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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.

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Updating the variance of a Normal Distribution Using Bayes [on hold]

I have a prior belief for the mean $\mu_p$ and std $\sigma_p$ of a normally distributed variable $X$. (no dataset). So I have the mean of these parameters, but NOT their variance (I'll just presume ...
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Learning prior distribution from data

Suppose I have a dataset. How can I learn the prior distributions of the parameters of a model from this data? I want to learn the prior from this data in order to use them in a Bayesian model. Sorry ...
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Is this Bayesian model averaging?

A classical example of Bayesian model averaging (BMA) is the regression setup where the choice of different sets of covariates corresponds to different models $\mathcal{M}_k$, $k = 1, \ldots, K$, ...
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Finding mode of posterior using Newton method in R

I am attempting to approximate the posterior $\tilde{\pi_{G}}(z|\theta,Y)$ which is the Gaussian approximation to the full conditional of $z$, and in order to do this I need to find the mode $z^{*} \...
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Best estimate of underdetermined system using prior

I have measured two variables which depend on the same set of four parameters. I want to know the parameters which best explain my measurements. Of course, I cannot solve for four unknowns from just ...
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Calculating Jeffreys Prior for geometric distribution

This question is already answered here, but I would like to know why it is worked out the way it is My lecture notes state the following: I am also given the following problem : Now, what I ...
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1answer
36 views

Posterior density for a linear regression model

Given a classical linear regression model $$y = X\beta + \varepsilon,$$ $$\varepsilon\sim N(0,\sigma^2I_n),$$ the posterior density is proportional to the product of the likelihood and the selected ...
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Some help interpreting posterior plots

I needed some help interpreting and comparing the plots that I created. I'm not really sure what is the most important thing to talk about when comparing these plots. I know for these plots that the ...
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25 views

Parameter Transformation Bayesian Learning

I read that given a parameter $\theta$ and a transformation $\phi = g(\theta)$ (where $\theta$ is the parameter of your prior distribution), the distribution of the transformed parameter would be: $...
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53 views

Elicit a Proper informative priors

I need help with eliciting,deriving and plotting two proper informative prior. One with mean 0.5 and standard deviation 0.25 Another with LQ=0.62 , UQ=0.715 and Median = 0.67 Is the first one ...
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Concrete example of a generative process using the joint distribution of (Dirichlet prior and Multinomial Distribution)?

I am not a statistician as it is not my field. Please bear with me and correct me when I am wrong. Your help is much appreciated. I know that when having a Dirichlet Prior and Multinomial ...
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Are there situations where improper priors can be avoided via a prior on a subset of the real line and a transformation?

There are many situations where improper priors are "permissable" (Berger, 2009). In many cases, these improper priors are improper because they are "flat" on the real line. A well known example is ...
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How can the prior distribution of bayes regression be estimated by empirical bayes?

Neither in Efron's book Large-scale Inference:Empirical Bayes Methods for Estimation, Testing and Prediction nor by Internet search, did I find a prior distribution estimation method of Bayes ...
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does Informativeness of the prior always decreases similarity of the posterior mean to the data mean? [closed]

I am looking for a proof of the statement "If the variance of the prior distribution is greater, the posterior is more affected by the data". More specifically, if X, X' are priors such that E(X)=E(X'...
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Understanding definition of informative and uninformative prior distribution

When using the "non-informative" prior $\pi(\mu,\sigma)\propto\frac{1}{\sigma^2}$ where $\pi(\mu)\propto1$ and $\pi(\sigma^2)\propto\frac{1}{\sigma^2}$ Where is the no information for the ...
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What should you consider while constructing a Gaussian prior for the binned fit? [closed]

I need to make Bayesian fit of the binned data. The model is simply linear function (background) + Gaussian (signal). I want to set a Gaussian prior on the mean of the Gaussian from the model. What ...
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Can the r-scale value (for Bayes Factor) be directly based on Cohen's d?

