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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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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50 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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65 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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Is Independent jeffreys prior different from independent reference prior?

I have a model involving two scalar parameters $\theta_1$ and $\theta_2$ and derive the Jeffreys prior for $\theta_1$ and $\theta_2$ independently (so for, e.g. $\pi(\theta_1)$, setting in the ...
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14 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
42 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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44 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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113 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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48 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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19 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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85 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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49 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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52 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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24 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
54 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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2answers
111 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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22 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
27 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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24 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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38 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
49 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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30 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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186 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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12 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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12 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 ...
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Generating priors form test statistics, and applying them to reported summary statistics

Hi a very specific question on generating sensible priors. I have estimates of the parameter $\beta$ in the following linear model, for a few hundred variables X separately (adding 1 X to the model ...
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1answer
22 views

Truncated prior leads to non-intuitive posterior

I am setting up a linear regression model for continuous data that is Normally distributed. For this model, I want to assume that my $\beta$ predictor is truncated to be positive, that is $$\beta \sim ...
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54 views

How to define prior for beta-binomial A/B test

I would like to run an A/B test using a Bayesian beta-binomial model whereby I would state probabilities such as P(pB>pA) in place of using a traditional T-test. I've read that the prior should be ...
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1answer
12 views

Odd Acceptance Ratio

Recently, I read some paper and sometimes they draw a sample $s\sim N(a,b)\times exp(d)$. But they defined the prior as $N(A,b) \times exp(D)$ with unknown $A$ and $D$. Therefore in the acceptance ...
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1answer
41 views

How to perform a sensitivity analysis in Bayesian statistics?

Bayesian inference is drawn from the posterior distribution or - in case we are interested in forecasting - from the predictive posterior distribution. However, these values are heavily affected by ...
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6 views

Include window function information as prior

I have time series of radial-velocity measurements for different stars. Because we can only observe at night, sometimes only when the moon is not in the sky, and only in the part of the year when the ...
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1answer
82 views

Posterior of the linear regression model with g-Prior

Assume a linear model of the form $$Y=X\beta+\epsilon$$ where $\epsilon$ has a multivariate Normal distribution with mean $0_N$ and covariance matrix $\sigma^2 I_N$. I would like to perform simple of ...
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22 views

finding non-informative prior for weibull and pareto distributions

How do I find/calculate the non-informative prior for the pareto and weibull distributions? (In particular, I want to find the non-informative prior for the parameters of the above mentioned ...
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232 views

Why is Laplace prior producing sparse solutions?

I was looking through the literature on regularization, and often see paragraphs that links L2 regulatization with Gaussian prior, and L1 with Laplace centered on zero. I know how these priors look ...
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1answer
38 views

Principled method for choosing the strength of a prior?

I'm working on an application similar to this one, where the intent is to sort a list of items with ratings according to the best estimate of their average rating. The solution proposed in this link, ...
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1answer
32 views

How to include Cohen's D from Meta-Analysis into JAGS/BUGS mean difference model

I was wondering if anyone knew of a way to use Cohen's D (and Standard Error) as an informed prior while building a BUGS model (to be tested in JAGS through R) that compares the mean difference ...
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69 views

Question about reparameterization and setting up a 'noninformative' hyperprior distribution

When I read Bayesian Data Analysis, 3rd ed. by Andrew Gelman, et al., I can not understand the problem of "choosing a standard parameterization and setting up a 'non-informative' hyperprior ...
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1answer
215 views

What would be the reason that the posterior distribution looks like the prior using MCMC

I am trying to use MCMC to obtain the posterior probabilities of the free parameters of a model. I have tried first to leave two free parameters for my model and I was able to estimate the posterior ...
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68 views

Linear regression with prior

Im trying to estimate the linear curve (y~x) where I know intercept must be normally distributed around -100, and slope always positive and normally distributed around 2 (blue continous line in plots ...
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100 views

Definition of improper priors

If a prior integrates to a finite constant that is not 1, is it still considered proper? Is a prior only improper if it integrates to an nonfinite value?
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1answer
41 views

Finding Distribution in R

I have a Bayesian Inference Question. Prior ~ Normal and f(x|theta)~Normal. Now, I want to get the distribution of h(x) in R. I mean is there any function or anything in R that could tell that ...
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48 views

Using Bayesian econometrics to forecast macro data (BVAR model)

I am in the middle of a Bayesian class. I have to make a project where I implement Bayesian statistics. I have chosen to do this on macro data. As far as I can see the optimal model to forecast ...
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40 views

Does a proper prior lead always a proper posterior?

Does a proper prior lead always a proper posterior? I cannot check whether the posterior is proper, so I was wondering if this assumption is always satisfied .
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17 views

Prior probability distribution question

Let, $ \mathbb H = (\theta_{1},\theta_{2},..,\theta_{k})$, where $'\mathbb H'$ denotes the parametric space. Let $ X_{1},X_{2},...,X_{n} $ be $'n'$ i.i.d. observations from a common density function, ...
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1answer
57 views

Prior distributions letting a small sample “speak”

I’ve got a general question. Let k be a parameter which must be estimated. It lies within the interval $[a, b]$, $a$ and $b$ being finite real numbers. Let us further assume we dispose of a series ...
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55 views

Problem with prior mean in MOE (Bayesian optimization)

I am playing with MOE package (yelp.github.io/MOE) - I try to optimize some function of one variable, adding one point for sample at a time. Here is the intermediate chart I got: Blue line is the ...
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1answer
121 views

Priors for Truncated Parameters - RJAGS

I would like to estimate the parameters of a specific model. The model specification is as follows: $p_t = k_t + B_t/(1-B_t) + \eta_t$, where $ \eta_t \sim N(0, \sigma^2)$ $R_{t+1} = R_{t} + R_t ...
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1answer
80 views

Gamma parameterization and how to randomly generate $\sigma$'s for use in `rnorm(n, $\mu$, $\sigma$)`

Say I have a normal distribution parameterized with a mean ($\mu$) and precision ($\tau = 1/\sigma^2)$. In JAGS, I would specify a prior for $\tau$ as ...
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1answer
53 views

Is this posterior probability integral right?

From Wiki: where , k is binomially distributed, and I'm not sure about u. I'm thinking that the second line should be: I mean, if we let X represent the toss of a die, then $P(X = 1, 2, ...
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133 views

Is this denominator of a posterior distribution the marginal distribution of Y?

From Wikipedia: , where Is the denominator (above pics are from Wiki) the marginal distribution of Y? Intuitively, it seems that way so that when we cross-multiply, LHS and RHS are mirrors. ...