In Bayesian statistics a prior distribution formalizes information or knowledge (often subjective), available before a sample is seen, in the form of a probability distribution. A distribution with large spread is used when little is known about the parameter(s), while a more narrow prior ...

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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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9 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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26 views

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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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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36 views

Return value of uniform distributions for MCMC simulations

I am confused about how what value should be returned from a uniform distribution when using MCMC simulations. The proper normal distribution is define as $$ p(\theta) = \left\{ \begin{array}{cc} ...
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24 views

Estimating von Mises Parameters for Angular Data

I want to model some angular data. Any input on how to incorporate the von Mises distribution and suggestions on appropriate priors in RJAGS for von Mises mean and concentration would be greatly ...
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15 views

Augmenting Kalman filter with parameter — what does the initial value mean?

It is a fairly standard trick to augment a Kalman Filter with unknown parameters and to propagate them forth with zero error to estimate them. I was wondering if anyone could tell me what the ...
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62 views

How to use information about likelihood of classes in a classifier?

General question: How can information about the likelihood of classes be used to improve a classifier? Suppose that the probability of each class is known quite precisely (from a very large sample), ...
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95 views

Is a spike-and-slab prior a proper prior?

Is a spike and slab prior a proper prior? (I am talking about a (product of Bernoulli) spike and Normal slab) If not, does it still lead to a proper posterior?
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20 views

Posterior predicted distribution, practical question

I'm new here to this place but I have already learned so much here. Yet I still remain with a large question involving my thesis in econometrics and medical scoence. For a starter, I have read ...
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27 views

Why using vague (or noninformative) priors? [duplicate]

In my Bayesian class, we are always required to specify vague (or noninformative) priors for bayesian modeling. I am quite confused about that. If I understand correctly, the main advantage of ...
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21 views

Variance of multinomial distribution that is product of 4 Beta random variables

I have a system of 4 binary random variables, $A$, $B$, $C$ and $D$. $A$, $B$ and $C$ are conditionally independent given $D$, and I'll call one set of samples $ABCD$ an event (e.g. $ABCD$ meaning all ...
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93 views

Can my Bayesian prior reflect what the data should say rather than what it could say?

Can my Bayesian prior reflect what the data should say rather than what it could say? For example, assume I collect data where $Y_i$ is whether or not student $i$ passed the test and $X_i$ is whether ...
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17 views

Prior for gamma distribution in “mean form”

I need to specify priors for the parameters of a gamma distribution. Normally the gamma distribution is parametrized in either the "rate-form'': ...
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19 views

Do mildly informative prior distributions tend to mitigate false positives (i.e. Type I error rates)?

I am curious if others have sources that speak to the matter that providing informative and/or mildly informative prior distributions on a parameter tend to mitigate false alarm rates? I know from the ...
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27 views

Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then ...
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72 views

How to select hyperprior distribution for Beta distribution parameter?

I have a parameter $\theta$ whose value should lie between $(0,1)$. Therefore, I am assuming the prior distribution of $\theta$ to be a beta distribution with hyper-priors $\alpha$ and $\beta$ ie. ...
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21 views

prior for integer-valued random variable taking values 1 or greater

In my model I have an integer-valued random variable which should only take values one or greater. I would like to specify an appropriate prior for this which has most of the mass say around 1 to 5 ...
3
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43 views

Is it possible to define the mean of a varying distribution?

Suppose $(p_1,\ldots,p_k)$ be the vector of multinomial parameters and $$(p_1,\ldots,p_k)\sim \mbox{Dirichlet}(\alpha_1,\ldots,\alpha_k).$$ Let's define a function $f(p_1,\ldots,p_k) \in \mathbb{R}$. ...
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40 views

How is data generated in the Bayesian framework and what is the nature on the parameter that generates the data?

I was trying to re-learn Bayesian statistics (every time I thought I finally got it, something else pops out that I didn't consider earlier....) but it wasn't clear (to me) what the data generation ...
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21 views

Why does Empirical Bayes work in my simple case?

I have a problem where I am trying to classify data into two groups using a single parameter. The distribution of this parameter is Gaussian for two groups, so what I'm dealing with is two overlapping ...
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11 views

Creating a model for a webshop

I'm going to create a Multi-armed bandit algorithm to handle recommendations for a large scale webshop. I'm going to use Thompson sampling (http://en.wikipedia.org/wiki/Thompson_sampling) and would ...
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47 views

Prior for the coefficients of a linear regression model

I have a linear regression model $\bf Y=\bf{X}\bf{\beta}+\epsilon$. I want to assign a prior on $\bf\beta$ in order to derive the posterior predictive model $p(y_{predictive}|\bf{y},\bf{X},\beta)$. ...
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25 views

Iteratively solving for prior probabilites.

I'm using Bayes theorem to classify data into two groups, where the conditional probability is known but the prior is not. So I assume that the ratio of prior probabilities is 1 and calculate the ...
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12 views

How can i get prior information using my few data set from the whole data? [duplicate]

I have a data set (x1...x500, y1....y500 ) I want to know about bayesian regression I want to know the prior information , few data set(400) from the whole data (500) using MCMCregress( packages in ...
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18 views

Is this notation for the improper uniform prior correct?

Can I write: $\mu \sim U(0,\infty)$ ? Or do I have to use the notation $p(\mu) \propto 1$? Thank you.
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Can improper priors be implemented in some way?

