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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Determining demographics by restaurant visits

Assume Person X likes to visit different restaurants at different times. We would like to determine the current estimate of their demographic distribution based on all prior information we know from ...
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Which model to predict air cleanness (air pollution) in daily-basis? [on hold]

How hard it is to predict air pollution? My friend is an agronomist: he is doing some research on some small plants. The plants are very sensitive to air pollution in urban areas [need deep ...
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Mean field variational inference

In Chris Bishop PRML book p.465 equation 10.6, the derivation doesn't explain why exactly the term $\int q_j ln(q_j) dz_j $ was generated, is not that term supposed to be multiplied by constant, did ...
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The econometrics of a Bayesian approach to event study methodology

Event studies are widespread in economics and finance to determine the effect of an event on a stock price, but they are almost always based on frequentist reasoning. An OLS regression -- over a ...
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Why Normalizing Factor is Required in Bayes Theorem?

Bayes theorem goes $$ P(\textrm{model}|\textrm{data}) = \frac{P(\textrm{model}) \times P(\textrm{data}|\textrm{model})}{P(\textrm{data})} $$ This is all fine. But, I've read somewhere: ...
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Recommendation: Fast algorithm for logistic random effects?

What is the fastest algorithm for fitting a simple logistic 'random effects' type model, with only one level of categorical predictors? Another way of putting it might be a logistic regression with ...
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41 views

Determining probability mass function (PMF) using Bayesian approach

For instance, a person is trying to determine the light intensity (unit: Candela) from a source that he knows must be coming from one of the 3 mediums (A, B, and C). From his experience he have ...
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51 views

What are the parameters of a Wishart-Wishart posterior?

When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$ \begin{align} \mathbf{x_i} &\sim ...
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Building Bayesian Models with insignificant independent variables

My main goal is to use four independent variables (y~x1,x2,x3,x4) and develop a Bayes Linear Regression model. However when I ran a simple linear regression model using these same four independent ...
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35 views

How to elicit prior distribution parameters?

A random sample of 300 women aged 60–69 years whose immediate families have had histories of cancer are to be screened for breast cancer. Let $y_i$ be 1 if woman i has a positive test, and 0 if not, ...
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Bayesian Correlation using PyMC [migrated]

I'm pretty new to PyMC, and I'm trying to implement a fairly simple Bayesian correlation model, as defined in chapter 5 of "Bayesian Cognitive Modeling: A Practical Course", which is as defined below: ...
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Marginal Likelihood from the Gibbs Output

I'm reproducing from scratch the results in Section 4.2.1 of Marginal Likelihood from the Gibbs Output Siddhartha Chib Journal of the American Statistical Association, Vol. 90, No. 432. (Dec., ...
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Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $. Over $ \mu $ I assume $ Gamma(\alpha_0, \beta_0) $ as a prior (where $ \alpha_0 $ and $ \beta_0 $ ...
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Why do we lose conjugacy when assuming unknown $\mu$ and unknown $\sigma^2$ in a normal distribution?

Model: The following model corresponds to samples drawn from a Gaussian distribution with unknown mean and unknown variance: \begin{align} x | \mu, \sigma^2 &\sim \mathcal{N}(\mu, \sigma^2 )\\ ...
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What exactly is a Bayesian model?

Can I call a model wherein Bayes' Theorem is used a "Bayesian model"? I am afraid such a definition might be too broad. So what exactly is a Bayesian model?
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Prior/Posterior predictive distributions

I have trouble understanding some notes I have from the lectures. We are in Bayesian linear regression and he explained how we can first introduce a prior probability distribution to the weights: ...
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1answer
105 views

Why the mixtures of conjugate priors is important?

I have a questions about the mixture of conjugate priors. I learnt and say the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such ...
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1answer
34 views

Probability of a model given an image

I would like to write the likelihood function for an image with respect to theoretically predicted values. Assuming uniform Gaussian noise, the pixels are statistically independent, and we can write a ...
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33 views

Proof that a density proportional to Gaussian is Gaussian [duplicate]

I try to develop Bayesian estimation for one dimensional Gaussian with unknown $\mu$ and known $\sigma$. I got \begin{align} p(x|D) &= \int p(x|\mu)p(\mu|D) d\mu \\ &=\int ...
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120 views

computing the posterior of two Gaussian probability distributions

I am a bit confused how to solve a Bayesian statistics problem. I have a parameter $\epsilon^s$ which is defined as following: $$\epsilon^s=\frac{\epsilon-g(\pi,z)}{1-g^*(\pi,z)\epsilon}$$ where ...
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31 views

Constructing Bayesian model for randomly picked points from a sine wave

I am trying to apply some data analysis on data which is generated by picking points from a sine wave with some noise added in. I am purposefully ignoring the time dependence, so just collecting data ...
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1answer
37 views

Bayesian Network or Logistic regression?

The Bayesian Networks and Logistic regression can be used to predict events or give to each customer the propensity to have a behavior. Which are the advantages or disadvantages of these 2 methods? ...
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Does data smoothness reduce the effective size of a dataset?

I would like perform de-blurring on a 3D data set where noise was added before the blur. The usual Bayesian way to do de-blurring (in the case of noise added after blurring) is first making a ...
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What would be an example of a really simple model with an intractable likelihood?

Approximate Bayesian computation is a really cool technique for fitting basically any stochastic model, intended for models where the likelihood is intractable (say, you can sample from the model if ...
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Does forward filtering result in OLS estimate for each time point?

I'm learning about dynamic linear models and was trying to think about the relationship between GLS and forward filtering (Kalman filtering where the state is the vector of parameters). Here's my ...
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2answers
167 views

How to calculate the Bayesian posterior analytically and by simulation?

