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 ...

learn more… | top users | synonyms

2
votes
0answers
12 views

Linear regression with prior on $\arctan \beta_1$

Suppose we have $\hat{y} = \beta_1 x + \beta_0$ (I ask only for the univariate case.) A typical Bayesian approach might involve Normal priors on both parameters. I was thinking today about a ...
3
votes
1answer
29 views

Relation between changing the prior and the effect of an additional data point

E. T. Janes writes the following in "Probability Theory: The Logic of Science": A useful rule of a thumb is that changing the prior probability $p(\alpha | I)$ for a parameter by one power of ...
0
votes
0answers
4 views

Prior knowledge and significance thresholding

I am curious whether the following is sensible for a translational or follow-up study: Knowing form a previous experiment that brain regions A and B show highly significant activation upon treatment ...
3
votes
2answers
49 views

Metropolis Hastings Algorithm - Prior vs Proposal vs Numerator of Bayes Theorem

I've been using this technique in 'black-box' form for a little while as a physics student. I have been struggling to understand what's happening under the hood for some time and I think I almost ...
0
votes
0answers
13 views

Math notation for bayesian hierarchical models with covariance matrix with LKJ priors

I am fitting a simple hierarchical bayesian gaussian model of the form: $$ Y = b_0 + b_1 X_1 + b_2 X_2 + e $$ $$ b_{0:2} \sim N(\theta_{0:2}, \sigma_{0:2}) $$ $$ \sigma_{0:2} \sim Gamma(.001 , .001) ...
1
vote
1answer
35 views

Reasoning regarding non-informative priors

I'm not sure whether this counts as a question. However, I'd be happy to receive feedback for the validity of my reasoning. Recently, I read a bit about Jeffreys' prior and the "problem" with using ...
1
vote
0answers
13 views

Reference prior for a three-parameter model and likelihood factorization

Let a (regular) statistical model with three parameters $\phi_1$, $\lambda_2$, $\mu$, and three observations $x_1$, $x_2$, $y$. Assume the likelihood has form $$ L(\mu,\phi_1,\lambda_2 \mid y, x_1, ...
0
votes
1answer
41 views

What is this prior distribution?

I know that the joint prior distribution is $p\left( {{a^2},{\beta ^2},\gamma ,\delta ,{\varepsilon ^2}} \right) \propto {\alpha ^{ - 2}}{\varepsilon ^{ - 1}}{\beta ^{ - 2}}$. However, I am confused, ...
0
votes
0answers
5 views

Bayesian approach for comparing the predictability of different datasets for another

Suppose I have three datasets A, B and C with not necessarily the same amount of data. Now, I want to know whether dataset A or dataset B is better in predicting C. I thought of using a Bayesian ...
0
votes
1answer
16 views

weighted glm model selection

Can AIC values between different weighed models be compared to select the best model (ie the model with the lowest weighted AIC)? For example, if my response variable is the 'Average Sales Per ...
0
votes
0answers
11 views

Interpreting mean of coefficients by accessing $Beta in BMR package in R

I've been using BMR (Bayesian Macroeconometrics in R) package to carryout BVAR(Bayesian Vector Auto Regression). When defining the Minnesota prior for my monthly dataset and have obtained mean of each ...
0
votes
1answer
74 views

Find a posterior distribution [closed]

I came across this task that I have no idea how to solve, because I'm not very good at statistics, so I was wondering if someone could help me understand it. 7 scientists with very different ...
2
votes
1answer
38 views

Forming a prior based on the solution to a linear system

On p. 115 of the 4th edition of Machine Learning a Probabilistic Perspective, we have the following: Let $\epsilon\sim N(0,\frac{1}{\lambda}\text{I})$ and let $L$ be a matrix of dimension ...
0
votes
0answers
13 views

Noninformative prior for variance, understanding and coding

I have three questions regarding the understanding behind and implementation of a noninformative prior for variance. I'm attempting to build a Metropolis sampler and I'm trying to sample from a ...
2
votes
0answers
28 views

Using empirical priors in PyMC

I'm using PyMC to sample the posterior distribution and I've run into a roadblock with using priors from samples, not models. My situation is as follows: I have some empirical data for a parameter ...
1
vote
2answers
309 views

Calculating SD for normal distribution with only mean and 5% and 95% quantile values

As part of a Bayesian method to estimate the divergence times of species, priors have to be set with values based on previous literature or known fossil dates. These priors can have different ...
2
votes
1answer
48 views

A simple question about MAP and MLE

I recently got this simple question from a friend. But I am quite confused about it. Suppose we toss a coin $N$ times, and got heads $m$ times. Assume the binomial distribution with $p$ which is the ...
2
votes
0answers
36 views

AR(1) model - which prior to use?

