# Tagged Questions

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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### Ratio of Gamma distributed variables with different parameters

I encounter a problem which I thought I can handle, however, I struggle a lot with finding a solution: The following setting applies: I want to compute the posterior probability of an event, which is ...
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### Classification: Selecting final label using prior information on class distribution

Using R, let's say that I have the following (dummy) data. ...
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### Does class weighting introduce bias in Random Forest classifier?

I want to use a Random Forest classifier to stratify a strongly imbalanced population of samples. During training I used class weighting to weight the vote for each class by considering its ...
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### setting log-uniform priors in Stan

I have been using Stan for a couple months now and I want to adopt a log-uniform prior on some parameter array real theta[N]. I want to do something like a ...
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### Incorporate an external estimate of probability as predictor in a logistic regression model

I am predicting a binary outcome (e.g., credit default) with logistic regression. For each observation, in addition to my own observed predictors, I have obtained a probability estimate from an ...
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### What are the hyperparameters? [duplicate]

I find the meaning of hyperparameters not always clear. The hyperparameters are defined as "the parameters of the prior". Suppose that one has prior information about a certain parameter $\theta$. ...
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### Bayesian derivation of unbiased maximum likelihood estimator

I was recently reading an old NIPS paper by Bishop and Qazaz where they claim that an unbiased estimator for variance, based on $N$ Gaussian $\rm i.i.d.$ samples with unknown mean and unknown variance,...
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### Verifying and/or changing priors assumptions on Bayesian ANOVA

I am performing a Bayesian analysis of around 1500 data, divided into 2 factors, one that I am interested x1, and the id-variable for the paired/within-subject x2. x1 has 15 levels, and x2 around 100 ...
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### choosing prior parameters for variational mixture of Gaussians

I am implementing a vanilla variational mixture of multivariate Gaussians, as per Chapter 10 of Pattern Recognition and Machine Learning (Bishop, 2007). The Bayesian approach requires to specify (...
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### Probability distribution transformation of variables question

Problem: Hi there, I'm stuck trying to derive an equation stated in a research paper relating to Bayesian statistics in Cosmology (the paper is: http://mnras.oxfordjournals.org/content/398/4/2049....
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### Implication of using independent priors for means of joint normally distributed random variables

I am using Bayesian methodology to estimate parameters of joint distribution(Multivariate normal) of random variables $(y_1, y_2) \sim N(\mu, \Sigma)$. I implemented the code for finding the posterior ...
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### How to find the posterior distribution and posterior mode of Beta given an exponential prior distribution and binomial data?

This is the question I'm working on: I have already completed part (a). For part (b) i, I modeled y1 given Beta and n1 as Binomial(n1, p1 = d1/(d1+Beta)). For part (b) ii, I showed that the ...
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### Comparing the rates of Poisson distribution using Bayesian inference

In the 'Theory of Probability' book by Sir Harold Jeffreys, (5.15), the form of the Bayes Factors that he derives for the comparison of Poisson rates is the same as that of Binomial rates. But I did ...
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### Risk and posterior expectation Bayesian Statistics

Consider $x\sim B(n,\theta)$ with $n$ known a)If $\pi(\theta)\sim Beta(\sqrt{n}/2,\sqrt{n}/2)$ give the associated posterior distribution and posterior expectation $\delta^\pi(x)$ b)Show ...
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### Why is computing the Bayesian Evidence difficult?

In Bayesian estimation, we need to compute the normalizing factor P(X). Say that our parameter space was y. Then in order to compute the Bayesian evidence we'd need ...
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### Incorporating prior class probability in decision trees

How do we incorporate prior class distributions in algorithms such as CART? I read that it would have an impact on the splitting of the tree (if the distribution is different than what we have in the ...
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### Examples of usage of community of priors or why aren't they used more commonly?

Kass and Greenhouse (1989) proposed using "community of priors" (see also Fayers et al, 1997; 2000). As described by Spiegelhalter (2004), they can be seen as a range of viewpoints that should be ...
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### Prior estimation for Dirichlet Process Clustering

I wrote this code for Dirichlet Process Clustering using Chinese Restaurant Process in which a parameter ...
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### Prior for Bayesian multiple logistic regression

I was wondering how I could incorporate a prior to form a posterior distribution for multiple logistic regression. More specifically, I am working with basketball data, where the response variable is ...
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### Dirichlet Process Clustering Prior

I'm doing dirichlet process clustering where dirichlet priors are used as: with CRP representation as: First customer will always choose first table. Second will choose already occupied table with ...
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### Do Bayesian priors become irrelevant with large sample size?

