Questions tagged [posterior]
In Bayesian statistics, the term 'posterior' refers to the probability distribution of a parameter conditioned on the observed data.
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How to interpret the population parameters of a Bayesian Hierarchical model?
This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.
This is incredibly ...
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Posterior Distribution in a Bayesian Multivariate Normal Model
I am currently working on a Bayesian inference problem and would appreciate some help on computing the posterior distribution of a hyperparameter within a specific multivariate normal model. Below, I ...
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Estimating expected value with respect to posterior
I have a neural network and I need to calculate the following:
$$\mathbb{E}_{P(\theta|D)}[f(\theta)]=\frac{\sum_\theta P(D|\theta)P(\theta)f(\theta)}{\sum_\theta P(D|\theta)P(\theta)}$$
Where $f$, ...
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prior and posterior predictive distributions, Bayes Theory
Consider the binomial sampling model with a Beta prior on $\theta$ and the prior predictive
distribution. Let $n$ be the binomial sample size.
\begin{align}
p(y^{new}) &= \int_{\theta}f(y^{new}|\...
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Confusions modeling times of related events
I am currently interested in modelling the time of related events, and I am currently confused as to how to incorporate all different sources of information in a single model.
Consider a toy example ...
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Show that the multinomial distribution with $k$ categories and Dirichlet distribution are conjugate prior
Problem: Show that the following distributions are conjugate priors for the corresponding densities..
The multinomial distribution with $k$ categories and
$$ p_{X|\theta_1 , \dots, \theta_k} (x_1, \...
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Bayesian hypothesis testing using posterior samples of estimated parameter
I'm modeling recruitment curves using a Hierarchical Bayesian model. There is a key parameter in my recruitment curve, let's call it $P$. I have two groups (A and B) of participants of respective size ...
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Updating a Beta prior based on observations from a product of two Independent Bernoulli variables
I'm working on a problem involving Bayesian updating with a Beta prior, but the data I observe comes from a slightly complex source.
Let $X \sim \text{Bernoulli}(p)$ and $Y \sim \text{Bernoulli}(q)$, ...
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Simulate posterior from a linear mixed model with categorical and continuous variables and their interactions
Currently, I am working on a data set containing isotopic ratios. To understand differences in the ratios I am fitting a linear mixed model in R using the lme4 package. I am then using the arm package ...
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prior distribution for iid gaussian, with a known variance
I have been reading Pattern Recognition and Machine Learning by Bishop, and I have a question regarding the prior distribution of an iid Gaussian with known variance.
The relationship $\dfrac{n}{\...
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Posterior of binomial and mixed prior
I'm currently studying posterior distribution with likelihood $y|\theta \sim B(n,\theta)$ and mixture of prior distribution $\theta \sim \pi Beta(\alpha_1, \beta_1) + (1-\pi)Beta(\alpha_2, \beta_2)$. ...
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Is the spectrum of a signal circularly symmetric if the signal itself is circularly symmetric?
Let’s consider a signal that is circularly symmetric complex Gaussian process (proper):
$$ s \sim \mathcal{CGP}(0, C, 0) $$, and,
the covariance has the following form:
$$ \mathbf{C} = \mathbf{C}_{rr} ...
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Why do T prior and likelihood make a bimodal posterior?
In this post, the author shows that when a likelihood and prior are both T-distributed with $2$ degrees of freedom, the posterior is bimodal. The given reason is that
The two modes persist - the ...
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Bayesian Analysis of Coin Toss with Three Outcomes: Incorporating a Fixed Probability of a 'Side flip' event
I'm working on a Bayesian analysis of a coin-toss scenario and have a conceptual question to clarify my understanding.
Background
Given a uniform prior on the probability that a coin lands tails over $...
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Calculate posterior distribution and full conditional of a HMM
Set up a Bayesian analysis of an hidden Markov model and calculate the posterior distribution and the full conditionals, given this assumptions:
The state space of the hidden process has size m
$Z_t|...
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Is it acceptable to take the mean of a bunch of median values?
I use a Bayesian latent variable model to construct a time series cross-sectional measure of corruption for all countries in the world from 1960 to 2010. For each country-year observation, I obtain a ...
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How do we obtain the posterior of a beta binomial mixture of continuous and a discrete density?
In section 3.6 of Jim Albert's 2009 book "Bayesian Computation with R" he describes a test of whether a coin is fair using a mixture of priors. The coin tossing follows a binomial ...