I'm new to bayesian inference. I've just discovered that improper priors can't be specified in WinBUGS/OpenBUGS. I was wondering if this is common or not in bayesian inference. Are there same cases in ...
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19 views

Definition of weakly informative prior [duplicate]

According to Gelman, a weakly informative prior is defined in the following way: We characterize a prior distribution as weakly informative if it is proper but is set up so that the information ...
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41 views

Doubt about conditional conjugate priors

I've just read the definition of conditional conjugate prior in this discussion but I have still some doubts. According to the definition given, it seems that the prior distribution of $\theta$, ...
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80 views

Bayes Linear regression- logarithmic transformation of prior distribution of the variance

I have a Bayesian version of a linear regression with 3 covariates. The model is given by \begin{align*} Y\sim N(\mu,\tau)\end{align*} \begin{align*} \mu=\alpha + \sum\beta_{i}x_{i}\end{align*} where ...
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31 views

Explanation that the prior predictive (marginal) distribution follows from prior and sampling distributions

While I have a vague intuition that this makes sense, I am interested in the formal demonstration that the prior predictive distribution in Bayesian inference is equal to the integral over $\theta$ of ...
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83 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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24 views

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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69 views

define prior probabilities in naive bayes with unbalanced classes and asymetric cost

I'm trying to apply Naive bayes to the following supervised problem: It's a binary classification problem The classes are unbalanced. The target class represents the 0.004266432 of the total and the ...
3
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39 views

What are examples of “flat priors”?

For example, for p as the parameter to a binomial or bernoulli, or a Poisson, what would a flat prior p be? What does it mean to be "flat" - does this refer to diffuse?
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87 views

Truncated Von Mises-Fisher distribution

I am putting a von Mises-Fisher prior on my data. The data does lie on a unit sphere, but the only problem is that my data is always positive. So I feel like I am wasting my prior on unnecessary ...
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33 views

p-values, prior probabilities

I've got a set of N normal independent normal distributions, each representing a signal. I also got a new data sample, a vector $v$ of size Nx1. Now let's say I compute the p-value using the ...
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35 views

Constructing gamma prior from Poisson

So if we have a Poisson distribution with a rate lambda we know that the prior is a gamma with alpha,beta. But suppose we didn't know that the prior was a gamma. How would we do the derivation please? ...
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13 views

The validity of using truncated PDFs as prior distributions?

I am trying to implement an ABC (Approximate Bayesian Computation) rejection-sampling algorithm in R. I am currently working with a six-parameter model and for each of the parameters I have specified ...
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79 views

Interpretation of priors in example

Suppose you have 3 variances $W_{1},W_{2},W_{3}$ that can be expressed as $W_{j}=q_{j}V$ with $j = 1,2,3$. According to one model, $W_{3}$ should be pronounced and $W_{1}$, $W_{2}$ should be small to ...
4
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119 views

Bootstrapping the data to set up a prior

I am using a Gaussian model with a conjugate Normal-Inverse-Wishart (NIW) prior, as described here. The advantage of this approach is that the marginal likelihood $p(y)$, which is what I am interested ...
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1answer
170 views

Undefined real result error at WinBUGS

I am currently working on my thesis and interested in estimating a multilevel differential item functioning model and I using at WinBUGS. Until I had done model check-up, there are no errors. However, ...
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18 views

Combining two estimates of p in a binomial estimation

I have an estimation problem for a binomial data. I got a sample and from that I can get an estimation. But I also have a kind of prior information about the p. But mind it, this prior is just a ...
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41 views

How can I use ratios to set priors on multinomial probabilities?

I have a vector, $k$, that determines allocation to five pools. I'd like to set priors on these probabilities, and I can provide informative priors on a few of the ratios, e.g.: $$ \frac{k1}{k2} ...
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31 views

Picking noninformative priors using pivotal quantities

In 'Bayesian Data Analysis' (Gelman, Carlin, Stern and Rubin) on page 64 it reads: "If the density of $y$ is such that $p(y-\theta|\theta)$ is a function that is free of $\theta$ and $y$, say $f(u)$ ...
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Selecting priors based on measurement error

How do you calculate the appropriate prior if you have the measurement error of an instrument? This paragraph is from Cressie's book "Statistics for Spatio-Temporal Data": It is often the case ...
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Covariance for a multivariate Bayesian Additive Regression Tree

Chipman, George, and McCullogh (2010) state that: One can also extend the sum-of-trees model to a multivariate framework such as: $$ (29) \qquad\qquad Y_i = h_i\left( x_i \right) + ...
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37 views

Empirical Bayes vs “non-informative” priors

I am familiar with the mechanics with both methods, but don't know what factors I should consider when choosing between these two approaches for adjusting a prior. I would imagine that, on a case by ...
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79 views

How does one interpret the distribution over parameters in bayesian estimation?

I am new to Bayesian estimation. The assumption that the parameters are random variables seems a little unsettling to me. For example when considering a model for data, what physical interpretation ...
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114 views

What is the mathematical difference between using a un-informative prior and a frequentist approach?

Un-informative priors are preferred in instances where bias is not acceptable (ie. courtrooms, etc.) However, it seems to me that it would just make sense to use a frequentist approach instead. Why ...