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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Estimating the Posterior Probability of Playing Coin 1
Suppose someone has two coins with probability $p_1$ and $p_2$ of getting heads with $p_2\le p_1$. The person is throwing the coins and may switch from playing coin 1 to coin 2 with probability $\...
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Which proposal function should be used in this particular case of the Metropolis-Hastings algorithm?
As part of my research, I would like to apply the Metropolis-Hastings in order to sample from some posterior distribution. More precisely, the data comes from a multivariate normal distribution in the ...
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Bayesian: Formally comparing prior and posterior distributions
Consider Bayesian inference in the regression model:
$$ \begin{align*} y &= \beta_0+\beta_1x_1+\beta_2x_2 + \varepsilon \\ \varepsilon &\sim
\mathcal{N}(0,\sigma^2) \end{align*}$$
Suppose we ...
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Sum to zero contrast that makes it easy to express equal uncertainty about each factor level
How do I need to set-up a sum-to-zero contrast so that it is easy to express equal uncertainty about each factor level? E.g. when I go with the default offered by R such as:
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Uniform prior and poisson likelihood, what posterior distribution will be produced?
If i have a uniform distribution over a fixed specified and a finite range, and a Poisson likelihood distribution, what posterior will be produced?
The likelihood has this form $$P(\pmb{X}|
\pmb{\...
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Bayes rule and prior distribution query
The following is obtained from a research paper (page 2):
Consider that we have discretized a continuous random variable $\theta$ such that $\theta_1,...,\theta_d \in [0, \pi]$. We then have the ...
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Normalized prior for discretized random variable
I would like to confirm my understanding of deriving a prior from a discretized continuous random variable and the correct expression for the normalization condition of the prior, please advise if the ...
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How to choose default uninformative prior in the R Package BAS
I'm conducting a Bayesian multilevel logistic regression based on the Rpackage BAS. I'm a beginner in Bayesian statistics.
But in bas.glm, I don't understand and I don't know how to specify my prior. ...
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What are the implications of setting off-diagonal elements of estimated covariance to 0?
I have sometimes seen in published work that when estimating covariance matrices, off-diagonal elements are set to 0. For example, in this paper, $N$ neurons are recorded and authors wish to use the $...
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how can predictive distributions be considered as expectations?
I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
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How to obtain likelihood ($P(B/R)$ given the prior $P(R)$ and the posterior $P(R/B)$
I am working on a topic related to multiple-choice response. I would like to measure the efficiency of the information source (or a student’s information search) and I believe Bayesian statistics is ...
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Can I use the Mean Squared Prediction Error to select the prior SD in a CausalImpact model?
I'm using the CausalImpact package (in R), and (as I expect is typical) the findings are very sensitive to the prior being used.
I have an OK understanding, I think, of what the prior is doing in this ...
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Estimating Markov Chain Probabilities with Limited Data
Suppose I have some data on transitions between states of a Discrete Time Markov Chain. Let's say that transitions between some events are observed more frequently from others. For example, in a 3 ...
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Does the example given correspond to a prior predictive check?
Could someone explain to me precisely what is meant by prior predictive check, in Bayesian inference? In some documents, one uses observed data (“in which we ...
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How to decide the parameters of a Gamma distribution for a Gamma-Poisson model?
In Bayesian inference, the Gamma-Poisson model uses usually a Gamma($\alpha$,$\beta$) prior on the $\lambda$ parameter of the Poisson distribution.
Are there any rules for setting appropriate values ...
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Strange Variance Term for Normal Prior $w^2\sigma^2$
I've attached two screenshots, one with the question and one with the answer. It seems to me that the prior is wrong and it should include $w^2$ not $w^2\sigma^2$
I apologise for, including such a ...
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How to choose between gamma and Gaussian given a choice of gauges?
I'm trying to make the choice between the gamma and Gaussian distributions as a prior distribution for some data. When I learned statistics a while ago, I was given the rule of thumb: if your data ...
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BIC with non-negligible priors
I want to do model selection based on the best-fit/MAP/marginal posterior I find from an MCMC and likelihood maximization. I have a likelihood $\mathcal{L}(X|\theta)$, some informative priors $\pi(\...
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Understanding PRIOR option in SCORE statement for PROC LOGISTIC (SAS)
Say I have a binary response which I want to model with logistic regression on covariates $x$. Fitting a model with PROC LOGISTIC will fit MLE coefficients for the model
$$
\text{logit}(\pi) = \alpha +...
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Avoid singular fits in mixed models in R with blme - checking layman's priors
While fitting linear mixed models, I would like to avoid zero random-effects (ranef(model)) and cluster-level SD estimates (...
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Turning a list of cost into categorical probability mass distribution
Background
Given a noisy dataset $D$, I have to solve a classification problem where the possible anserwer is $i\in\{1,\dots,N\}$. So far I can get pretty decent result with an algorithm that, based ...
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How should uncertainties be treated when scaling data for optimisation
I have a large dataset for which I am using Bayesian statistics for parameter estimation and model selection (using MultiNest for more detail).
This involves setting a prior over which the nested ...
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How is data generated when using an improper prior
Let $X$ be an $\mathcal{X}$ valued random variable. We are doing Bayesian statistics. Suppose that $\theta$ is a $\Theta$ valued random variable with known prior distribution $\Pi$ and that the ...
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Can we solve by hand the early exit multi-class classification problem? [closed]
Problem: Find a solution $\hat{\varepsilon}$ of the following minimization problem
\begin{align*}
&\min_{\varepsilon \in \mathbb{R}^M} \sum_{h=1}^M \varepsilon^h \hat{R}^h+\beta \sum_{h=1}^M \...
