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

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Identify distribution change [on hold]

I have a categorical variable Product that can have one of $4$ possible values, ${x_1,x_2,x_3,x_4}$ The current distribution is ...
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28 views

Kernel of a Normal Distribution

From Wikipedia , The kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the ...
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sampling from distribution [duplicate]

In Monte Carlo Markov chain (Gibbs or Metropolis-hastings) samples are drawn from posterior distribution. In layman terms, how sampling is done from a distribution?
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HELP: Bayesian Multi-level model with seasonality

I am trying to define a Bayesian Multi-level model which has seasonality in BUGs. I have defined the model (below).I have attached a graphical representation of what im trying to model. eventually ...
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37 views

Bayesian meta analysis: implementation in BUGS/JAGS/STAN

I would like to conduct a meta analysis in order to collate the information from a number of studies. The parameter of interest is a probability $\theta$. In each of the studies, the observed data ...
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1answer
33 views

Neural network & Bayesian in this machine learning algorithm

I am new to machine learning etc and found this comprehensive algorithm: http://scikit-learn.org/stable/tutorial/machine_learning_map/ . However, I am not able to make out any reference to neural ...
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8 views

R package for quadratic minimization Subject to a Nonlinear Constraint [on hold]

I search for a R-package for quadratic minimization Subject to a Nonlinear Constraint. Do you know one of such packages? Thanks.
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1answer
37 views

PyMC3 Implementation of Probabilistic Matrix Factorization (PMF): MAP produces all 0s

I've started working with pymc3 over the past few days, and after getting a feel for the basics, I've tried implementing the Probabilistic Matrix Factorization model. For validation, I use a subset ...
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19 views

Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline. With that assumption, the mean and trend ...
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19 views

EM algorithm: With prior vs. not prior

I have a working EM algorithm without prior. I am asking for some advice on how to add prior on latent variables. Define: $t_i \in \{ +1, -1 \} $: variables of interest to be predicted $p_j \in ...
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Multipliers on Top of Binomial Rate Estimates

I was wondering if anyone has come across a similar question to the following. I have data of the form $s_{x,y}, t_{x,y}$ (successes and trials) for varying groups with $x \in X, y \in Y$. I also ...
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1answer
41 views

Construct the likelihood with asymmetric uncertainties

I want to study the correlation between 2 parameters, this is done by fitting a straight line. I have uncertainties on both parameters. I want to solve my problem using the Bayesian approach, i.e. I ...
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26 views

Bayesian Monte Carlo modeling and selecting priors [duplicate]

Could anyone recommend some not-too-mathy introductory texts to Bayesian regression and Monte Carlo modeling? I am neither a statistician nor an econometrician. The frequentist perspective makes ...
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1answer
49 views

Evaluate posterior predictive distribution in Bayesian linear regression

I'm confused on how to evaluate the posterior predictive distribution for Bayesian linear regression, past the basic case described here on page 3, and copied below. $$ p(\tilde y \mid y) = \int ...
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1answer
218 views

Can a Multinomial(1/n, …, 1/n) be characterized as a discretized Dirichlet(1, .., 1)?

So this question is slightly messy, but I'll include colourful graphs to make up for that! First the Background then the Question(s). Background Say you have a $n$-dimensional multinomial ...
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1answer
27 views

D-separation in a Bayesian Network [closed]

The above question asks to see if Radio is D-Separated from Petrol given certain evidence. For evidence (i), why would this mean D-Separation? If Battery is true, we have a inactive triple. If ...
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23 views

Questions about Linear regression vs bayesian way of parameter estimation

I have several questions about the difference between linear regression and bayesian way of parameter estimation using MCMC. I know their difference in mathematical stuffs and I can implement them ...
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14 views

Bayesian Inference for Poisson process 2

How do I calculate Bayesian posterior distribution of Poisson likelihood function with Pareto prior distribution?
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21 views

Expected ratio of probabilities--is there a term for it?

I recently came across the following quantity when I played around with some information theoretic quantities and Bayesian learning. Given three probability distributions $q(z), p(z)$ and $p(z|x)$. ...
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1answer
36 views

Why do we use Gamma($\epsilon, \epsilon$) as non-informative prior for precision and Normal prior for betas in Linear Regression

Suppose my regression model is $$Y_i = \beta_0 + \beta_1X_{i1} + \epsilon_i $$ In most books I am seeing that the prior used for precision $\tau = 1/\sigma^2 $ is $Gamma(\epsilon, \epsilon)$. However ...
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1answer
46 views

How to assess if a model is good in multinomial logistic regression?

I have some ordinal response $y$ that I modeled using both ordinal logistic regression and multinomial logistic regression (to avoid the proportional odds assumption), using two continuous variables ...
2
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1answer
35 views

Translating a bayesian model from MCMCpack (hregress) to JAGS

I am trying to convert a hierarchical model that currently works with the R package MCMCpack, into a JAGS version. I have experience working with regular models in JAGS, but not with hierarchical ...
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1answer
45 views

Estimating mean of Normal with unknown variance and then predict the future observation

I am trying to estimate population mean of 9 observations when the variance is unknown. I marginalized the posterior and understand that the t- distribution would give me the distribution of ...
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29 views

Bayes estimator, poisson distribution with exponential prior

If X ~ Poisson($\theta$) and $\pi(\theta)$~$exp(1)$(prior). Find the Bayes estimator for $P_\theta(X=0)$ with respect to quadratic loss $f(\theta|x)=\frac{e^{-n\theta}\theta^{\sum x_i}}{\prod x_i}$ ...
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18 views

Bayesian process with noise?

