# 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.

932 questions
Filter by
Sorted by
Tagged with
1 vote
37 views

### Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?

Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"? Suppose you assume a probability distribution. You ...
1 vote
16 views

### When to specify multivariate versus univariate priors on parameters?

Suppose a linear regression model: $$y \sim Normal(\beta X, \sigma)$$ For our purposes, assume $y$ is a univariate outcome and $X$ is a design matrix containing an intercept and one additional ...
17 views

### How to improve the predictions of a model when we have too few predictor variables?

I tried to use a linear model to explain a variable "age" with two variables "x1" and "x2". I can clearly see a decreasing slope inside my scatterplot for age vs x1, or ...
• 1
11 views

### Using a Generalized Beta Distribution of the Second Kind as a Prior in Stan Linear Regression

So I'm considering a simple linear regression model with $p = 1$ predictor $$y = \beta x + \epsilon$$ where $\epsilon \sim N(0,\sigma^2)$. I want to use a generalised beta distribution of the second ...
• 249
1k views

### Priors that do not become irrelevant with large sample sizes

This may be a weird question. My colleagues and I are working on a medical estimation problem, where relevant prior knowledge regarding plausible values of some physiological parameters exists. In ...
• 1,484
22 views

### Bayesian replication, but with new variables

Suppose I have data I've collected containing predictor variables $X_1, X_2$, and $X_3$. I build a main effects statistical model predicting $Y$ from these predictors and estimate the relevant ...
• 569
1 vote
90 views

### How to use priors on the parameter number with an information criterion (AIC, BIC, …)?

Example The example is made up because I hope that it’s more accessible than my actual problem. I want to determine the number of planets of a star. I have: data for some astronomical observable of ...
• 2,257
100 views

• 133
55 views

### How to find the marginal prior distribution?

Suppose that $\beta$ has the following prior $$\beta|\zeta \sim f(\beta,\zeta)$$ Then I know that the marginal prior distribution of $\beta$ is given by $$\int f(\beta,\zeta) d\zeta$$ However, ...
• 273
1 vote
60 views

### Is it possible to estimate the parameters of a superposition of Poisson processes through Bayesian inference from a binarized sequence?

My question is complementary to a previous problem : Bayesian inference on binarized Poisson distribution. I retake the previous notations. Problem description : I am counting the number of balls ...
1 vote
31 views

### Creating a better prior based on past observations

Based on this post, In plain english, update a prior in bayesian inference means that you start with some guesses about the probability of an event occuring (prior probability), then you observe what ...
• 1,609
1 vote
32 views

### Bayesian Prior definition [closed]

The prior of an inference problem where we try to infer $x$ from observations $y$ is defined as $P(X)$. Often (e.g.) I see another definition where the prior is defined as $P(X|Q)$, what exactly is $Q$...
1 vote
50 views

### In Bayesian hierarchical models, what is the difference between an Empirical Bayesian approach to parametrising priors vs using flat hyperpriors

Say I have a simple hierarchical model, where: $y_{g,i} = \beta_g x_{g,i} + e_{g,i}$ where $g$ represents the group, $i$ represents the individual within the group, and $e$ is the error. So the ...
• 51
34 views

### Inverse or reciprocal distribution of a discrete random variable

https://en.m.wikipedia.org/wiki/Inverse_distribution Above wikipedia article only talks about continuous random variables. If Y=1/X, where X is strictly positive, and density function for X is given ...
44 views

### Non-Dirichlet Prior for $Cat(\theta)$ parameter that can tractably be integrated out (for Latent Dirichlet Analysis)?

In LDA Topic Models, it is standard to 'integrate out' the $\theta$ parameter, which contains a document's Categorical probabilities of drawing each topic. QUESTION If one uses the standard Dirichlet ...
• 111
32 views

### Bayesian statistics: what is the variable we are integrating in?

This is a screenshot from Bayesian Data Analysis by Gelman. I am a little bit confused by Equation 1.4 (first and second lines), having read Equation 1.3. In Equation 1.3, the variable of integration ...
• 8,155
30 views

Say we start with a linear regression model of the form $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon, \quad \epsilon \sim N(0, \sigma^2)$$ with the conjugate prior \begin{align*} &\sigma^... 0 votes 1 answer 19 views ### Prior selection in Gaussian - an application to height measurement Say I have just purchased ACME's Tree Height Measuring Device (THMD). ACME states that the error \epsilon in tree height measurement from this device can be modelled as a normal distribution with ... 1 vote 1 answer 81 views ### The PDF of the Data (Marginal Likelihood) Given the Prior of a Gamma Distribution with Prior on the  \beta  Paraneter Given a model where  x_i | \beta \sim \mathcal{Gamma} ( \alpha, \beta )  where  \beta \sim \mathcal{Gamma} ( \alpha0, \beta0 ) , is there a closed form formula for the PDF of  x_i ? Namely, what'... • 135 2 votes 2 answers 98 views ### The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean Given a model where  x_i | \mu \sim \mathcal{N} ( \mu, \sigma^2 )  where  \mu \sim \mathcal{N} ( \mu_0, \sigma_0^2 ) , is there a closed form formula for the PDF of  x_i ? Namely, what's  p (... • 135 0 votes 1 answer 59 views ### Is there any strong argument about objective/non-informative improper prior? Decades ago improper objective priors - e.g. \pi(\sigma) \propto \sigma^{-1}, \sigma > 0, for a scale parameter - were considered problematic because some authors thought they were leading to the ... • 41 0 votes 0 answers 13 views ### Beta distribution equivalence with two redondant parameters [duplicate] context In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive ... 1 vote 1 answer 26 views ### Kl Divergence between factorized Gaussian and standard normal Given two distributions, one a parameterized gaussian and the other a standard normal gaussian: q(x) \sim \mathcal{N}(\mu,\sigma) p(x) \sim \mathcal{N}(0,I) We want to compute the KL Divergence ... 0 votes 1 answer 40 views ### Bayesian statistics Assuming I have that Y_i\mid \mu is an iid ~ N(\mu,\sigma^2), for i \in (1,\dotsc,n) with \sigma_i known and improper prior \pi(\mu)=1 for all \mu. i. How can I derive a formula for the ... 0 votes 0 answers 35 views ### Bayes: How to use results from a single study to shape data-driven priors One way to construct informative priors for a subsequent Bayesian analysis is to carry out a meta-analysis of previous studies. Here, substantial research has been done. However, what to do when there ... • 491 0 votes 0 answers 51 views ### Derivation of posterior distribution under Dirichlet prior distribution: suppose that \mathbf{y}=(y_1, y_2, \cdots, y_n) is a vector of n observed sample points drawn from a mixture of g components, and \mathbf{z}=(z_1, z_2, \cdots, z_n) is a vector of latent ... • 131 1 vote 1 answer 71 views ### Partially specified Bayesian prior? In bayesian linear regression for example, we may specify a model as:y_i \sim N(\beta_0 + \beta_1 x_i, \epsilon^2) \\\\ \beta_0 \sim N(0, \tau_0^2) \\\\ \beta_1 \sim N(0, \tau_1^2) \\\\ \epsilon \...
• 121
If we are given a standard geometric distribution $(1-p)^{x-1} p$, with $0<p<1$ what would be a suitable non-informative prior for this?
Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$, ...