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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Bayesian Gaussian mixture - is my prior correct?
I'd like to sample from the Bayesian Posterior of a Gaussian mixture model, but I am not sure about the correct Bayesian formulation of the latter. Is the following correct?
I consider the 1-...
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A correct understanding of z-score confidence intervals with improper, informative priors [duplicate]
Please check that my understanding of hypothesis testing, confidence intervals, and their relation to the prior on population mean $\mu$ is correct.
Let $X_i\sim N(\mu, \sigma^2)$ be IID samples for $...
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Correlation for priors in BLRM with EWOC in R package OncoBayes2
I'm trying to reproduce results in this protocol with OncoBayes2 package to learn BLRM with EWOC (Neuenschwander 2008) in dose finding studies. In section 14.2.2.1 (appendix 2) of the protocol it ...
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Derive the prior on variance scale if uniform prior placed on logarithm scale
In page 64 of Bayesian Data Analysis by Gelman et.al. they write
... sensible vague prior density for µ and σ, assuming prior independence of location and scale parameters, is uniform on ($\mu$, $\...
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Reduce Variance of monte carlo estimator using guess of mean
Suppose you have a random variable $X$ and black-box function $f$. Suppose you also have prior estimates $m$ and $s$ of the mean and standard deviation of $f(X)$.
How can we use this prior information ...
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Bounded uniform prior in R
I have been fitting a bayesian GLM using brms. The code works well but when I loop this over several data and make it a bit more complex, R encounters a fatal error and crashes. This seems to be ...
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Parameter distribution of $\theta$ from a rectangular matrix multiplication $C\theta$
I am struggeling to see where this problem fits - i.e. what topics this problem relates to, so I am not able to find the right literature. I want to use some particular information as a prior to a ...
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Importance sampling for a parameterized family of distributions using a wide distribution from the same family
I'm motivated here by a problem for robust Bayesian analysis. Let $l(Y|X)$ be the likelihood and let $\{p_\xi(X)\}$ be a parameterized family of prior distributions where $\xi$ denotes the ...
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Laplace approximation from a log-posterior in R
I would like to perform a Laplace approximation of a log-posterior.
The evolution of a cancer cell at given time $t_j$, $j = 1,\cdots,n$ for an experiment $i$ follows the following Poisson ...
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Specifying priors over softmax outputs
Looking to train a simple single-layered NN with a N-dim softmax output (and a relatively small feature vector size, ~2-10) in a streaming fashion accumulating K samples in a buffer and then ...
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Computing log-posterior for large variance priors
Let's say that some quantity is modelled by a time-dependent Poisson distribution,
$$
y(t) \sim \text{Pois}(\mu(t))
$$
where
$$
\mu(t) = \alpha_0 \exp(-\alpha_1 e^{-\alpha_2 t})
$$
and $\alpha_k > ...
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Light tailed symmetric distribution
Is there a family of distributions that resemble the normal distribution (symmetric, spanning all real numbers, and approximately bell-shaped) but have lighter tails than normal distribution?
I'm ...
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Informative priors for Bayesian chi-squared test
A colleague recently presented results from a chi-squared test that used a Bayesian method for estimation. The results seemed promising, but when I looked up the main function ...
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on the choice of prior distribuion in multi level model
I'm comparing the performance of the Restrected ML estimator and the Baysean method for estimating a multilevel model by Monte Carlo simulation. As I'm a beginner in Baysean analysis, I don't know how ...
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Some questions about the posterior distribution when the marginal distribution is zero
Let $\{f(\cdot|\theta): \theta \in \Theta \}$ be a family of pdfs and let $\pi: \Theta \to \mathbb{R}$ be a prior. According to Bayes' theorem (as stated in, e.g., Casella and Berger), the posterior ...
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Distribution families whose likelihoods integrate to $+\infty$ for some sample values
I've recently started learning about Bayesian statistics, and I came across this very nice answer by Xi'an https://stats.stackexchange.com/a/129908/268693, which [in my slight paraphrasing] says the ...
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Beta prior for the Clopper-Pearson interval
A binomial sample of $n$ trials consists of $k$ successes. The distribution of $k$ is
$P(k|\theta, n) = C_n^k \theta^k(1-\theta)^{n-k}$
We would like to construct a confidence interval for the ...