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Correction variance estimation from the posterior in a Bayesian framework
My question is quite basic, I have posterior distributions for some parameters derived from an arbitrary Bayesian framework. Since I know that the posterior variance under-estimates the true variance, ...
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Estimating posterior of proportion of positives in population from per-observation probabilities
I have a sample from some population of 0s and 1s and need to estimate the posterior of the proportion of 1s in this population. But the catch is: for each observation in the sample I only have ...
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How do I evaluate correlation of model parameters using MCMC posterior samples from a rstan fit?
Is there a better way to do so than simply by taking posterior parameter estimates and calculating the Spearman or Pearson correlation between them? Anything specific to having posterior samples from ...
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Is it correct to use the posterior distribution from a Bayesian model in other analysis?
I have written a Bayesian model in JAGS that I use to calculate the growth rates of several plant populations as well as their variance while taking into account the observation error during the ...
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Some Problems in Auxiliary Particle Filter
recently I am studying PF. And I am stuck in APF for a few days, though I derived many times. Here is my question:
I followed the framework of this paper. The APF is defined in Algorithm 1:
The ...
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Why the Pitman estimator is given by the sample mean of X and Y?
Let $(X,Y)$ be bivariate normally distributed with $E[X] = E[Y] = \theta$, $Var[X] = Var[Y] = 1$
and $cov[X, Y] = \rho, |\rho| < 1$, where $\rho$ and $\theta$ are unknown. Find the minimum risk ...
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Maxdiff Approach - Claims comparison , how to compare product claims of two different surveys Maxdiff gives Preference shares or count based analysis
My business objective: I want to create a MAX diff approach where I will have multiple surveys my output will be Claims and its Posterior Probability , count of best and worst selection.
For more ...
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Is Inverse-Wishart a conjugate prior for Wishart likelihood?
Suppose I have a noisy observation $Z$ of a covariance matrix $F$, given a prior on $F: p(F)$, I would like to find the posterior of $p(F|Z)$, does the following specification forms conjugacy?:
$$ F \...
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Bayesian VAR: Derivation of predictive distribution for reduced form VAR
I have a standard reduced form VAR of type without intercept: $y_t = A_{1}y_{t−1} + \ldots + A_py_{t−p} + e_t$, $e_t \sim N(0,Σ)$.
I need to derive the predictive posterior distribution $p(y_{T+h}|y_{...
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Bayesian Gaussian mixture - is my prior correct?
I'd like to sample from the Bayesian Posterior of a Gaussian mixture model, but I am not sure about the correct Bayesian formulation of the latter. Is the following correct?
I consider the 1-...
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Could one use mixtures of Gaussians to turn MCMC posterior samples into a new prior?
Theoretically in Bayesian inference one could use one experiment's posterior as another experiment's prior, such that knowledge of the parameters accumulates from $p(\theta) \rightarrow p(\theta|\...
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Where does the uncertainty of the "true" $p_{*}(y|x)$ come from?
You'll often see the goal of a statistical estimation problem as being to fit a model such that it $\approx p_{*}(y|x)$ where $p_{*}(y|x)$ is the "true distribution of the data".
My question ...
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Computing a posterior distribution
I need to compute the posterior distribution of a parameter $\theta$ conditionally on signal $t$. $\theta$ is uniformly distributed in $[0,1]$, while $t=\theta+\eta$ where $\eta$ represents a noise, ...
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Proving that a function is always increasing
Given that $X_1, \ldots, X_n$ are conditionally independent and identically distributed random variables and that, given a value of $\theta$, $X_i \mid \theta \sim \operatorname{Bernoulli}(\theta), i =...
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Laplace approximation from a log-posterior in R
I would like to perform a Laplace approximation of a log-posterior.
The evolution of a cancer cell at given time $t_j$, $j = 1,\cdots,n$ for an experiment $i$ follows the following Poisson ...
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Sampling from an approximate distribution to estimate posterior mean
Suppose I have a model with parameters $\theta = (\theta_1,\theta_2)$ and observed data $x=(x_1,x_2,\ldots,x_k)$. I want to estimate the posterior mean $\hat{\theta}_1 = \mathbb{E}[\theta_1|x]$.
The ...
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Computing log-posterior for large variance priors
Let's say that some quantity is modelled by a time-dependent Poisson distribution,
$$
y(t) \sim \text{Pois}(\mu(t))
$$
where
$$
\mu(t) = \alpha_0 \exp(-\alpha_1 e^{-\alpha_2 t})
$$
and $\alpha_k > ...