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Random sequence generator algorithm non informative piror distribution
I want to conduct a Bayesian statistical analysis of a sequence generation phenomenon.
The sequences generated contain elements from a known alphabet.
Working on that, I have tried to define the prior ...
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How to interpret a noninformative joint prior?
I am currently working on a homework assignment and have the following question:
$\theta_1$ and $\theta_2$ are parameters of interest and $y_1$ and $y_2$ are the likelihood functions which are $\text{...
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How to derive conditional destribution of MVN variable
I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147):
$$Y \sim MVN(X\beta, \sigma^2I)$$
$$\beta|\sigma^2 \...
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Full conditional posteriors
so up to now I dealt with posteriors in the form of:
$$p(\theta|x) \propto p(x|\theta) p(\theta)$$
No we started to model a linear regression with the bayesian approach:
$$Y \sim MVN(X\beta, \sigma^2I)...
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How to sample from the prior predictive distribution with the BSTS R package?
Assuming I have to use the bsts package in R, I'm trying to understand the degree to which my prior distribution choices (implicit or otherwise) are consistent with ...
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Credible intervals with parameter near boundary
When doing Bayesian inference on a parameter that is bounded, often we use priors that approach 0 as the parameter approaches the boundary. For example, when estimating $(\mu, \sigma^2)$ for normal ...
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How to select a proper prior to control the time dependent structure of variable?
I am new in analyzing RCT data and not familiar with the techniques that are always used in RCT analysis.
I am analyzing a dataset of a study: An RCT study with 50 participants; the data was collected ...
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Do we ever use the prior predictive distributions of Bayesian Statistics?
As my question states, I am wondering if there is any chance we use the prior predictive distribution. I am studying Bayesian Statistics and have understood what it is. It is a must to go through in ...
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Understanding the Binomial likelihood notation
Let $X \sim Bin(n,\pi)$.
I don't understand why the binomial likelihood is then given by $f(x|\theta)=\binom{n}{x} \theta^x (1-\theta)^{n-x}$. Shouldn't it be $B(x|\pi,n)=P(X=k)=\binom{n}{k} \pi^k (1-\...
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Bayesian linear regression: How to enforce constraint on the sum of coefficients?
I have a linear regression problem in which my $X$ matrix is not full rank. Here is a small example:
$$X =
\left[\begin{array}{rrrr}
-1 & 0 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 &...
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Trouble understanding priors
The following comes from a book called Bayesian Statistics by Ben Lambert:
Assuming the following model for $r$ disease-positive people out of $n$ people:
$$Pr(Z=r, \theta) = {n\choose r}\theta^r(1-\...
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Choosing Bayesian Priors [duplicate]
I am fairly new to Bayesian Modeling, however I am experimenting with such framework in order to produce several estimates.
The part I am struggling the most with is the selection of prior ...
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How subjective are prior beliefs?
I find the term 'prior-beliefs' to be a little bit vague, what is acceptable to class as a prior belief in Bayesian analysis? For example, I could not look at any data and decide to myself "I ...
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How to match my prior beliefs to beta distribution?
I have some data that I believe comes from the binomial distribution. I also have some old data from a past-experiment that I would like to base my prior beliefs on. The old data observations are: $$6,...
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Using old posterior as new prior given new data [duplicate]
Suppose I have some data, and use this data to create a posterior distribution.
Now suppose I have some new data that I believe is from the same population as the data before. Can I now use my old ...
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Confusion forming prior beliefs
Let's say, for the sake of ease, I have a sample of data that I believe comes from the binomial distribution. Let's say I have 100 observations in my sample. Previous to this experiment, I have ...
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Why would I pick a beta prior for binomial data? More generally, how would I pick any prior distribution? [duplicate]
Obviously, being a conjugate prior is useful as it saves performing tricky integrals or using a simulation, but why would I want to use a beta distribution? More generally, how does one pick an ...
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Picking parameters for beta prior
I have some data that I believe come from a binomially distributed population. A beta prior seems like an appropriate choice, but I don't have any very strong prior beliefs. I could use a less ...
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Estimator of variance for a binomially distributed sample
I want to run some Bayesian analysis on some data. Suppose we have a sample that we believe comes from a binomial population. We have $m$ observations
$$X_i \sim \text{Bin}(n,p)
\quad \quad \quad
\...
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Posterior probability for $\theta$ with a discrete prior
I'm trying to find a posterior probability for this model but I can't find the solution. Help would be appreciated!
Prior distribution: $\theta$ follows a discrete probability function: $\mathbb{P}(\...
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Derivation of acceptance probability from Linero, Yang (2018)
I am wondering how this paper Bayesian Regression Tree Ensembles that Adapt to
Smoothness and Sparsity by Linero & Yang (2018) derived the acceptance probability for $\sigma$.
The authors give $\...
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Bayes estimator of possion distribution with Pareto prior
Consider a random sample of size $n$ following the possion distribution with parameter $\ln \theta$, that is
$$
f(x|\theta)=\frac{(\ln\theta)^x}{\theta x!}, x=0,1,2,\cdots
$$
and the prior of the ...
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Centering Priors on MLEs vs. Using MLEs as Initial Conditions for MCMC [duplicate]
Here:
Centering prior distributions on MLE/OLS estimates
I ask about centering priors on MLEs in the context of a logistic regression (in my case with only categorical predictors), which I've seen a ...
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posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision
What is the posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision. Thus, what is the distribution of x given:
\begin{equation}
x \sim \mathcal{N}(x; \...
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prior distribution for iid gaussian, with a known variance
I have been reading Pattern Recognition and Machine Learning by Bishop, and I have a question regarding the prior distribution of an iid Gaussian with known variance.
The relationship $\dfrac{n}{\...
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