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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29 views

Inverse gamma posterior

I am working on bayesian analysis and I have a normal likelihood function and an inverse gamma (IG) prior for the parameter $\lambda$. I have the following posterior: $$ \propto ...
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55 views

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 i.i.d. samples with ...
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42 views
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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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13 views

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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1answer
93 views

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: ...
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17 views

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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20 views

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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18 views

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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1answer
30 views

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 $\delta^\pi(x)$ ...
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58 views

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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9 views

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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19 views

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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7 views

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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1answer
46 views

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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1answer
31 views

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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2answers
274 views

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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44 views

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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45 views

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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56 views

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 ...
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23 views

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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31 views

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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241 views

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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20 views

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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1answer
30 views

Posterior from a Poisson likelihood and prior

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: ...
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1answer
53 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 ...
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46 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 ...
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86 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 ...
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118 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 ...
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63 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 ...
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37 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 ...
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1answer
47 views

Updating Bayesian prior & likelihood for A/B test

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 $1$ ...
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1answer
54 views

Why does probabilistic PCA use Gaussian prior over latent variables?

I am currently reading papers about probabilistic PCA and I am wondering why is Gaussian prior (and not some other prior) chosen for the latent variables? Is it just because it's simple or is there ...
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123 views

How does one formalize a prior probability distribution? Are there rules of thumb or tips one should use?

While I like to think I have good grasp of the concept of prior information in Bayesian statistical analysis and decision making, I often have trouble wrapping my head around its application. I have ...
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151 views

Specifying the priors for multivariate MCMCglmm mixed model in R (Poisson distribution)

I am trying to build a model using MCMCglmm. Ideally, I would use a negative binomial distribution for my response; however, this is not an option in MCMCglmm. I don't know of any open-source ...
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20 views

Reestimating prior probabilities after making assumptions about them

Imagine that I have $M$ observations, and each of them can be classified in two different ways: it belongs to one of the classes $A = \{A_1, A_2\}$ and to one of the classes $B = \{B_1, ..., B_n\}$. ...
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94 views

What does Jaynes mean by “mollusk-like quality”?

I am trying to read Prior Probabilities (1968), by Edwin T. Jaynes. In two sections he discusses mollusk-like qualities [of parameter spaces]: The real problem, therefore, must be stated rather ...
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1answer
52 views

Is assuming a vague prior different than assuming nothing?

If you use a vague prior in a Bayesian analysis so that $\Pr(A) = 1\mathbin/n$, then \[ \Pr(\pi\mid x) = \frac{\Pr(x\mid \pi)\Pr(\pi)}{\Pr(x)} = \frac{\mathcal{L}(\pi;x)}{n\Pr(x)} \propto ...
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58 views

Incorporating prior knowledge in machine learning?

What are recommended methods for incorporating prior knowledge in machine learning tasks? By prior knowledge I am referring to information like which features are a priori considered more important or ...
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26 views

Is the inductive bias a prior?

Wikipedia defines it like this: The inductive bias (also known as learning bias) of a learning algorithm is the set of assumptions that the learner uses to predict outputs given inputs that it has ...
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1answer
63 views

Linear regression with t-distribution prior for beta coefficients

Having: $$y\sim N_n(X\beta, \sigma^2 I_n)$$ with prior distributions: $$\beta\sim t_\nu(\beta_0, B_0)$$ and $$\sigma^2 \sim IG(\alpha_0/ 2, \delta_0/2)$$ What would be the conditional posterior of ...
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119 views

Can prior distributions incorporate uncertainty AND variability in parameters?

I am trying to understand how to interpret Bayesian prior and posterior distributions in situations where there is believed to be variability in model parameters (due to variation in the population ...
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25 views

Posterior distribution dependent on two variables make inferences about one

If i have some model for X that depends on THETA1, THETA2 and has a posterior P(THETA1,THETA2 | x1,...,xn). How would I make inferences just about THETA1? What I am thinking so far is just to ...
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1answer
34 views

exponential likelihood with normal prior

If I have a likelihood function based on the exponential distribution with parameter $\lambda$ , why would a normal prior with a very large variance be inappropriate for $\lambda$?
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28 views

exponential likelihood uniform prior

Say I have a sample $x_1,...,x_n$ from an exponential distribution where $x_i$ is i.i.d exponential with parameter $\lambda$. 1) Suppose the prior for $\lambda$ was a uniform $0$ to $\beta$, what ...
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76 views

Set G in prior for MCMCglmm in R

I am new to the MCMCglmm package in R, and rather new to glm models in general. I have a dataset of species traits and whether or not they have been introduced outside of their native range. I would ...
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1answer
56 views

Include prior knowledge in regression model

I've a classical dataset with real attributes and I want to perform a regression. But, not all the entries in the training dataset are trustworthy; there is an attribute that I can turn into a ...
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35 views

Prior pdf decay in Recursive Bayesian Estimation

I'm doing Recursive Bayesian Estimation numerically. I have a state vector, x, that I'm trying to estimate by regularly taking noisy measurements, z. I use Posterior = Likelihood x Prior / ...
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286 views

Why is Lasso penalty equivalent to the double exponential (Laplace) prior?

I have read in a number of references that the Lasso estimate for the regression parameter vector $B$ is equivalent to the posterior mode of $B$ in which the prior distribution for each $B_i$ is a ...
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16 views

Jeffrey's Prior for Dirichlet

As I understand, Jeffreys' Prior for any distribution is an objective one, and in the wiki, the Jeffreys' prior is written to be 0.5. On the other hand, I've found these notes which state that the ...
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14 views

priors in weighted least squares

I have read mostly about weights in weighted least square that deal with outliers. But, lets assume that we have prior understanding of the process from which data is generated. And we can see that ...