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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Hyper-prior for negative binomial in hierarchical model using JAGS/BUGS

Below I'm using a negative binomial because it is more flexible than a simple poisson model. The data are counts $y$ of events for 16 individuals $x$. There are 14 counts (i.e. counting periods) for ...
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24 views

Univariate priors for the parameters of a Beta distribution

I need a rather a prior on the parameters of a Beta distribution (i.e. $\alpha$ and $\beta$). I have an external constraint that requires me to use univariate priors, one for $\alpha$ and one ...
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7 views

Applying Beta distribution for calculation latent variables

I would like to find the probability distribution function for the below scenario,its similar to Computer Adaptive technique (IRT) I need to estimate the ability of a student from answered questions ...
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30 views

Non-informative prior for Poisson/gamma density

In the Albert book on Bayesian computation with R, exercise 4.8.5 (p.83), it is suggested to use $$ p(a, b) \sim (a \times b)^{-2} $$ as the non-informative prior for the Poisson/Gamma model: $$ ...
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23 views

Calculating posterior and prior odds

Question: Now, I'm confused about assigning probabilities here. I find $P(A^c|E) = (.001)(.99) = .00099$ and $P(E|A) = .99$, but what about the first two sentences? Does that mean that $P(E) = .001$ ...
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59 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. $\text{Data} \sim \text{Gamma} (\alpha, \beta)$ Parameters $\alpha \sim \text{Gamma} ...
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18 views

Marginal Likelihood Prior

I have a model with probability matrix for a distribution of $x$, $y=\{0,1\}$, $p(x,y|w)$ where $w=[w_1,w_2,w_3,w_4]$ $p(x=0,y=0)=w_1$, $p(x=0,y=1)=w_2$ $p(x=1,y=0)=w_3$, ...
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45 views

How can a uniform prior make the posterior mean different from the MLE?

I read the following in Machine Learning: A Probabilistic Perspective: How can a uniform prior move the posterior mean? Isn't a uniform distribution supposed to not bias the result? Are there any ...
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76 views

Log odds as prior

I have this problem: In betting situations one is often interested in odds, referring in the thumbtack tossing $\theta / (1 - \theta)$. Alternatively one may consider the log-odds: ...
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74 views

Prior Gamma distribution: Select appropriate alpha given beta and median

I am trying to programatically select a prior distribution from the Gamma family of distributions. The primary criteria that I need to satisfy is that the median of the distribution should be a given ...
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32 views

Prior predictive with discrete prior

I'm working with a uniform distribution as a prior, defined as: $\pi(\theta) = \begin{cases} \frac{1}{7} & \text{if } \theta\in\{0,\frac{1}{6},\frac{2}{6},\ldots,1\} \\ 0 & ...
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36 views

Probabilistic modelling MCMC question with pyMC

This is my first post and I am a newby in pymc. I am trying to model a non-linear system (see below for a further explanation). I create my synthetic data with: ...
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50 views

Explicit prior versus implicit prior

I am reading a paper where they talk about keeping a prior explicit as opposed to an implicit prior. To be honest, I have never came across the terms explicit/implicit in context of priors and I was ...
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38 views

Figuring out quantiles in quantile regression

Suppose I have a dataset $\{y_i,x_i\}$ $i=1,2,...n$. For the response variable, $y_i$ as per quantile regression I have the following likelihood: $$p(y_i|\beta,\alpha_i,\sigma) ...
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77 views

what does “a distribution over distributions” mean?

I am reading a pdf about Dirichlet Process, and it said "A Dirichlet Process is also a distribution over distributions." anyone could explain this in plain English what does it mean by that? thanks ...
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47 views

Why not use Beta(1,1) as boundary avoiding prior on a transformed correlation parameter?