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How to account for measurement noise when calculating significance of fit for linear regression?
I want to build a model that will allow prediction quite far from a few available measurements (i.e. extrapolation) using prior knowledge of the true system to make predictions when there is ...
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How to put a prior over model parameters?
I am doing a problem in variational inference. I have some data $y$ and I want to understand which distribution it came from. I have the ELBO defined as - $$\text{ELBO}(\phi, D) = \sum_{n=1}^{N} E_{q(...
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Using Jeffreys prior for Bernoulli distribution to find the prior of a transformation on p
The question goes like this: Use Jeffreys prior for Bernoulli distribution and find the prior for $\eta$ where: $$\eta(p) = \left(\frac{p}{1-p}\right) $$
So $\eta$ here is some kind of a ...
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How does an ideal prior distribution needs a probability mass on zero to reduce variance, and have fat tails to reduce bias?
I am reading this article about the horseshoe prior and how it is better than lasso and ridge priors. The author makes several points that I don't understand. One of them is "The ideal prior ...
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How informative should a Gaussian Process prior to be?
I recently started learning about the Gaussian Process for a GP machine learning project so my understanding is relatively limited. However, from what I have read/watched so far you have a prior GP ...
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Power of Bernoulli likelihood in Jags (R2jags) [closed]
In a fixed power prior model, the model is set up as:
$$ \pi(p_i \mid \alpha,\mathcal{D}_0) \propto L(p_i\mid \mathcal{D}_0)^{w} \pi(p_i) $$
Suppose that the event follows a Bernoulli distribution ...
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Same relative spread of posterior means in Bayesian Linear Regression
I'm doing a Bayesian Linear Regression based on some marketing data, and pretty much following the tutorial outlined here. To summarise: I aim to predict revenue based on a bunch of different ...
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What is the best way to encode the prior of a Gaussian Process model in this application?
I'm using Gaussian Process regression for the first time to model the unknown energy efficiency of a compressor which I know is a smooth, non-linear relationship that looks something like the line in ...
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Bayesian statistics -- interpreting the mean and standard deviation of priors
I have just started reading about Bayesian statistics and have a question regarding the general interpretations of priors. Let’s say I have recorded the reaction times (RT, in ms) every time a single ...
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Parameterization of inverse gamma prior in Bayesian methods
For a prior of $\sigma^2 \sim IG(0.01, 0.01)$, often recommended as an uninformative prior for the variance parameter in MCMC approaches and other Bayesian methods, which parameterization does this ...
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Is it fallacious to sample a hyperprior without updating it in Bayesian estimation?
Intro
Uniform priors can lead to lots of divergences in Monte Carlo methods, but sometimes I really don't have external/prior knowledge about what parameter value is (even slightly) more likely.
One ...
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How to choose priors for bounds on circular truncated distributions?
I am considering choices of priors for truncated distributions on a circle. Let's take the truncated normal distribution on the unit circle as an example. It has parameters $\mu \in [-\pi, \pi]$ and $\...
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Jeffreys prior of a multivariate Gaussian
I have found two different expressions for the Jeffreys prior of a multivariate Gaussian. Eq. (3) in this article states that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+2)/2}$$
However in page 73 of ...
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How to calculate the posterior distribution with a normal likelihood function and a prior that involves sigma
In the problem, the data X follows a normal distribution, or $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{1}{2}(\frac{x-\mu}{\sigma})^2)$. Let's say I know the value of $\sigma^2$ and ...
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Show that $\hat{p}=0$ is the Bayes rule with respect to some prior $f$
We are given that X ~ Bin(n,p) and that the loss function is $L(p,\hat{p}) = \bigg( 1- \frac{\hat{p}}{p}\bigg)^2$ and that we are allowing $\hat{p} = 0$. Even though this falls out of the parameter ...
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Possibility priors in Bayesian analysis?
A couple of trains of thought have come together for a model I am designing. Let's start with the first part: Bayesian inference doesn't update strongly enough.
One of the parameters $\theta$ is an ...
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Better default prior for non-negative canonical polyadic decomposition of counts than Exp(1)?
