Questions tagged [prior]

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 distribution represents a greater degree of information.

Filter by
Sorted by
Tagged with
2 votes
2 answers
3k 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 ...
1 vote
1 answer
351 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 & \...
2 votes
1 answer
258 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: ...
1 vote
2 answers
1k 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 ...
  • 4,450
1 vote
0 answers
92 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) =\frac{\alpha_i(1-\...
  • 1,153
6 votes
2 answers
7k views

What does "a distribution over distributions" mean?

I am reading a paper about Dirichlet Processes, and it said "A Dirichlet Process is also a distribution over distributions." What does that mean?
8 votes
1 answer
961 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 ...
26 votes
8 answers
5k 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 ...
2 votes
2 answers
430 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 ...
16 votes
3 answers
5k 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 ...
  • 975
11 votes
0 answers
973 views

Negative binomial Jeffreys prior [duplicate]

The negative binomial distribution NB($m,r$) is defined as $$\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, \...
  • 161
7 votes
2 answers
155 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 ...
7 votes
2 answers
2k 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 ...
  • 20.4k
5 votes
0 answers
255 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?
  • 521
1 vote
1 answer
551 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 ...
  • 291
2 votes
1 answer
80 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 ...
  • 4,450
7 votes
2 answers
3k 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 $I$ ...
24 votes
3 answers
33k views

How to choose prior in Bayesian parameter estimation

I know 3 methods to do parameter estimation, ML, MAP and Bayes approach. And for MAP and Bayes approach, we need to pick priors for parameters, right? Say I have this model $p(x|\alpha,\beta)$, in ...
  • 3,175
7 votes
0 answers
2k views

Modeling prior probability as a delta function [closed]

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 ...
2 votes
1 answer
179 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 ...
  • 175
1 vote
1 answer
404 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 ...
  • 4,673
0 votes
1 answer
121 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 $\...
6 votes
1 answer
6k 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 ...
  • 4,673
7 votes
1 answer
4k 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 ...
  • 1,711
5 votes
1 answer
1k 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 ...
5 votes
1 answer
1k 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: ...
  • 4,673
1 vote
3 answers
166 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 ...
  • 1,425
1 vote
2 answers
281 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 $\lambda_C$ ...
11 votes
2 answers
6k 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?
  • 1,580
1 vote
0 answers
82 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: $y_i|...
11 votes
1 answer
17k 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 ...
7 votes
1 answer
2k 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 ...
2 votes
0 answers
158 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 ...
1 vote
1 answer
1k 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 ...
  • 21
3 votes
0 answers
71 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 $...
  • 13.1k
7 votes
1 answer
5k 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 ...
  • 71
3 votes
0 answers
206 views

Question about foundations of the uniform shrinkage prior

I am collecting papers about the uniform shrinkage prior for hierarchical Bayesian model. In "A prior for the variance in hierarchical models" of Michael J. Daniels it is stated at the end of page two ...
  • 4,673
15 votes
2 answers
9k views

Flat, conjugate, and hyper- priors. What are they?

I am currently reading about Bayesian Methods in Computation Molecular Evolution by Yang. In section 5.2 it talks about priors, and specifically Non-informative/flat/vague/diffuse, conjugate, and ...
  • 862
3 votes
1 answer
1k views

Prior selection for Gaussian Processes (GP)

I am trying to select a prior for the covariance parameters of my Gaussian Process (GP) and have been running into numerical problems with my MCMC code. My model is the following: $$Y = D\beta + GP(...
's user avatar
1 vote
0 answers
111 views

Getting Bayes Prior

We want to predict how many points per game LeBron James is going to get per game. Assume there is some underlying theta that does not change game to game, that predicts how many points he will get. ...
9 votes
1 answer
20k views

How do I choose parameters for my beta prior?

Suppose today I'm going to flip a coin. I believe that 9 of 10 flips will come up heads. I flip the coin and 8 of 10 are heads. Is my distribution of belief beta(9+8, 1+2) beta(1+9+8, 1+1+2) beta(...
  • 3,066
4 votes
1 answer
195 views

Estimating probability or frequency with low N?

I am trying to estimate the probability of an event using a low number of observations. The naive estimator $\hat{p} =\frac{\text{number of positive observations}}{\text{total number of observations}}...
2 votes
0 answers
213 views

Non-informative prior for integer restricted parameters

Does anyone have advice/references on choosing a minimally informative prior for the shape parameter of an Erlang distribution (gamma distribution with shape parameter restricted to integer space), or ...
  • 709
3 votes
1 answer
304 views

Bayesian estimation of tridiagonal covariance

I want to estimate covariance of a multivariate normal distribution from data using a Bayesian method. I want to force the result to be tridiagonal. I am looking for an appropriate prior or method. Is ...
  • 31
3 votes
2 answers
784 views

why isn't the the marginal distribution needed when using a conjugate prior?

What is a good explanation as to why you wouldn't have to integrate to find the posterior when you use a conjugate prior. Most examples (like for instance: http://www.youtube.com/watch?v=0XD6C_MQXXE) ...
8 votes
1 answer
958 views

Choosing non-informative priors

I am working on a model relying on an ugly parametrized function acting as a calibration function on a part of the model. Using a Bayesian setting, I need to get non-informative priors for the ...
  • 4,673
5 votes
0 answers
886 views

Variance of marginal posterior distribution

Suppose $Y_1,\dots,Y_n\mid\mu,\sigma^2 \sim \text{ iid } N(\mu,\sigma^2)$ and suppose the priors $\mu \mid \sigma^2 \sim N(\mu_0, \sigma^2 / \kappa_0)$ and $1/\sigma^2 \sim \text{gamma}(\nu_0/2, \nu_0 ...
  • 186
2 votes
0 answers
110 views

priors for strictly positive index or score types of variables

Is there a prior that's commonly used for "index" or "score" type variables that are user-defined as a weighted sum of a small number of variables (sometimes with pre-defined interaction contributions)...
  • 2,524
4 votes
2 answers
8k views

What are typical values to use for alpha and beta in Latent Dirichlet Allocation?

Specifically in the case where I don't know anything about the documents I'm working with. I'm looking a specific number or number range.
0 votes
1 answer
303 views

Help with Bayesian inference in OpenBugs

I have a task that involved Bayesian inference and could use some pointers and hints. I've already got some parts figured out but others remain blurred. Also, my OpenBUGS abilities are frankly limited ...
  • 1,491