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.

Filter by
Sorted by
Tagged with
0
votes
1answer
29 views

Does quadratic loss find the median of the prior distribution?

Does quadratic loss find the median of the prior distribution? Someone told me linear loss finds the mean, all-nothing loss function finds the mode of the prior.
0
votes
2answers
97 views

Pick a prior for my bayesian generalised linear model with binary outcomes

I need help in my choice of a prior for a bayesian model. I have data from a set of participants responding to a set of yes/no questions. Answers are correct or incorrect. I suspect some questions ...
16
votes
2answers
3k views

What's a good prior distribution for degrees of freedom in a t distribution?

I want to use a t distribution to model short interval asset returns in a bayesian model. I'd like to estimate both the degrees of freedom (along with other parameters in my model) for the ...
2
votes
1answer
51 views

No operational difference between a prior density $f(\theta)$ and $f(x \vert \theta)$?

I am currently studying the textbook In All Likelihood -- Statistical Modelling and Inference Using Likelihood by Yudi Pawitan. Section Bayesians versus frequentists of chapter 1 says the following: ...
1
vote
0answers
92 views

Multiple priors in Bayesian estimation

Typical Bayesian estimation equation is: Estimate = ( SampleSize * SampleEstimate + PrioriEstimateWeight * PrioriEstimate) / ( SampleSize + PrioriEstimateWeight ) Typically, the PrioriEstimate is ...
0
votes
1answer
161 views

Would a posterior distribution with a flat prior look identical to the likelihood?

Graphically, let us assume that we have a flat prior for a normal distribution (a horizontal line at y=1 over all real numbers). Then, we have a likelihood function that resembles a normal ...
7
votes
3answers
333 views

Putting prior on a function of parameters

Suppose that we have a likelihood for a conditional distribution $p(y|X,\theta)$. For clarity purposes we can consider linear regression with homescadastic errors. It is clear to me how one will put a ...
1
vote
0answers
33 views

How to formally express low confidence in data in Bayesian framework?

I am trying to build a Bayesian regression model, and I am looking for a formal way to express low confidence in noisy data. I know that should be reflected in the prior specification.. I am using ...
2
votes
1answer
139 views

Posterior distribution dependent on two variables make inferences about one

If I have some model for X that depends on $\theta_1, \theta_2$ and has a posterior $P(\theta_1, \theta_2 | x_1, ... x_n)$, how would I make inferences just about $\theta_1$? What I am thinking so ...
0
votes
1answer
217 views

Prior, Posterior and Bayes rule for discrete random variables. Calculating Posteriors?

For the discrete case in the image below below, could someone explain why a density, $f(x)$, is used rather than a pmf, $p(x)$. My notes say that, for most cases the value of the parameter takes ...
0
votes
0answers
15 views

Covariance of two random variables with random parametres with a prior distribution

Let $N_1, N_2$ be conditionally independent random variables with probability distributions $Pois(t_1\theta)$ and $Pois(t_2\theta)$ respectively. Constants $t_1, t_2$ are known. The parameter $\theta$ ...
1
vote
0answers
92 views

Bishop: Understanding the prior and posterior for a curve fitting example (1.2)

In Bishop's Pattern Recognition and Machine Learning Book, he uses an example of fitting a polynomial to data collected from a sinusoidal curve with Gaussian noise. The goal is to find the most ...
1
vote
1answer
272 views

Bayesian Homework: Uniform Prior

Suppose posterior density of parameter $\theta$ is $$\pi(\theta|\mathbf x)=\frac{\Gamma(5)}{\Gamma(3)\Gamma(2)}\theta^{3-1}(1-\theta)^{2-1}.$$ Now I have to find which of the two hypotheses $H_1:\...
3
votes
1answer
46 views

Calculating conditional probability $P(\Theta \le c | Y=0)$

Let $Y$ be a random variable with $Pois(\theta)$ distribution and the parameter $\theta$ be a realization of a random variable $\Theta$ with a priori distribution $Exp(\lambda)$. The task is to ...
5
votes
2answers
425 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 ...
7
votes
3answers
451 views

Which pdf to choose for the prior of an angle?

