuracy mean here?

I'm doing an independent study in Bayesian Statistics following some chapters from BDA3. When solving the first question from Ch 10 I got stuck. It says: [If] a scalar variable $\theta$ is ...
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Sampling methods and parallelization

A couple of years ago I learned about recent work in parallelizing slice sampling methods. More recently, I have read great things about NUTS and Hamiltonian Monte Carlo methods (HMC) in general (e.g. ...
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Bias Variance tradeoff from a Bayesian perspective

I know the general question about bias variance has been asked before. I understand the frequentist approach and the concept of model selection and the impact of bias and variance on "accuracy" of a ...
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933 views

Bayesian model selection in PyMC

I'm trying to do model selection using PyMC (v2.2), but having difficulty assessing the models using various Information Criteria and/or Bayes Factor. My model is similar to a typical regression, with ...
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186 views

Running regularized logistic regressions on very large datasets

I want to run a regularized logistic regression on a dataset with 25 million observations and about a 1000 mostly non-sparse columns with non-ignorable weights. My first choice would be BayesGLM, ...
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597 views

How do I sample from the posterior distribution with gamma likelihood with unknown alpha and beta?

I realize that this Wikipedia page provides the proportional form of the conjugate prior to the gamma distribution with unknown $\alpha$ and $\beta$ parameters, as well as the posterior values of $p$, ...
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Laplace smoothing parameter choice for Markov chain transitions

Let $Y_{t}$ be the state of the process at time $t$, ${\bf P}$ be the transition matrix then: $$ {\bf P}_{ij} = P(Y_{t} = j | Y_{t-1} = i) $$ Since this is a Markov chain, this probability depends ...
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130 views

MCMC direct comparison of difference of two parameters

Say I have run a Hierarchical Bayesian model in STAN (or JAGS or BUGS) and I have the posterior samples of two slope parameters that I want to compare: $\beta_1$ and $\beta_2$. The model appears to ...
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182 views

Combining evidence using Dempster-Shafer theory

Can someone post a simple explanation of Dempster-Shafer theory? There are lot of links available but the reading material in those sites is academic in nature and time consuming to read and ...
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673 views

Variance of marginal posterior distribution

Suppose $Y_1,\dots,Y_n\mid\mu,\sigma^2 \sim \text{ iid } N(\mu,\sigma^2)$ and suppose the priors $\mu \mid \sigma^2 \sim N(\mu_0, \sigma^2 / \kappa_0)$ and $1/\sigma^2 \sim \text{gamma}(\nu_0/2, \nu_0 ...
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Finding a prediction interval for an lmer model via mcmcsamp versus simulate

I've created a model with lme4's lmer and wanted to create a prediction interval around my model fit. I figured I could do it ...
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70 views

How to infer a prior belief after observing a behavior

My participant goes through a maze made of 32 T intersections. At each intersection he must choose whether to go either to the left or to the right: one option will lead to another T intersection, ...
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What was the fundamental error of Bayesian statistics according to Fisher?

Fisher wrote: "the theory of inverse probability is founded upon an error, and must be wholly rejected" I wonder what was Fisher's reasoning and what error he means in particular. The quote is ...
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76 views

Maximizing the information gain on a Gaussian RV with a noisy comparison question

The question Let $X \sim \mathcal{N}(0,1)$ be a random variable denoting the location of a target on the real line. $Y_a$ be a binary random variable encoding the (noisy) answer to the question: "is ...
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334 views

Fourier transform of a Gaussian process

I would like to discuss and ask a question regarding the Fourier transform of a Gaussian process, if it makes sense. For that purpose, let me describe the following situation. Let $z(s)$ be a ...
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122 views

Bayesian Multivariate Normal-Normal Model when covariance depends on the mean

Let $y$ denote a D-dimensional multivariate normal random variable with density $$y \sim N(\mu, \Sigma(\mu)) $$ such that the covariance $\Sigma$ is a deterministic non-linear function of $\mu$. ...
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Including feature-dependent priors on output class, in bayesian logistic regression

When doing logistic regression with data $D_N = \{(x_i, y_i)\}_i^N$ with $x_i \in \mathbf{X}^N$ (each data point has N features) and $y_i \in \mathbf{Y}$ being assigned output classes, in a Bayesian ...
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119 views

Run MAP estimates before MCMC in most cases?

I am learning Bayesian statistics. I found that this pymc3 introduction sometimes uses MAP to estimate the parameters for the MCMC input (the regression example), but the intro doesn't run MAP for ...
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A puzzling observation by Bradley Efron in his article in Science regarding Bayes’ Theorem in the 21st Century

Mr. Effron has published an interesting article in Science magazine with the enticing title "Baye's Theorem in the 21st Century". The article is quite short and can be found here: http://web.ipac....
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246 views

Confidence Interval vs Credible Interval for the Variance

I understand the conceptual difference between confidence and credible intervals. But I have difficulties applying these concepts to my application. I would like to know the concrete difference ...