I am working with this model: Prior: $P(\lambda)$~ N(0, 1), only the positive part likelihood: $P(x) = 1 - e^{-\lambda x}$ or $P(\vec{x}|\lambda)=\prod(1-e^{-\lambda x})$ Posterior: ...
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Explanation of an example of the Bayes Estimator

In section 4.4 of 'Introduction for Machine Learning' by Ethem Alpaydin the following example of estimating a prior density us given: For example let us say that we are told that [the random ...
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Incorporating Risk Aversion in Bayesian Expected Loss functions

In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory: $\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$ Where ...
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What is the meaning of the conditional $y|b$

I think I'm confused about a very simple thing. When we say that some variable is distributed as a Poisson distribution and we write $y \sim \text{Pois}(\lambda)$, is this the same that saying ...
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Bayesian optimization for known objective function with high dimension

I was wondering if one can use Bayesian Optimization algorithm for KNOWN, high-dimensional, expensive objective functions? If the answer is yes how efficient is that in terms of the quality of the ...
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1answer
33 views

What is the correct order of these hierarchical priors?

I'm quite new to Bayesian data analysis, so this is most likely am easy question. I have the following model: a function f has two exponential rate parameters $\lambda_1$ and $\lambda_2$ and for some ...
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2answers
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Direct parametrization of Cholesky decomposition of spatial covariance matrix

In spatial data analysis, a simple way to model the covariance stucture between spatial observations is via a covariance function like $cov(y_i,y_j) = C e^{-rD_{ij}}$, based on some (euclidean) ...
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46 views

Bayesian linear regression with continuous and binary covariates

I am interested in learning more about applying Bayesian linear models for covariates some of which are continuous and some are binary. What is the appropriate terminology for such models so that I ...
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35 views

Posterior probabilities with decision trees or decision forests

Is there a way to get posterior probabilities $P(C | \vec{x})$ (probability that a data item $\vec{x}$ belong to one of the given classes) in a multiclass classification problem using decision trees ...
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33 views

What convergence diagnostics are appropriate for a Bayesian hierarchical logistic regression model?

Using WinBUGS, I fit several Bayesian hierarchical logistic regression models for the mean of a binary response variable conditional on a set of criteria. I am now using CODA in R to determine if my ...
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Group mean versus subject means in hierarchical bayesian analysis

I'm using stan in R to estimate several parameters of interest. Doing so requires using hierarchical models. In some basic testing of what I'm trying to do, I got results where the estimates of all ...
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How to compare predictions from MLE-based regression Vs. predictions from bayesian regression?

Say I have two linear regression models that I want to use for predictions. Linear regression: \begin{equation} \mathbf{y} \sim \mathcal{N}(\mathbf{X^Tb}, \Sigma_y) \end{equation} Bayesian linear ...
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1answer
119 views

Convergence of likelihood implying convergence of marginal likelihood?

I will ask my question through a toy motivating example. It is well known that a Poisson process is the continuous time analog to a Bernoulli process (for example: ...
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2answers
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What is going on in an MCMC chain?

I'm currently running some data through the MCMCglmm package in R. I was wandering what actually happens in the chain that is created? I have a multiresponse model (as a quick overview of the data: ...
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Do Bayesians accept Kolmogorov axioms?

Usually probability theory is taught with Kolgomorov axioms. Do Bayesians also accept Kolmogorov axioms?
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36 views

Clarification on a paper regarding estimating N from a Binomial Distribution

I was wondering if someone could clarify the following for me. In the paper "Inference for the binomial $N$ parameter" by Adrian Raftery, his first example outlines the posterior of $N$ given $x$ as ...
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With posterior inclusion probability how do I settle on the final predictive model?

After using the spike-and-slab prior to perform Bayesian model selection, I get the posterior distribution of my variables, from which I calculate the inclusion probability for each variable. How do ...
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Is MLE with regularization a bayesian method?

It is usually said that priors on bayesian statistics can be regarded as regularization factors since they penalize solutions where the prior places low density of probability. Then, given this ...
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25 views

Bayesian Logistic Regression Likelihood Computation for Binary Data

For computing the likelihood function for a Bayesian Logistic Regression model, I understand the original form to be $$p(y_i|\beta,X) = \prod_{i=1}^{k}\left( \frac{ \text{exp} \{ \beta^{\prime} x_i ...
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likelihood in bayesian linear regression

I was going through the derivation for the likelihood of Bayesian linear regression http://en.wikipedia.org/wiki/Bayesian_linear_regression#Posterior_distribution I did not understand this step where ...
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1answer
25 views

Mean square error for Bayesian estimate

I am trying to work on Bayesian linear regression. i have Classical and Bayesian regression estimates, now i want to find the Mean square error (MSE) for both approaches. Is the formula to find MSE ...
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1answer
45 views

Skewed posterior distribution on constrained parameter space for Bayesian inference of MCMC. Advice on what to do?

I am running a fully Bayesian MCMC procedure to estimate some time series models, and my model has a lot of parameter estimates. In particular, one of these parameters, $\phi$, is $\in [-1,1]$. The ...
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1answer
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Posterior of a simple Bayes linear regression

In the context of simple linear Bayesian regression, why or when is it appropriate to define the posterior as $p(\beta, \sigma^2|y)$ and not $p(\beta, \mu|y)$?
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find posterior distribution parameters by simulation

I've calculated the posterior distribution parameters of a variable X analytically and by simulation. But doesn't mach. X ~ Normal(mu,s=6).And the prior distribution of X is a Normal(mu=100,s=20). As ...
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Best linear unbiased estimator

I have a sample of N stocks. I have the following information: For each stock i, I have an estimate of variance (of returns) $\hat{\sigma}^2_{i}$. I also have a fitted variance, denoted by ...