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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Defining the exponential prior with jumping order of magnitude in parameter space

I want to define an Exponential prior for a parameter of my MCMC code with the following condition for the probability of parameter $M$: $$p(M)\varpropto\exp(-M/10^{15})$$ The parameter $M$ also ...
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17 views

Defining constraint on prior with potential class

I have written an MCMC code in order to estimate parameters Xpos, Ypos, MASS and concentration with a set of input data gal_pos, ...
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145 views

What does it mean to integrate over the posterior?

I have been reading a book that cites an example where a uniform distribution is the initial prior, and then a person scores 9/10 on a test. Then the resulting posterior becomes the prior ...
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1answer
54 views

How to construct a reasonable prior and likelihood for Bayes modelling?

To apply Bayes inference for data analysis or machine learning, we have to construct prior and likelihood, right? But if I fail to come up with a reasonable prior and likelihood, then the Bayes model ...
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33 views

learning hyper parameters: are we allowed to touch the prior parameters after observing the data?

There are many algorithms/applications that aim to learn the hyper parameters i.e. the parameters of a prior distribution from the observed data. A typical algorithm works in an iterative function ...
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38 views

Use of the Jeffreys prior in multidimensional models

Suppose a model, $$x_{i} \sim N(\theta_{i}, \phi), \text{ for } i=1,\ldots,n$$ Furthermore, suppose the variance parameter, $\phi$, is some known constant. The multidimensional Jeffreys prior is ...
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31 views

Gibbs sampling with Log-Normal observations

I am writing a Gibbs sampler for data that is Log-Normal (LN) distributed, with unknown mean and variance. There is a wealth of information on inference for LN models when either the mean or variance ...
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13 views

Prior over transition matrices of markov chain

I want to be able to marginalize over the transition matrix of a markov model. The goal is to get the marginal likelihood on the number of states necessary to explain the data. Something that would ...
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3answers
287 views

How can an improper prior lead to a proper posterior distribution?

We know that in the case of a proper prior distribution, $P(\theta \mid X) = \dfrac{P(X \mid \theta)P(\theta)}{P(X)}$ $ \propto P(X \mid \theta)P(\theta)$. The usual justification for this step is ...
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43 views

Are discrete single value prior distributions always lost in MAP estimation?

I’d like to illustrate my problem with a little (heavily abbreviated) excercise. I think it will help a lot to stress my point. Meet Mary, Tom and Jane. They all are programmers. Mary is a decent ...
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1answer
47 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...
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33 views

How should I construct a prior distribution with a particular kind of count data

For context I will first explain the overall problem that I am working on. I am given a catalog of product names and I am also given a large text dataset that may contain mentions of these catalog ...
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206 views

You observe k heads out of n tosses. Is the coin fair?

I was asked this question with $(n, k) = (400, 220)$ in an interview. Is there a "correct" answer? Assume the tosses are i.i.d. and the probability of heads is $p=0.5$. The distribution of the ...
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245 views

Ridge regression – Bayesian interpretation

I have heard that ridge regression can be derived as the mean of a posterior distribution, if the prior is adequately chosen. Is the intuition that the constraints as set on the regression ...
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1answer
37 views

Distribution of a Mean and Variance

Say we have observations $x_1 \dots x_n$ and we have some sort of Bayesian framework where we would like to estimate a distribution for the mean $\mu$ of our observations and the variance $\sigma^2$ ...
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107 views

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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1answer
47 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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11 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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46 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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25 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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70 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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21 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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52 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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84 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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95 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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37 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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1answer
74 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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58 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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47 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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90 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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75 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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199 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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73 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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189 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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22 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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16 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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122 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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54 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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267 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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38 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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1answer
64 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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56 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
160 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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104 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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1answer
61 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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134 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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48 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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1answer
282 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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1answer
125 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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1answer
130 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 ...