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.

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Best estimate of underdetermined system using prior

I have measured two variables which depend on the same set of four parameters. I want to know the parameters which best explain my measurements. Of course, I cannot solve for four unknowns from just ...
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58 views

Posterior density for a linear regression model

Given a classical linear regression model $$y = X\beta + \varepsilon,$$ $$\varepsilon\sim N(0,\sigma^2I_n),$$ the posterior density is proportional to the product of the likelihood and the selected ...
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81 views

What would be an ignorance prior of AB, given the probabilities of A and B?

Let us have two events, $A$ and $B$ whose probabilities are $P(A)$ and $P(B)$. In the absence of any other information, what would be a reasonable probability to assign to $AB$, that is, $A$ and $B$ ...
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What are the limitations on the flat prior in the BAT framework?

By default, the BAT framework sets the flat priors on the parameter. My question is: how in this case allowed range for a parameter of the flat function is estimated?
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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 ...
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28 views

Prior/degree of belief/degree of lack-of-information/algorithms/complexity

For a long time I had a bit of difficulty understanding what "degree of belief" means. Recently I had some thoughts about it and I wonder if they make any sense, or is there some literature about ...
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406 views

What is the interpretation for the priors in the derivation of Laplace smoothing?

Laplace smoothing has a generalisation that can be justified with the use of Bayes formula. Let $f(x;\alpha,\beta)$ be the (non-normalised) beta distribution, i.e. $$f(x;\alpha,\beta) = x^{\alpha-1}(...
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128 views

Biassing priors to improve confusion matrix

I have a text classification problem to solve. I need to classify a given sample of text into one of two classes A or B. My training set has about 30% A and 70% B. This is my prior. Now, when I ...
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Applying Bayesian Gaussian movement question

I have a question from my stats class that I am confused about how to proceed with. I have a general idea of what I am to do but I am not sure how to start. The question is about a car that is moving ...
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48 views

Restriction in bayesian likelihood expression

I'm doing MCMC simulation but I'm confused in some part of my model. I dont know which of my likelihood expression is right. My model is as following. $\gamma$ = $(\gamma_{1},\cdots,\gamma_{K})$ and ...
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Informative priors

I have a general query regarding informativeness of priors, since my laptops gone down and not able to run this on Stan to check (but from previous runs I think this was the case). If the priors used ...
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140 views

Equivalent of using a Poisson prior in terms of a penalized regression?

I know that most penalized regressions have also a Bayesian interpretation, e.g. ridge least squares regression corresponds to the MAP estimate obtained under a Gaussian prior in a Bayesian regression,...
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How constraints about the posterior can be used in bayesian inference?

I'm referring to bayesian inference as described by E. T. Jaynes in "Probability theory: The logic of science": $$P(\theta \mid X, I) = P(\theta\mid I) \cdot \frac{P(X \mid \theta, I)}{P(X \mid I)}$$ ...
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Is a mean of 0 and sd of 0.5 acceptable scaling for a normal prior?

I have thus far been fitting a Cauchy prior (centred at 0; scale of 2.5) on my parameters, as recommended by Gelman et al. (2008). They recommend scaling variables (predictors, and in the case of a ...
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746 views

Correcting an unnormalized posterior for the weighting of the prior

Bayes' rule tells us that, $P(h|d) = \frac{ P(d|h) P(h) }{ \sum_i P(d|h_i) P(h_i) }$. Let's say we have four hypotheses: $h_1$, $h_2$, $h_3$, and $h_4$. The likelihood over hypotheses looks like ...
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Basic Bayesian updating priors

I have a question about simple Bayesian reasoning. Suppose there are N datasets, two mutually exclusive models, priors for models can be chosen as 0.5 to 0.5. The datasets have different ...
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61 views

Bayesian improper prior on product of parameters

I'm interested in two sets of variables coming from bivariate normal distribution. $X_i$'s iid ~ N($\mu_x$,$\sigma^2$) $Y_i$'s iid ~ N($\mu_y$,$\sigma^2$) and Cov(Xi,Yi)=$\sigma_{xy}^2$. Now $\mu_x$= $...
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94 views

How to restrict positive posterior?

Is there a way to restrict posterior, say, to only positive values or an interval of values? Let's say I want to estimate a linear model, y = a + Xb, using Bayesian techniques. I specify priors for a ...
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37 views

Hierarchial Bayesian approaches versus simple prior based approaches

The point of Hierarchical Bayesian models is that you can get parameters for different "hierarchies" within your data. For example, if you have 10 data points for one person, 10 for the next and so on,...
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254 views

Specification of priors for multivariet hierarchical regression using MCMCglmm

I'm analyzing data from experiment, where people had to select a point in plane. I'm trying to asses which atributes of the task and personality are asociated with the outcome. Becouse we used ...
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32 views

Improved state of knowledge due to measurement

In a set of notes, the following stated: "If we consider a measurement where the initial state of knowledge (or prior) $P(x)$ is mistaken and not equal to the true probability, then the measurement ...
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250 views

Normal distribution prior and posterior

Consider a razor company that is testing a new men's facial razor before releasing it for sale. They want it to have a certain performance rating (out of 100) before releasing it. Based on previous ...
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292 views

Translating user-defined joint-distribution from PyMC to PyMC3

I'm attempting to set up a simple beta binomial hierarchical model with an uninformative prior in PyMC3. I've read that the uninformative prior for this model should have alpha and beta hyper-...
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246 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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76 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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392 views

Set prior for logistic regression in R when using unequal group sizes (29 versus 48 cases)

I have 29 cases for negative outcome (0) and 48 cases for positive outcome (1). I fit my data with logistic regression model ...
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178 views

Why does Empirical Bayes work in my simple case?

I have a problem where I am trying to classify data into two groups using a single parameter. The distribution of this parameter is Gaussian for two groups, so what I'm dealing with is two overlapping ...
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96 views

Interpretation of priors in example

Suppose you have 3 variances $W_{1},W_{2},W_{3}$ that can be expressed as $W_{j}=q_{j}V$ with $j = 1,2,3$. According to one model, $W_{3}$ should be pronounced and $W_{1}$, $W_{2}$ should be small to ...
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32 views

Combining two estimates of p in a binomial estimation

I have an estimation problem for a binomial data. I got a sample and from that I can get an estimation. But I also have a kind of prior information about the p. But mind it, this prior is just a ...
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101 views

learning hyper parameters: are we allowed to touch the prior parameters after observing the data?

There are many algorithms/applications that aim to learn the hyper parameters i.e. the parameters of a prior distribution from the observed data. A typical algorithm works in an iterative function ...
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223 views

Weighted Distributions in Predictive Model

I'm not a statistician but do work with large datasets and have a problem I'd like to use a predictive model for. I have two datasets that I'd like to use together to build predictions. The first ...
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156 views

What Bayesian prior for normally distributed data with knowledge observations are too high

I'm just learning Bayesian statistics with stan, so bear with me. Let's say we have one observed data points: ...
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72 views

How are these priors generated? [closed]

I am trying going through an exercise, I don't understand how the information provided in the text below transitions into the parameters displayed in the beta priors. How are these informative priors ...
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34 views

Bayesian estimation Prior adaptation [closed]

I have a dataset of 1 dimensional 20points as prior information, so assuming prior distribution to be Gaussian distribution we can easily find its variance and mean. Now we will use this prior finding ...
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1answer
31 views

Some help interpreting posterior plots

I needed some help interpreting and comparing the plots that I created. I'm not really sure what is the most important thing to talk about when comparing these plots. I know for these plots that the ...
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148 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...

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