I want to use the following univariate model: $y_t = \mu_t + \epsilon_t, \ \epsilon_t \sim N(0,1)$ $\mu_t = \phi \mu_{t-1} + \omega_t, \ \omega_t \sim N(0,\sigma_\omega^2)$ That is, $\mu_t$ follows ...
0
votes
0answers
15 views

Variational Bayes with non-symmetric priors

I recently developed a Variational Bayes (VB) model with the intention encoding priors on a latent variable that was previously estimated using EM. In particular the VB model has two latent variables, ...
0
votes
0answers
23 views

A linear model with prior information

Suppose I have this experimental data: I have measurements of drug response from patients (let's say its blood pressure). Specifically, I have measurements after being treated with drug A (30 ...
0
votes
1answer
70 views

Equal weight between prior probabilities

While constructing a model hierarchy for Bayesian analysis, I have two parameters: $\theta_0$ ~ Uniform(80, 90) $\theta_1$ ~ Normal(0.093, 0.002) I take the $ln$ of the pdf for the parameter's ...
13
votes
2answers
477 views

What is the problem with empirical priors?

In literature I sometimes stumple upon the remark, that choosing priors that depend on the data itself (for example Zellners g-prior) can be criticized from a theoretical point of view. Where exactly ...
0
votes
1answer
60 views

Define own noninformative prior in stan

In the simple case of normally distributed data with unknown mean and variance, Jeffrey's prior is given by $$p(\mu, \sigma^2)=\frac{1}{\sigma^2}.$$ How can I define such a prior in the Stan ...
4
votes
2answers
114 views

Jeffreys Prior for normal distribution with unknown mean and variance

I am reading up on prior distributions and I calculated Jeffreys prior for a sample of normally distributed random variables with unknown mean and unknown variance. According to my calculations, the ...
0
votes
0answers
12 views

Issue in hierarchical model specification

I have an issue while trying to specify a model in a Bayesian framework for the following problem: With the goal of increasing statistical power when detecting associations between an outcome $y$ and ...
2
votes
1answer
23 views

Selecting correct priors for Bayesian A/B Testing

I'm running an A/B test. The test is a funnel that, at some step in the funnel, sends half the population to experience A, the other half to experience B. Traditionally everyone saw experience A at ...
0
votes
0answers
23 views

Set prior for logistic regression in R when using unequal group sizes (29 versus 48 cases)

I have 29 cases for negative outcome (0) and 48 cases for positive outcome (1). I fit my data with logistic regression model ...
1
vote
0answers
68 views

Relationship between low identifiability and prior weight in Bayesian model

I'm trying to get intuition into the relationship between low identifiability and prior weight in Bayesian model. Is it true to say that in lowly identifiable model + data the prior will have a higher ...
0
votes
0answers
11 views

Alpha Parameter Specification Dirichlet Prior

I have a straightforward Dirichlet-Multinomial model with code that is running in RJAGS. The data are a collection of 200 2 x 2 contingency tables. The multinomial counts are those of a 2 x 2 ...
0
votes
0answers
43 views

How to exploit relationships between independent variables?

Data: Each instance (representing a document) is a bag-of-entities (like BOW, except they're Wikipedia entities instead of words), so each feature is a binary or tfidf-like score based upon the ...
0
votes
0answers
35 views

Meaning of the prior and loss parameters in rpart in R

Could someone please explain to me what specifying priors and/or loss parameters in R's rpart actually do? I found R's documentation completely unhelpful. For example, let's suppose I have a highly ...
1
vote
1answer
91 views

Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
1
vote
0answers
39 views

Stochastic Block Model Priors

In the generic stochastic block model (binary edge data, no degree correction, etc.), if an uninformed prior is used for the Bernoulli coefficients i.e. Beta with $(a,b) = (1,1)$, will the model ...
-2
votes
2answers
1k views

Priors in Bayesian MCMC

I am trying to understand how the choice of priors affects a Bayesian model estimated using MCMC. At a basic level I understand that the product of the prior and the likelihood are proportional to the ...
0
votes
0answers
28 views

Jeffrey's Prior for normal distribution with mean = 0

How would I go about calculating Jeffrey's Prior for a normal distribution with mean = 0, So far I get: But then don't know where to go next. Any help much appreciated
7
votes
2answers
219 views

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 ...
0
votes
0answers
23 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 ...
2
votes
2answers
76 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? ...
0
votes
1answer
81 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 ...
0
votes
2answers
43 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} ...
1
vote
1answer
41 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 ...
1
vote
2answers
56 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 ...
3
votes
0answers
66 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), ...
4
votes
1answer
358 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?
0
votes
0answers
22 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 ...
0
votes
1answer
33 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 ...
0
votes
0answers
34 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 ...
2
votes
2answers
104 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 ...
1
vote
1answer
24 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'': ...
2
votes
0answers
23 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 ...