I want to set an r-scale value for a Bayesian t-test (i.e. to calculate Bayes Factor likelihood ratio) based on previous results (i.e., from a posterior). But I simply cannot find a straightforward ...
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Coming up with decent priors for Bayesian Online Changepoint Algorithm

I am trying to identify changepoints in time series data using this online changepoint detection algorithm. It seems that there are a couple of user defined parameters that I have to give the model: ...
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Information about parameters using priors distributions [duplicate]

When using the "non-informative" prior $\pi(\mu,\sigma)\propto\frac{1}{\sigma^2}$ where $\pi(\mu)\propto1$ and $\pi(\sigma^2)\propto\frac{1}{\sigma^2}$ Where is the no information for the ...
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1answer
42 views

What would be an ignorance prior of AB, given the probabilities of A and B?

Let us have two events, $A$ and $B$ whose probabilities are $P(A)$ and $P(B)$. In the absence of any other information, what would be a reasonable probability to assign to $AB$, that is, $A$ and $B$ ...
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What are the limitations on the flat prior in the BAT framework?

By default, the BAT framework sets the flat priors on the parameter. My question is: how in this case allowed range for a parameter of the flat function is estimated?
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How to Change prior probabilities for predicted variable in neural networks and other methods in SPSS Statistics

i am trying to find right model for predicting categorical variable with two values. Problem is that ratio of cases in group 1 and group 2 is not equal but rather in ratio of 2:1. When i try to find a ...
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Prior distribution for shape, scale, rate, mean and standard deviation

What are the standard normal prior that suitable to be use for each parameters like shape, scale, rate, mean and standard deviation. Thank you
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Gaussian Process regression: does there exist a conjugate prior over hyperparameters?

When adopting a fully Bayesian hierarchical setting in Gaussian Process regression is there a choice of kernel (covariance) function such that there exist a conjugate prior? If so which?
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PyMC3: up-to-date implementation of Price is Right example?

So, getting into PyMC3 a lot more and working through examples, I found I cannot implement in an up-to-date form an example from Cameron Davidson-Pilon's Bayesian ...
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Importance of the prior

The maximum a posteriori objective can be written as $$\widehat{\theta}_\textrm{MAP} = \operatorname*{argmax}_\theta \log P(y\mid\theta) + \log P(\theta)$$ where $\log P(\theta)$––the prior––is a ...
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Defining custom Bayesian priors in R (BayesFactor package)

I'm performing some Bayesian analyses in R using the BayesFactor package, and was wondering whether it is possible to specify priors for the alternative not centered on zero (the current defaults ...
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Prior predictive distribution usage

I understand the mechanics and math behind prior predictive distributions, but I don't understand its practical uses. Theoretically and application wise, what is its purpose?
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What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions?

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions? The purpose of finding this information is to enlighten me on how should one decide on a prior for a positive-...
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1answer
472 views

Joint posterior distribution of $(\mu,\sigma^2)$ in the Normal model

Find the joint posterior of $(\mu, \sigma^2)$ given Normal data. I've found the joint prior of $\mu$ and $\sigma^2$ (where $\displaystyle\sigma^2\sim\chi^{-2}(v_o,v_os_o^2)$ and $\mu\mid\sigma^2\sim ...
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Gaussian Processes, Prior function of constant, Linear and Polynomial Kernel

In my university course slide, regarding the exponential and Gaussian kernel there are these two relevant pictures. They show the prior function of each kernel: You can see that the resulting prior ...
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1answer
33 views

Gaussian Processes, basic question about how the prior is computed

I'm approaching the topic of GP, and I have a question regarding how functions are sampled. On my textbook is stated that to represent a distribution over a function (the prior): we only need to ...
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Prior/degree of belief/degree of lack-of-information/algorithms/complexity

For a long time I had a bit of difficulty understanding what "degree of belief" means. Recently I had some thoughts about it and I wonder if they make any sense, or is there some literature about ...
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Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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Why is the normal distribution a default choice for a prior over a set of real numbers?

