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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Hierarchical Bayesian model with heterogenous errors

I have an experiment where I repeatedly show subjects two lights, and I ask which light is brighter. I am interested in whether error rates decrease over time, holding all else constant. I also ...
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What is the predictive distribution of Bayesian supervised Learning? (rigorous argument)

I was trying to understand the posterior predictive distribution for any supervised predictor (by that I mean any classifier or regression predictor $f$). The exact equation I am unsure of is: $$ ...
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In Bayesian analysis, how to sample from full conditional given uniform prior and normal data likelihood?

In Bayesian analysis, assume a simple linear regression model with two straight lines that meet at a certain changepoint $c$. The basic setup is as following. \begin{align*} Y_i \ & \sim \ ...
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Setting up posterior and likelihood of Bayesian for more than one model

If I have a data-set and I would like to fit a model and determine its two or three free parameters, while I know that I can fit twice or three times the model to my data and obtains the free ...
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OpenBUGS example: Stagnant, a changepoint problem and an illustration of how NOT to do MCMC! - Why is the second parameterization better?

I am working on an Bayesian problem from an OpenBugs example: Stagnant, a changepoint problem and an illustration of how NOT to do MCMC!. This is a changepoint problem. Basically we assume a model ...
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In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer? [duplicate]

I am working on a Bayesian project based on Stagnant data from a OpenBugs example, which is a changepoint problem. Basically we assume a model with two straight lines that meet at a certain ...
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2answers
16 views

Bayesian AB testing with time lag

I'm creating an AB testing framework using Bayesian methods. It's a conversion based test, so users land on the site, randomly get assigned one of two experiences (i.e. group A or group B) and then ...
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Update (Bayesian?) selection probability based on rankings

I am looking for a framework to help solve a prioritisation/selection problem but am not sure what the technical name for this type of problem would be (knowledgable edits welcome!). Main problem ...
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20 views

Meaning of the prior and loss parameters in rpart in R

Could someone please explain to me what specifying priors and/or loss parameters in R's rpart actually do? I found R's documentation completely unhelpful. For example, let's suppose I have a highly ...
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1answer
240 views

what do we mean by hyperparameters? [duplicate]

Can anyone give me full details about what we mean by hyperparameters, and what in the Dirichlet distribution are called hyperparameters? A practice example for the estimation of those parameters ...
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40 views

How to find Bayesian average

I am trying to understand Bayesian concept. Apparently, the final value is dependent on prior estimate of value. So, for the simplest situation of finding average of following series of numbers, how ...
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full conditional posteriors for bayesian lasso

I am reading the original Bayesian Lasso paper, and its follow up; They look straightforward to implement, mainly because of the conditional posterior probability for the gibbs sampler; however, I ...
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1answer
31 views

Finding covariance matrix for weight priors for bayesian regression with feature space mapping of inputs

I want to implement Bayesian regression which returns the MAP estimate for given aggregation of columns of design matrix mapped into the feature space $\Phi(X)$, responses as a column matrix $y$ ...
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1answer
12 views

Probability of binary outcome based on observed values of correlated variable

How should one approach the following problem? Suppose an object has an unknown binary attribute X in {0, 1} (for example it is only possible to be either ...
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14 views

WinBUGS/JAGS code for calculating Bayesian p-value from negative binomial model

I have a working negative binomial model written in BUGS code, but am not sure about the appropriate Bayesian p-value code to test goodness of fit. Specifically, I would like to calculate Pearson's ...
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1answer
71 views

How to respond to reviewers asking for p-values in bayesian multilevel model?

We were asked by a reviewer to provide p-values as to better understand the model estimates in our bayesian multilevel model. The model is a typical model of multiple observations per participant in ...
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1answer
50 views

Introductory examples in computational statistics class

I'm looking for an example of Bayesian inference for a class with the properties: The problem is easy to state, and the model & prior are both pretty reasonable, and R can't really calculate the ...
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28 views

Posterior Conditional on Beta in Bayesian Linear Regression with Factor Analysis

This should be an easy question if you're familiar with the terms involved. I am performing some research using a hierarchical Bayesian regression model that incorporates factor analysis into the Beta ...
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1answer
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Representing a Suite of Hypothesis with Stan

After having read the excellent Think Bayes from Allen Downey, I'm now diving deeper into Bayesian Analysis and learning MCMC with Stan. The dice problem in Think Bayes goes like this: Suppose I ...
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1answer
32 views

Prediction uncertainty intervals for predictions of machine learning algorithms

Assume I have a regression problem. I fit models on a train data set and tune their hyperparameters using CV. I then run the models on the test set. What is the best way to calculate prediction ...
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24 views