Does anyone know of any research considering Dirichlet processes (or other Bayesian nonparametric models), where sample points have a known gaussian noise attached to their input? I.e. I have a set ...
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How to derive type-2 maximum likelihood method for RVM regression

According to PRML book(7.85,7.86,exercise 7.12), the marginal likelihood for RVM regression is $$ \ln p(y|X,\alpha,\beta)=−1/2\{N\ln2π + \ln|C| + y^TC^{−1}y\} $$ $$ A=diag\{\alpha_1,..,\alpha_D\} $$ ...
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BIC for Bayesian ANOVA

I am doing a Bayesian ANOVA as follows: BIC0 = -2 * logLik0 + k0 * log(N) # null hypothesis BIC BIC1 = -2 * logLik1 + k1 * log(N) # alternate hypothesis BIC ...
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1answer
23 views

“Multiple definitions of node p[1]” Error using WinBUGS [closed]

I have written the following code in WinBUGS and every time I try to compile the data after loading in the data I get the same error which is "Multiple definitions of node ...
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1answer
233 views

calculation threshold for minimum risk classifier?

Suppose Two Class $C_1$ and $C_2$ has an attribute $x$ and has distribution $ \cal{N} (0, 0.5)$ and $ \cal{N} (1, 0.5)$. if we have equal prior $P(C_1)=P(C_2)=0.5$ for following cost matrix: $L= ...
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A question on Bayesian updating (Han Solo and the Bayesian priors)

http://www.statslife.org.uk/the-statistics-dictionary/2148-han-solo-and-bayesian-priors in the preceding article, there is a bit of Bayesian updating that does not make sense to me. We have a prior ...
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14 views

Matching a query distribution to a family of template distributions

I was turning over a hypothetical question in my head: Suppose I have a set of template probability distributions, let's say each giving the probability of the occurrence of certain objects like ...
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49 views

Bayesian probability, verifying stupid examples

A wikipedia article on Bayesian inference offers an informal example in which a friend has had a baby, about which I know nothing more, and a (presumably reliable) mutual friend shows me a picture of ...
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40 views

Gibbs Sampler - Sample mean convergence

To simulate from the posterior distribution $p(\theta|Y)$ where $\theta = (\mu,\lambda_1,\lambda_2)$, I run a Gibbs sampler to draw approximately random values from $p(\theta|Y)$. This Gibbs sampler ...
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1answer
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Gibbs Sampler output: how many Markov chains?

When running a Gibbs sampler (for $n=200$ Iterations) with two full conditionals, I get the output $\mathbf{x} = (x_1^{(n)},x_2^{(n)})_{n =1,...,200}$. So $\mathbf{x}$ is the realizations of a Gibbs ...
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Multi-Level (Sparse) A/B Testing

I've been reading some articles about Bayesian A/B testing such as: http://engineering.richrelevance.com/bayesian-ab-tests/ but my application requires a framework for handling sparsity and group ...
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19 views

Calculating Marginal Data Density for VAR Model

I am currently estimating Bayesian vector autoregressive (BVAR) models and I would like to do model comparison with Bayes factors. I have read about the Gelfand-Day method, the Geweke (1999) modified ...
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1answer
33 views

Full conditionals - Gibbs Sampler

i want to draw samples from a 5-dimensional posterior distribution $f(k,\theta,\lambda,b_1,b_2|Y=y)$. From Bayes-Theorem there is the following relationship between posterior and likelihood: ...
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1answer
58 views

MCMC/EM limitations? MCMC over EM?

I am currently learning hierarchical Bayesian models using JAGS from R, and also pymc using Python ("Bayesian Methods for Hackers"). I can get some intuition from this post: "you will end up with a ...
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1answer
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Implementing Gibbs sampler in R from posterior distribution

I am referencing a follow-up idea from something I posted earlier (Zero-inflated Poisson and Gibbs sampling, proofs and sampling). I want to implement the Gibbs sampler, by generating a large ...
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Bayesian Inference applied to a time series

I am trying to better understand how Bayesian Inference can be used. Let's assume I am measuring a given property, $\theta$, of some material. I have done 100 measurements. These had a Gaussian ...
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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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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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Bayesian inference with sampling and mixture models

I'm having some trouble doing Bayesian inference on an experience I have in hands. I apologize in advance if it is too complex, but I couldn't find a trivial way to split it in several parts. Let ...
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1answer
36 views

When is mu_a used in this STAN example?

I'm looking at an example of a random effects model with 2 random effects fit by Peter Li demonstrating how get models fit in lmer into STAN. The code for this and the accompanying data are stored ...
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1answer
43 views

Log likelihood function for binary classification

I need help with this following task. There is a binary classification problem where each observation xn is belong to one of two classes (t = 0 and t = 1). The training data points are sometimes ...
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Predicting with Posterior PMF(discrete non-pram dist) ? Predictive posterior distribution?

Sorry in advance if my question is abit awkward, I'm somewhat confused because most of the tutorials on the Internet mention that you should use Posterior-predictive dist to predict new data. The ...
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Why does adding a lag effect increase mean deviance in a Bayesian hierarchical model?

Background: I'm currently doing some work comparing various Bayesian hierarchical models. The data $y_{ij}$ are numeric measures of well-being for participant $i$ and time $j$. I have around 1000 ...
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Factorizable time evolution in a dynamic stochastic process

I have a stationary dynamic system which at each given time $t$ is in state $x_t \in \mathcal{X}$. The set of states $\mathcal{X}$ is assumed to be finite but too large to be enumerated by a practical ...
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Cluster Assignment in Bayesian perspective

I am going to study clustering methods in the Bayesian perspective. I understood how k-means works, and I found it pretty clear, due to the notion of distance and assignments to specific centers. I ...