In Bayesian Data Analysis, chapter 13, page 317, second full paragraph, in the modal and distributional approximations, Gelman et al. write: If the plan is to summarize inference by the posterior ...
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153 views

Posterior very different to prior and likelihood

If the prior and the likelihood are very different from each other, then sometimes a situation occurs where the posterior is similar to neither of them. See for example this picture, which uses normal ...
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1answer
43 views

Specification of logical node (with distribution?) in WINBUGS

For a piece of homework I have an assignment using WINBUGS which I must admit confuses me to say the least. Tangential to my question but I have a few stochastic nodes that are to be gamma ...
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108 views

Why does nobody use the Bayesian multinomial Naive Bayes classifier?

So in (unsupervised) text modeling, Latent Dirichlet Allocation (LDA) is a Bayesian version of Probabilistic Latent Semantic Analysis (PLSA). Essentially, LDA = PLSA + Dirichlet prior over its ...
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21 views

Classification model that allows changing priors at prediction time

I would like to know if there's a way to build a classification model in R that would allow me to change the class weights at prediction time. The scenario where I would want to do this: I have a ...
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13 views

generating covariance matrices from multiple priors

In many optimisation problems, one typically uses many forms of regularisations over the parameters that is being estimated. For example, a typical cost function (to maximise) may look like this: $$ ...
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98 views

Negative binomial jeffreys prior

The negative binomial distribution is NB($m,r$), $$\Pr(X = k) = \left(\frac{r}{r+m}\right)^r \frac{\Gamma(r+k)}{k! \, \Gamma(r)} \left(\frac{m}{r+m}\right)^k \quad\text{for }k = 0, 1, 2, \dots.$$ I'm ...
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44 views

Could prior elicitation by actually drawing the density of the prior be sensible? Has it been done/discussed?

Somebody mentioned, I don't remember who, that "there are many ways to specify the prior, you could even draw it!". It is clear to me that it is possible to actually draw the density of the prior ...
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160 views

Example of “real life” use of Bayesian inference on $\mu$ from a normal distribution?

A classic example for students, when teaching Bayesian statistics, is to make inference on the mean parameter $\mu$ of a normal distribution, when it has a prior normal distribution. I would like to ...
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31 views

Prior elicitation with Normal-Gamma or Normal-Inverse-Gamma

I am looking for a way to have experts elicit a prior for a Normal-Inverse-Gamma Bayesian linear regression model. Is there any material suggesting intuitive ways to go about this?
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48 views

Finding the posterior pdf

Suppose $X$ has probability density function $$f(x, \theta) = \theta e^{-\theta x}$$ when $x > 0$ and $\theta > 0$, and $0$ otherwise; given $\Theta = \theta$. Suppose the prior probability ...
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52 views

Help with understanding this covariance setup

I have been reading a paper that formulates the problem of image registration as a generative model and I have been having a lot of trouble understanding some concepts and I was wondering if someone ...
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1answer
124 views

Jeffreys prior for continuous uniform distribution

A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by: $f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ ...
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87 views

Modeling prior probability as a delta function

I'm using approximate Bayesian computation to find the true value of a parameter. My prior distribution is uniform over $(0, 1)$. I was watching this video on Bayesian learning and the lecturer ...
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48 views

Posterior distribution of the parameter knowing the prior

I have exponentially distributed probability of event $E$ $$P(E|a) = a \exp(-aE),$$ where $a$ is the rate parameter of the exponential distribution. Now the probability distribution for $a$ is a ...
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1answer
124 views

When Jeffreys prior “fails”

Context: I am working on a calibration problem involving a 1D function of parameter $\theta$ for which I derived a Jeffreys prior (in fact a 2D but I have an informative prior for one of the ...
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1answer
44 views

Posterior distribution

I read the following in some document: Let $Y$ be a random variable with distribution $\mathcal{N}(\theta,\sigma^{2})$. The variance $\sigma^{2}$ is known. Let $p(\theta) = 1$ a flat prior on ...
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166 views

half-cauchy prior for scale parameter

I am looking for a prior for a scale parameter for which I have prior knowledge such that: "$\sigma$ typically does not exceed 100." ("typically" meaning that occasionnally this can happen). In such ...
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101 views

How do I combine multiple prior components and a likelihood?

Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component ...
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117 views

Why use ${1/\sigma^2}$ as a prior for $\sigma^2$?

In a lot of cases, the prior for $\sigma^2$ is chosen so that it is proportional to ${1/\sigma^2}$. I have a few queries re this: What is the intuition for this choosing this prior? What is the ...
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98 views

Use the improper prior $p(v) \propto 1/v$ into Jags

I know that one can approximate this density ($p(v) \propto 1/v$) using its truncated version and implement it this way: ...
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85 views

MCMC for an explicitly uncomputable prior?

I am trying to sample from a posterior distribution and I only have an explicit formula for likelihood but I can sample from the prior distribution. How can I sample from the posterior distribution ...
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80 views

How can I get the prior of a random variable that's a function of a random variable in Bayesian data analysis?

I have a model which includes the following priors: $\lambda_C \rightarrow \dfrac{1}{\sigma_C^2}$ and $\sigma \sim \text{uniform}(0,500)$ Where $\sigma$ is the standard deviation and $\lambda_C$ ...
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99 views

What exactly is weakly informative prior?

Is there a precise definition of weakly informative prior? How is it different from a subjective prior with broad support?
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35 views

posterior of uniform distribution. I keep getting 1, but that is not right

Suppose 100 animals are classified five ways: $y = (50, 10, 25, 15)$ with corresponding probabilities $(\frac{\theta_1 + \theta_2}{5}, \frac{4(1 - \theta_1)}{5}, \frac{1-\theta_2}{5}, ...
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31 views

Finding the full conditonal distribution when there are multiple distributions involved

6 neighboring countries have the following disease instances: $y = (y_1, y_2,...,y_n)$ with a population of $x = (x_1, x_2,...,x_n)$. The following model and prior distributions are considered: ...
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41 views

Closed form solution for posterior with scale mixture normal prior

Is there a closed form for the posterior mean and variance of an observation drawn with known measurement error if the prior is a scale-mixture normal? To be more clear, here's the implicit model: ...
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104 views

Finding Jeffreys prior for the multivariate normal distribution. Conditional probability algebra

For the multivariate normal model, find Jeffrey's prior of the form $p_j(\theta, \Sigma|y1,...,yn)$, $p_j(\theta|\Sigma, y_1,...,y_n)$, and $p_j(\Sigma|y_1,...,y_n)$. Attempt: I know that ...
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280 views

How does the beta prior affect the posterior under a binomial likelihood

I have two questions, Question 1: How can I show that the posterior distribution is a beta distribution if the likelihood is binomial and the prior is a beta Question 2: How does choices the prior ...
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1answer
150 views

Which distributions are parameterization invariant when based on the Jeffreys prior?

I understand that the Jeffreys prior provides a method for constructing a prior distribution over parameters for a given model (likelihood function) such that the prior distribution is "invariant ...
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61 views

Convenient posterior distribution for homogeneous bivariate Gaussian model

For the model given by some independent pairs $(x_i,y_i)$ identically generated from a bivariate Gaussian distribution, there is the convenient semi-conjugate family of "Normal-Wishart" prior ...
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26 views

Guidelines to conduct a prior sensitivity analysis

I will be interested in conducting a prior sensitivity analysis on a hierarchical model. I looked for some references but did not manage to find a clear guideline. Maybe some of you have some hints?
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63 views

Priors and Loss in R

I am fairly new to R and data mining concepts and am trying to understand the rpart package in R. I am a bit confused about the role of priors and loss in the ...
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33 views

How to include prior knowledge that a model might be able to figure out itself

I have a problem where I want to predict the outcome of a sequence given another sequence online. Let $(x_1, x_2, ... x_T)$ be denoted by $x_{1:T}$, then I am estimating: $$ p(y_T|x_{1:T}) $$ where ...
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247 views

How to specify the Wishart distribution scale matrix

I'm running the below Bayesian mixing model in R using the rjags package, but I am having difficultly in specifying the scale matrix for the Wishart distribution. Essentially, I want Sigma.inv to be a ...