When performing Bayesian inference, we operate by maximizing our likelihood function in combination with the priors we have about the parameters. Because the log-likelihood is more convenient, we ...
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### Priors on variable ordering and/or percentile ranking

Consider a set of variables $\mathbf{X}$ = $X_1 \ldots X_n$ where each variable is $\in [0,1]$. I am modeling an inference problem on these variables. Among other things, I have the following prior ...
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### Incorporating Prior Class Probability Distribution in Logistic Regression

I am amazed that I can not find any articles / lectures about how one can incorporate Prior Class Probability Distributions in classifiers like Logistic Regression or Random Forest. So my question ...
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### Horseshoe priors and random slope/intercept regressions

I'm interested in using the horseshoe prior (or the related hierarchical-shrinkage family of priors) for regression coefficients of a traditional multilevel regression (e.g., random slopes/intercepts)....
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### conjugate prior for exponential distribution

If there is an exponential distribution $$p(x | \theta) = \theta\,e^{-x\theta}\mathbb{I}_{x>0}\, ,$$ what is a good conjugate prior? Also, will the posterior mean is a convex combination of prior ...
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### Prior on Precision or variance

I've been reading this tutorial on variational bayes which talks about sparse Bayesian learning (Relevance Vector Machines if you prefer). In the paper they put a Gamma prior on the precision ...
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### Gaussian likelihood + which prior = Gaussian Marginal?

Given a Gaussian likelihood for a sample $y$ like $$p(y|\theta) = \mathcal{N}(y;\mu(\theta),\Sigma(\theta))$$ with $\Theta$ being the parameter space and $\mu(\theta)$, $\Sigma(\theta)$ arbitrary ...
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### Ising-Like Priors with Fractal Boundaries (Application to Image Processing)

Overview: I'm interested in looking for priors that "look a little like" the Ising model, but have different large-scale behaviour. In particular, I'm looking for priors that give rise to large ...
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I have the following Poisson mass function: $$p(y| \theta) = \frac{\theta^y e^{\theta}}{y!}$$ Which has a corresponding likelihood for n independent realizations of y as follows: $$\frac{e^{-n\... 1answer 81 views ### How to fit a Pareto distribution via Bayesian estimation (with a Pareto prior)? I don't know Bayesian statistics very well, so I don't know if the question makes sense. Let me give an example. We assume that the income distribution of a country is a Pareto distribution (the ... 0answers 88 views ### How Do I choose parameters of prior on regression coefficients in a Bayesian linear model? I'm trying to perform a linear regression in a Bayesian way. The response is normal,the prior I would like to put over Beta (vector of regression coefficients) and Sigma^2 (variance of the error ... 2answers 110 views ### priors for Gamma shape and scale parameters I have a random variable X that is Gamma distributed with unknown parameters \alpha and \beta:$$ X\sim \text{Gamma}(\alpha, \beta)  I now want to estimate $\alpha$ and $\beta$ from samples $... 1answer 148 views ### How to determine posterior distribution of the parameter in a binomial Assuming that I performed n iid tests, and the total number of test is n which is a fixed value, and the observaton of 1 which corresponding to successful results is X observations yeild with ... 0answers 70 views ### Is Independent jeffreys prior different from independent reference prior? I have a model involving two scalar parameters$\theta_1$and$\theta_2$and derived the Jeffreys prior for$\theta_1$and$\theta_2$independently (so for, e.g.$\pi(\theta_1)$, setting in the ... 0answers 55 views ### How to write unnormalized posterior when prior is a mixture of continuous and discrete Suppose I want to do bayesian inference on the regression problem$\beta$for Y = X$\beta$+$\epsilon$for$\epsilon_i \sim N(0,\sigma^2)$. The complication is that the prior for each component$\...
I'm fairly new to bayesian. I'm trying to edit a bayesian python code for $A/B$ test analysis. I'm using uninformative priors as a beta distribution, so my $\alpha$ & $\beta$ parameters are $1$ &...