I have a system in which one uncertain variable is a direction in two dimensions. If I want to define a prior for this, is there an elegant way to reflect the fact that the parameter space dimension ...
1
vote
1answer
341 views

appropriate use of prior weight in glm

I am modelling the effects of several variables on the number of scallops caught. The variables are: Stratum (categorical, 23 levels), Vessel (categorical, 3 levels) , Density of scallops (continuous)...
5
votes
2answers
109 views

Frameworks for modeling prior knowledge other than Bayesian statistics

It is my understanding that one can easily model prior knowledge about variables or even models with Bayesian statistics. In a certain way, Bayesian stats "forces you" to think about prior knowledge ...
2
votes
2answers
149 views

Can an improper prior distribution be informative?

I have just worked through an example where, with an improper prior, the bayesian estimator equals the maximum likelihood estimator, leading me to believe that improper priors are uninformative. But ...
0
votes
1answer
49 views

Does frequentist statistics support the use of priors?

My understanding of the main difference between Frequentist vs Bayesian stats is that the former treats parameters as variables with fixed but unknown values whereas the latter treats them as random ...
0
votes
0answers
52 views

how to determine a priori probability distribution of sigama2 in montecarlo simulation?

1、the monte carlo simulation code in SAS: Example1: https://support.sas.com/rnd/app/stat/examples/BayesStd/new_example/index.html ...
3
votes
0answers
94 views

Understanding the influence of the prior distribution on the original parametrization

Be $y_1,y_2,..y_2$ a simple random sampling from $p(y|\theta)$. Be $\theta$ a parameter with a given prior distribution $p(\theta)$. A way to understand how much informative is $p(\theta)$ is to plot ...
8
votes
2answers
3k views

When does the maximum likelihood correspond to a reference prior?

I have been reading James V. Stone's very nice books "Bayes' Rule" and "Information Theory". I want to know which sections of the books I did not understand and thus need to re-...
1
vote
1answer
303 views

Choice of prior and combination with likelihood, sample from exp distribution

I am taking a course in Statistical Theory of Science. We have an exercise where we are comparing "Classical" and Bayesian parameter estimation. We have a sample $(x_1,...,x_5) = (0.28,0.30,0.94,0.42,...
6
votes
1answer
5k views

half-cauchy prior for scale parameter

I am looking for a prior for a scale parameter for which I have prior knowledge such that: "$\sigma$ typically does not exceed 100." ("typically" meaning that occasionnally this can happen). In such ...
1
vote
1answer
49 views

Deriving Marginal Distribution of Poisson [duplicate]

How do you find the marginal distribution of a Poisson distribution given a gamma(a,b) prior?
3
votes
0answers
31 views

Informative priors from frequentist regression

Say I run a standard frequentist regression on a subset of my data (for example using lm in R) and obtain some values for the coefficients of the model, I have fairly large data set ~100k samples. Now ...
1
vote
0answers
197 views

Quantify the output variance of a neural network classifier

Lately at work we are dealing with a theoretical problem concerning the output variance of a neural network classifier. To set the scene, suppose you have an image classifier, which takes an image as ...
4
votes
2answers
107 views

Accounting for uncertain information (few observations) in a prior (empirial Bayes)

I did not really know how to choose an adequate title for this question, so please feel free to change it. I have a weird case wherein frequentist and Bayesian philosophies come together. I am ...
0
votes
1answer
41 views

How to Change prior probabilities for predicted variable in neural networks and other methods in SPSS Statistics

i am trying to find right model for predicting categorical variable with two values. Problem is that ratio of cases in group 1 and group 2 is not equal but rather in ratio of 2:1. When i try to find a ...
20
votes
1answer
885 views

What are the best ways to generate Bayesian prior estimates using beliefs of non-statisticians?

I work with a lot of qualitative researchers and designers. Many of whom interact with users and develop strong, often accurate intuitions about how the data should look. I frequently try to quantify ...
0
votes
0answers
20 views

Sampling a proposed value with a limited range target when running MCMC [duplicate]

I want to do an MCMC algorithm and need to sample a proposed value from a proposed distribution. In the Metropolis algorithm, people usually use a normal distribution as proposal. But if the prior ...
1
vote
0answers
20 views

Linear regression - Bayesian Predictive distribution

I am trying to answer a question about linear regression but i am stuck: $y=w \cdot x + \epsilon, \epsilon \sim N(0,\alpha)$ i am also given a prior: $w\sim N(0,\beta)$ from which i was able to ...
2
votes
1answer
280 views

Random-walk prior with ridge-like regularizarion?