Stochastic Block Model Priors

In the generic stochastic block model (binary edge data, no degree correction, etc.), if an uninformed prior is used for the Bernoulli coefficients i.e. Beta with $(a,b) = (1,1)$, will the model ...
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46 views

Evaluating integral to obtain marginal PDF related to Tikhonov Regularization

I am attempting to derive the marginal PDF for an application of the Gibbs Sampler. My joint PDF contains: $P(b,x) = \frac{1}{\sigma^{n}}\exp \left( -\frac{1}{2\sigma^2}\left\lVert ...
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2answers
955 views

Priors in Bayesian MCMC

I am trying to understand how the choice of priors affects a Bayesian model estimated using MCMC. At a basic level I understand that the product of the prior and the likelihood are proportional to the ...
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What are good values for autocorrelation, Gelman, and cross-correlation in rjags?

I don't want to post my whole code since it is long, so I will only post part of it: ...
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Estimating bias in surveys

Say a company runs a survey across random N cities independently in some country estimating the fraction of males and females on each city. E.g.: Males = $X_1$% ...
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1answer
31 views

Prediction based on bayesian model

I have created a bayesian model that estimates 6 parameters using rjags from R. Now i want to do some predictions based on new data in R. Can anyone help me with an example. ...
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1answer
28 views

Bayesian Mixture Model Gibbs Sampler for two linear relationships

I am attempting to use a Gibbs Sampler to model a mixture of two groups, where the group membership is defined by a linear relationship conditional on x. Both groups have the same slope and intercept, ...
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15 views

Jeffrey's Prior for normal distribution with mean = 0

How would I go about calculating Jeffrey's Prior for a normal distribution with mean = 0, So far I get: But then don't know where to go next. Any help much appreciated
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What are some advanced algorithms in bayesian networks? [closed]

What are some advanced algorithms in bayesian networks? I am familiar with the conventional algorithms of network construction and inference in bayesian networks. What are some algorithms that provide ...
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How does one use Bayes theorem with a continuous prior?

If my prior is modelled as a continuous probability distribution, say, a beta distribution skewed to reflect my bias towards certain models, how can I calculate the posterior probability? The ...
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20 views

Likelihood of hypothesis in live data

Bayes rule is $P(H|E)=\frac{P(H)P(E|H)}{P(E)}$ I have a prior distribution from categorical data prior={'a':0.2,'b':0.6,'c':0.1,'d':0.1} Which forms my ...
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Posterior predictive for Gamma distribution with unknown scale and shape

I have a question that needs clarification. The posterior predictive distribution can be described as the distribution that a new i.i.d. data point $\tilde{x}$ would have, given a set of $N$ existing ...
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1answer
19 views

Log likelihood for inverse gamma

For a gamma distribution, the answer to this question shows that you can just use the log of the gamma distribution density function. Is the same true for inverse gamma? It is the same as the log of ...
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When should I be worried about the Jeffreys-Lindley paradox in Bayesian model choice?

I am considering a large (but finite) space of models of varying complexity which I explore using RJMCMC. The prior on the parameter vector for each model is fairly informative. In what cases (if ...
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2answers
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Marginal likelihood vs. prior predictive probability

In the Bayesian framework, to me, it seems that the marginal likelihood and the prior predictive distribution/probability are equal. Is that the case? Or maybe this just holds for single data points? ...
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Bayesian Risk and Subjectivity

I am studying the differences in bayesian and frequentist approaches to point estimation. I understand that there are objective and subjective approaches to Bayesian and some people don't like the ...
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Bayesian Analysis of Box-Cox Transformation

This problem is problem 5 in Chapter 7 of Bayesian Data Analysis, 3rd edition. Consider the Box-Cox transformation: $y_i^{(\lambda)} \sim \mathcal{N}(\mu, \sigma^2)$ where $y_i^{(\lambda)} = ...
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1answer
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Example Bayesian resolution of the Two Envelopes Problem [closed]

What is a concrete example of a Bayesian resolution to the Two Envelopes Problem?
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Bayesian Probabilistic Matrix Factorization (BPMF) with PyMC3: PositiveDefiniteError using `NUTS` [migrated]

I've implemented the Bayesian Probabilistic Matrix Factorization algorithm using pymc3 in Python. I also implemented it's precursor, Probabilistic Matrix ...
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Is this how a Bayesian bootstrap works?

I am a bit new to the whole nonparametric and Bayesian idea, so tell me if this is correct: to estimate, say, the mean of a dataset's population we do the following: We define a function $f(x)$ that ...
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1answer
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JAGS Error: Invalid Parent Values on last observation

I am using R2jags to fit a model in R using JAGS. Here is my code: ...
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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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1answer
102 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
48 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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1answer
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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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24 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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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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33 views