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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Jaynes' $A_p$ distribution

In Jaynes' book "Probability Theory: The Logic of Science", Jaynes has a chapter (Ch 18) entitled "The $A_p$ distribution and rule of succession" in which he introduces the idea of $A_p$ distributions,...
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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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761 views

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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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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234 views

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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178 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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526 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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263 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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1answer
73 views

Why does Bayesian p-value involve the parameters in addition to the data?

On page 146 of Gelman's Bayesian Data Analysis, Gelman discusses Bayesian p-value as a way to check the fit of the model. The idea is to compare the observed data ($y$) with data that could have been ...
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2k views

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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1answer
4k views

Unscented Kalman filter-negative covariance matrix

I have recently started working on the unscented Kalman filter. I coded the numerically stable version (i.e., square root Kalman filter) and used MATLAB for implementing. In the final update step, ...
7
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1answer
506 views

Hypergeometric: how do I construct a credibility interval around K (population successes) in R?

I have a problem for which I believe I should use the hypergeometric distribution, but I can't figure out how to do it in R. Say I have a bag of marbles with known number ($N$) of marbles, but the ...
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60 views

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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91 views

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 "Bayes' Theorem in the 21st Century". The article is quite short and can be found here: http://web.ipac....
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456 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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155 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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1k views

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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108 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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748 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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141 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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84 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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1k views

Combining posterior probabilities from multiple classifiers

I am new to machine learning and can't get my head around this problem. I have two patient datasets, the first ($D_1$) contains $Y,Z,X$ that convey blood-sample information and the second ($D_2$) ...
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1answer
5k views

What does “def” above an equals sign mean?

I am reading this: https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf and on equation (17), there is a def on top of the equal sign. What does this mean?
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29 views

Posterior variance vs variance of the posterior mean

This question is about the frequentist properties of Bayesian methods. Suppose we have data ${\bf y}$ generated from a distribution with a single parameter $\theta$, equipped with a prior $\pi(\...
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575 views

Bayes-Poincaré solution to the Behrens-Fisher problem 2: calculations for Jeffreys’ priors

In a previous post Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?, the classical Bayesian and likelihoodist solutions to 2-sample tests for comparison and the ...
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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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1answer
129 views

How does the marginal distribution become the prior distribution?

Bayes' Theorem for densities/pmf's states that, given, say, two univariate random variables $x,z$ we have $$p(z\mid x) = \frac {p(x\mid z)\cdot p(z)}{p(x)}$$ This is part of the core of the ...
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124 views

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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54 views

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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243 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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347 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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320 views

Prior for $\lambda$ is LASSO prior?

I have a regression model with regression coefficients $\beta_j$, $j=1,...,n$, and I would like to use a LASSO prior for $\beta_j$, this is: $$\beta_j \sim Laplace(0,1/\lambda),$$ where the Laplace ...
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164 views

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 ...
5
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1answer
96 views

Examples of usage of community of priors or why aren't they used more commonly?

Kass and Greenhouse (1989) proposed using "community of priors" (see also Fayers et al, 1997; 2000). As described by Spiegelhalter (2004), they can be seen as a range of viewpoints that should be ...
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175 views

Cox's Theorem: ignorance, objective priors, and the Mind Projection Fallacy

I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered ...
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2k views

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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458 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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461 views

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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386 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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249 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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1answer
141 views

How to do inference over two steps in a graphical model simultaneously?

I have observed data $D$ about a physical object described by $M$. I would like to determine the posterior distribution of $M$ given $D$, or $p(M|D)$. Now I can't infer this directly because unknown ...
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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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2answers
450 views

Credit Risk and Concentration

I am working with a UK credit-union and we are looking to build a model to assess our credit risk and changes to this over time. We have a number of loans to borrowers who each have a credit rating (...
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101 views

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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755 views

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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941 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 ...