I am working with a model that contains a large number of coefficients, arranged in an ordered vector $\beta_1, \dots, \, \beta_N $. I have some prior knowledge that could be used to improve the ...
1
vote
0answers
22 views

Clarifying a proof of a particular paper on Steins Estimator

I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases. The doubt is a s follows: Given $$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
4
votes
1answer
56 views

What prior would lead to $\ell_\infty$ regularization of model weights?

Gaussian prior on weights of a GLM lead to Ridge / $\ell_2$ squared regularization. Laplace prior leads to $\ell_1$ regularization Question What prior would lead to $\ell_\infty$ regularization ?
5
votes
0answers
802 views

Bayes-Poincaré solution to the Behrens-Fisher problem 2: calculations for Jeffreys’ priors [closed]

In a previous post Bayes-Poincaré solution to k-sample tests for comparison and the Behrens-Fisher problem?, the classical Bayesian and likelihoodist solutions to 2-sample tests for comparison and the ...
7
votes
2answers
2k views

Jeffreys prior for continuous uniform distribution

A nonnegative random variable $x$ has a continuous uniform distribution in the interval $(0,\theta)$. Therefore, the likelihood is given by: $f(x|\theta) = \frac{1}{\theta}I(x\leq\theta)$, where $I$ ...
0
votes
1answer
44 views

How to select variables when using shrinkage priors?

I am fitting a linear regression model using shrinkage priors (Horseshoe and Laplace/LASSO). This shrinks many of the variables close to zero, but I would like to select the variables. Can I use the ...
2
votes
0answers
324 views

Why is Half-Cauchy, Half-Student-t as prior for variance parameters better than a normal distribution?

Gelman often refers to using half-cauchy or half-student-t distributions for variance parameters. Why is it better than using a vague normal distribution such as N(0,10)? Can somebody explain me the ...
3
votes
1answer
148 views

Why in Hamiltonian MCMC do we multiply the posterior distribution by the likelihood?

So maybe I am misunderstanding what the author is staying, but I am reading Chapter 14 of Kruschke's Doing Bayesian Analysis. I am reading about the software Stan and how it uses the Hamiltonian MCMC ...
0
votes
1answer
22 views

What kind of a priori distribution for the Markov Switching models?

Why in the Markov-Switching models is chosen as prior distribution for the probability of the transaction as follows: $$f(P) \propto \prod_{i=1}^K \left(\prod_{j=1}^K p_{i,j}\right) I \left\{0 < ...
3
votes
2answers
16k views

Uniform vs Beta(1,1) prior

Is there any difference in applying a uniform prior or a Beta(1,1) prior for your Bayesian analysis ?In which conditions is one preferred over the other ?
5
votes
2answers
1k 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 ...
1
vote
0answers
54 views

What do these equations on Bayesian regression (MAP) from Chapter 3.3 in PRML by Bishop mean?

This was taken from Ch 3.3 on Bayesian Linear Regression from Pattern Recognition in Machine Learning by Bishop. Apparently the posterior can be described by eq 3.49. Eq 3.48 represents the prior ...
1
vote
1answer
52 views

Explanation of Equation 5.3 from Gaussian Processes for Machine Learning

I am currently reading through C. E. Rasmussen & C. K. I. Williams' Gaussian Processes for Machine Learning and was going through chapter 5. I could not exactly understand the derivation of ...
4
votes
2answers
1k views

Shrinkage priors

I am building a Bayesian model where I to put shrinkage priors such as spike and slab, horseshoe prior, etc on some parameters for feature selection, but I am not able to decide which one is the best. ...
4
votes
3answers
510 views

Bayesian prior via cross validation

I have a particular problem where I am using Bayesian techniques to estimate parameters of a distribution of a random variable. I would like to use an external source of data to determine an ...
1
vote
1answer
144 views

Bayesian Linear Regression, trouble with posterior. Variance equal identity

I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $ \pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...
37
votes
6answers
5k views

If a credible interval has a flat prior, is a 95% confidence interval equal to a 95% credible interval?

I'm very new to Bayesian statistics, and this may be a silly question. Nevertheless: Consider a credible interval with a prior that specifies a uniform distribution. For example, from 0 to 1, where 0 ...

1
3 4
5
6 7
17