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Why is the quotient distribution of two probability distributions angular?
Background
I am working with posterior probability distributions for parameters obtained from a Bayesian binomial generalised linear model with a logit link function. The parameters returned by the ...
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Regarding the bayes rule derivation of posterior distribution, $p(\omega|x,y),$ for a given dataset $D$ over $\omega.$
So I was going through this paper and under Uncertainty modeling it says
So I tried deriving it on my own and I got
$p(\omega | X, Y) = \frac{p(Y | X, \omega) \cdot p(X,\omega)}{p(Y | X) \cdot P(X)}$
...
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Some questions about the posterior distribution when the marginal distribution is zero
Let $\{f(\cdot|\theta): \theta \in \Theta \}$ be a family of pdfs and let $\pi: \Theta \to \mathbb{R}$ be a prior. According to Bayes' theorem (as stated in, e.g., Casella and Berger), the posterior ...
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Distribution families whose likelihoods integrate to $+\infty$ for some sample values
I've recently started learning about Bayesian statistics, and I came across this very nice answer by Xi'an https://stats.stackexchange.com/a/129908/268693, which [in my slight paraphrasing] says the ...
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how do we select the parameter values to run model from the posterior draws [closed]
after obtaining the posterior draws(N=1000) for parameters for a bayesian VAR Model. how do we select the single parameter among 1000 draws that best depicts the posterior distribution
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Posterior predictive distribution for Bernoulli (and categorical)
I'm trying to confirm something I've tried to figure out about the posterior predictive distribution for Bernoulli vs. Binomial (and categorical vs. multinomial) random variables after a Bayesian ...
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Posteriori - How to determine the normalization coefficient without integrating the marginal [closed]
How can I deduce the posterior probability, up to a multiplicative coefficient, and how can I determine, without integral calculus, the normalization coefficient?
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Covariance between two posterior predictive distributions?
Suppose we have three features $x_i \sim N(0, 1)$ for $i=1,2,3$. We then use Bayesian linear regression with interpolant $f(x, w) = wX$, such that we model y as $N(f(x, w), \beta)$, i.e., with a ...
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intepretating mean of posterior with dependency on likelihood and prior variance
$p(\theta | x)$ is the unnormalised posterior distribution of interest.
Let's suppose the likelihood function for this posterior $p(\theta | x)$ is $L(\theta | x) = N(\theta | \mu, \sigma^{2}) exp[\...
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Quantiles of the posterior predictive distribution of a Gumbel random variable under the degenerate prior $\pi(\mu,\sigma) = \sigma^{-1}$
I need to find an automatic way to calculate with good precision the quantile of the posterior predictive distribution (ppd) of a random variable following a Gumbel law, under the degenerate prior $\...
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Posterior of exponential likelihood and gaussian prior
Likelihood function: $$L(\zeta) = \zeta^{\alpha} \exp\left[\zeta\sum_{i=1}^{\alpha}(x_i - x)\right]$$
Prior function: $$p(\zeta) = \frac{1}{(\sqrt{2π}σ_\zeta)} \exp\left[-\frac{(\zeta-\zeta_0)^2}{(...
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How informative should a Gaussian Process prior to be?
I recently started learning about the Gaussian Process for a GP machine learning project so my understanding is relatively limited. However, from what I have read/watched so far you have a prior GP ...
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Logistic vs. linear regression for "inherently continous" variable - comparing probability
This is a situation that arises commonly in my area (medicine).
Suppose there is an inherently continuous variable $y$
Suppose there is some normal range for this variable, say 80 - 120
Suppose there ...
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Gaussian Process posterior distribution
I'm trying to find a way to get the posterior covariance function for a mgcv::gam fit. Assuming I have a simple model y ~ s(x), ...
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ABC model selection from posterior samples
I would like to know if there is a general scheme to do model selection based on the posterior samples from a set of ABC (Approximate Bayesian Computation) runs for a given set of models.
Particularly ...
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Bayesian posterior density estimation, given data and a prior: Can the denominator be evaluated in two different ways?
Assume the simple, well-known scenario: Data = $(x_j, y_j)_{j=1}^{n}$, and that, as usual, the $n$ data points are drawn iid. The $x$'s may be considered non-random, but the $y$'s are observations ...