I found this question without any context while preparing for a job interview. The only reason I can think of is that the Normal distribution's support are the real numbers. Why wouldn't we use a ...
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How to justify using Beta distribution as a prior distribution in the following problem

Let $\theta$ be the proportion of people who are ready to quit smoking within 6 months. Let's say we perform a survey in $2017$ with a $n$ volunteers who ask people this question until they obtain yes ...
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What prior distribution of parameters in the Bayesian estimation of a GARCH model?

In the case of the Bayesian estimation of GARCH(1, 1) model with Student–t or a Skewed distributions for innovations, is it more correct to assume a uniform distribution for the parameters or to ...
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What is the interpretation for the priors in the derivation of Laplace smoothing?

Laplace smoothing has a generalisation that can be justified with the use of Bayes formula. Let $f(x;\alpha,\beta)$ be the (non-normalised) beta distribution, i.e. $$f(x;\alpha,\beta) = x^{\alpha-1}(...
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How to show that $\frac{1}{\theta}$ is a flat prior for $\log\theta$? [duplicate]

I do not really understand what the statement even means. To my understanding, a prior $p(\theta)$ is said to be flat if $p(\theta) =$ constant $\forall \theta,$ where $p(\theta)$ is the prior ...
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Biassing priors to improve confusion matrix

I have a text classification problem to solve. I need to classify a given sample of text into one of two classes A or B. My training set has about 30% A and 70% B. This is my prior. Now, when I ...
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Incorporating prior knowledge into feature selection in the setting of multicollinearity?

Background: I'm trying to find the optimal combination of two parameters for finding the first peak meeting some criteria in a signal. The filtering is a bit simplistic, there's a threshold (...
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Ockham's Razor in Bayesian Modelling

This question might be a little philosophical / generate discussion. I hope, there may still be some useful answers. I am currently thinking about how Ockham's Razor relates to Bayesian statistical ...
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1answer
86 views

Binomial distribution as likelihood in Bayesian modeling. When (not) to use it?

I am currently trying to figure out some strangeness about using the Binomial distribution in Bayesian modeling to define the likelihood. To make an example assume I have two conditions, and in each ...
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2answers
102 views

Bayesian approach to report simulation studies?

I am running a simulation study where we want to estimate a proportion $p$. We are reporting the coverage of credible intervals with a uniform prior, and we are doing $500$ Monte Carlo simulations. We ...
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1answer
44 views

How to derive the noninformative prior for location parameters and scale parameter?

I am reading this paper, in it: I have a lot of confusion reading it, I will list it one by one: Let $X$ be distributed as $f(x-\theta)$, which is a location invariant density. Q1: The sentence <...
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Dirichlet distribution as conjugate prior to Multinomial distribution [closed]

I stumbled upon the following exercise: Show that Dirichlet distribution with parameter $\alpha$ is a conjugate prior to the Multinomial distribution as likelihood. Derive the parameters to the ...
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53 views

Understand VAE with VamPrior

I am currently reading this paper. The authors propose to use as an prior this expression: $$ p_\lambda(z) = \frac{1}{K} \sum^K_{k=1} q_\phi (z\mid u_k) $$ where $q_\phi$ is the encoder, and $u_k$ is ...
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Doing cumulative bayesian analysis in R: shouldn't results converge?

I'm in the process of designing a Bayesian simulation of a cumulative bayesian analysis. In short, I am attempting to use the posterior information from a t-test to provide the priors for a subsequent ...
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Objective Bayesianism: Jeffreys priors vs reference priors vs principle of transformation groups

According to this answer, José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data by maximising the ...
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Choice of Gaussian process in non-parametric regression

I have been trying to understand non-parametric regression using Gaussian processes (GP), which are used to represent prior distributions over the space of functions. The linear model considered is $$ ...