Suppose I have a instance of a random $k$-mode tensor $X_{n_1 \times \ldots \times n_k}$ of count data. I would like to perform non-negative canonical polyadic decomposition of this tensor using ...
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Choice of base measure in nCRP (for validity and computation)
I'm trying to apply the nested Chinese restaurant process (nCRP) for structure learning. nCRP is a nested extension of CRP, with each table in CRP uniquely pointing to the restaurant on the next level....
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Eliciting a Gamma informative prior in a Gamma–Poisson Bayesian problem
I employ the Gamma–Poisson conjugate family for my statistical model.
I want to use an informative prior.
From theory, I know that the values of the Gamma-distributed random variable lie within the ...
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Correctness of product of densities representing parts of information as prior density in Bayes inference
suppose I've got data $X$ from a model driven by parameter $\theta$. Model of data is represented by conditional density (likelihood function)
$$f(x|\theta).$$
Suppose the prior density of $\theta$ is ...
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Is it ok to widen a prior during an MCMC which did not converge yet?
I am calibrating parameters of a process model. The runtime of the model is high and the calibration already ran for more than two weeks with many cores on a HPC.
After almost 150k iterations I ...
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Distribution of the sample variance given that $\sigma^2$ is unknown
By Cochran's theorem, if $y_1,....,y_n\sim\mathcal{N}\left(0,\sigma^2\right)$ independently with a known variance $\sigma^2\in\mathbb{R}_{>0}$, then
\begin{equation}
(n-1)\frac{S^2}{\sigma^2}\sim\...
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Joint posterior distribution of differences
I have data $x_1,...,x_n$, $y_1,...,y_m$ and $z_1,...,z_p$ where
$$x_1,...,x_n\sim N(\mu_x,\sigma^2_x)$$
and
$$y_1,...,y_m\sim N(\mu_y,\sigma^2_y)$$
and
$$z_1,...,z_p\sim N(\mu_z,\sigma^2_z)$$
Now let'...
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Bayesian Linear Regression: Is there significant difference between setting a gamma prior or flat prior over hyper parameters?
I am reading Bishop’s Pattern Recognition and Machine Learning 2006 and I am confused about this claim that defining conjugate gamma prior distributions over hyper parameters alpha and beta leads to ...
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Bayesian Analysis in the Absence of Prior Information?
I have always wondered - how confident do researchers tend to be in their "prior" information when deciding to create statistical models using a Bayesian Approach vs. a Frequentist Approach?
...
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Does the beta negative binomial (BNB) distribution have a conjugate prior?
BNB distribution is constructed using negative binomial and beta distributions, which are both exponential family, so my guess would be yes, there shoudl exist a conjugate prior in theory. But what is ...
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Nesting Priors within Gaussian Process Model in PyMC
I'm new to Gaussian Processes, and I have some questions about the model.
I understand that when using GPs for regression, GPs are a prior distribution of a set of functions on some unknown regressive ...
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Popular Methods for Choosing Hyperparameters in Bayesian Statistics
I'm wondering which methods are commonly used to estimate hyperparameters for priors in Bayesian statistics, and how they work? The setting I'm working with is Bayesian linear regression, so I'm ...
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How to incorporate prior knowledge into a CNN?
I'm pretty new to Bayesian inference and machine learning, so I think I'm just lacking the right words to search for a paper that addresses this topic, so here goes:
I'm trying to do image ...
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How do I combine two beta priors?
The problem is,
Manufacturer receives 30% chips from A, and 70% chips from B.
Both A and B has Beta prior distribution for defective rate.
During inspection of 100 chips, 10 chips were found to be ...
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Dependencies in Multivariate Horseshoe Prior
I have a multiple regression problem, where the label generation mechanisms $p(y \mid x)$ are (partly) similar to each other, i.e. if factor $k$ is inactive in one mechansim it's likely that it is ...
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fitting a gam with constraints on parameters to deal with separation in parametric terms
I am trying to fit a gam using mgcv which has a mix of smooth and parametric terms. The model is for some count data on fish catches. I am modelling variation in location and time, but also ...
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Choosing between Gaussian/Laplacian prior distributions for MCMC regression
When doing a linear regression using MCMC, you have to specify prior distributions for the values of the regression coefficients of the independent variables. If all of the priors are Gaussian ...
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