Questions tagged [posterior]

Refers to the probability distribution of parameters conditioned on data in Bayesian statistics.

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Bayes classifier using maximum likelihood

Derive the estimate of the Bayes classifier from m data points $(x_1, y_1), . . . ,(x_m, y_m)$ using Maximum likelihood. Do so under each of the following assumptions. $Y ∈ {−1, +1}$, $X ∈ \mathbb{R}...
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What is the intuitive meaning of these statements in the context Bayesian prior and posterior?

I am now familiar with the Bayesian thinking process of using a prior and then getting the posterior once we observe data using the prior. I read the following statements which I am trying to get my ...
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30 views

If we take estimated parameters from an MCMC and plug it back into the likelihood to draw new observations, what does the histogram approximate?

Suppose we have the following set-up and we conduct an MCMC on it. Likelihood: $$ X\sim ~ Gamma(\alpha,\beta) $$ Prior: $$ \alpha \sim Unif(0,10) $$ $$ \beta \sim Gamma(0.5,0.5) $$ Assume the ...
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Determining overdispersion of count variable in bayesian model (brms)

I am trying to determine whether my response count data are too overdispersed for a (brms) Bayesian poisson model. I constructed a poisson-generated response variable with low and high levels of noise/...
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21 views

Marginal posterior distribution, likelihood mean sum of two standardnormal priors

How would I compute the marginal posterior distribution of $\mu_1$ and $\mu_2$ if the likelihood $(y | \mu_1,\mu_2) \sim N(\mu_1+\mu_2,1)$ and $\mu_i \sim N(0,1)$
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Posterior probability of hypothesis distributions

Suppose I have $K$ classes with distribution $\theta$ over $\{1,...,K\}$ and an underlying domain $D$ on which each class defines a categorical distribution $\phi_i$. Given a draw $i\sim\theta$ and $...
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20 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 ...
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33 views

Posterior becomes infinity for bayes-theorem interpretation

posterior $=$ $\frac{likelihood * prior}{evidence}$ If evidence = 0, then posterior is infinity or doesn't exist. What does that mean in terms of posterior, likelihood, prior, etc? What happens to ...
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Gibbs Sampling for Order Restricted Bayesian Inference

I am trying to learn Gibbs Sampling for Bayesian Inference, I am very confused about the following setting. Suppose we observe $X :=(x_1 , \dots, x_N)$ where $x_i $ is binomially distributed ...
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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 ...
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Computing posterior distribution $\theta|x_{1:n}$

Assume that $X_1, ..., X_n, X_{n+1}| \theta \sim \text{iid } Exp(\theta) $ and the posterior is $\theta \sim Gamma(\alpha, \lambda)$. Task is to compute the posterior $\theta|x_{1:n}$. $\pi(\theta|x_{...
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Which is the decision rule in a gda classifier?

On the text book there is the following formula for the prediction rule, but I don’t understand where it comes from: The textbooks says it had been derived from I suppose pi and theta are the ...
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How can I plug in the value of parameter found by Maximum a Posterior?

Suppose I have 1 heads and 4 tails from 5 coin tosses. To find out the probability of 1 heads and 4 tails in my coin toss experiments, I decided to use Binomial Probability Mass Function for the ...
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Bayesian factor analysis: Help with posterior derivation

I am trying to derive the Bayesian factor analysis model described on page 10 of this paper. In brief, consider the model (simplified from the paper) $y_i \sim N(M \boldsymbol{f}_i, S), \quad i=1,\...
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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 $\theta$ ...
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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 ...
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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:\...
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Sampling from unknown probability distribution [duplicate]

I'm reading about Monte Carlo methods. Suppose that $X_1,...,X_n$ are i.i.d $p(x_i|\theta)$, where $\theta$ is an unknown parameter of interest. My textbook states: Suppose we could sample some number ...
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Simplying Bayes Theorem expression: SIS particle filter posteriori

In the book Beyond the Kalman Filter: Particle Filters for Tracking Applications on page 39 the weight update equation for the particle filter is derived. The derivations begins by introducing the ...
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Maximum A Poseriori - Regression - Dependence between feature coefficients and features?

I was going through some lectures on regression when I came across these statements. Wouldn't the expansions, respectively, be $P(w|y,X) = \frac {P(y|w,X)*P(w|x)} {P(y|X)} $ and $ P(y,w|X) = P(y|w,X)...
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would it be possible to pick a different likelihood model achieve the same posterior estimation?

Take the coin flipping example. When we decide to use the Bernoulli distribution to model a coin flip, of course with and without a conjugate prior would make some difference for estimation. Would it ...
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Proposal Metropolis distribution for complex Bayesian models

In the book Uncertainty Quantification: Theory, Implementation, and Applications, by R.C. Smith, there is a chapter about Bayesian inference. The likelihood is Gaussian, with error variance $\sigma^2$ ...
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Posterior predictive of Gaussian

Given is a Gaussian $\mathcal{N}(\mu,\sigma^2)$ with unknown $\mu$ and known $\sigma^2$. There is only one data point $x$. What is the parameter for the mean of the posterior distribution, i.e. what ...
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Why does thinning work in Bayesian inference?

In Bayesian inference, one needs to determine the posterior distribution of the parameters from the prior distribution and the likelihood of the data. As this computation might not be possible ...
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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 ...
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posterior - many data points as well as one?

I used to thinking of the posterior as being the probability distribution over parameters consistent with a particular data point, $$ p(\theta|x) \propto p(x|\theta) p(\theta) $$ where $x$ is some ...
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Classification of Bayesian posterior probabilities

I have run a series of Bayesian models with flat priors in which I obtain a posterior probability distribution for my coefficient of interest. The reviewer of my paper wishes us to classify these ...
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Choosing Prior Distribution for AR coefficients in TAR/SETAR model

I am trying to choose the prior distribution for the AR coefficients for a general TAR model. I am tone between a truncated normal and uniform priors. Are there books or resources you can suggest for ...
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20 views

Sampling posterior distribution of a function

I have the following problem: let's say I have a function $y=f(x)$. Let $f$ be defined for all $x$ but it it might not be invertible. Further assume $x \sim p(x)$ with some probability density $p(x)$. ...
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144 views

Posterior mean with MCMC

Let's say we have a posterior distribution: $$\pi(\theta_1, \theta_2, \theta_3 | \bf{y})$$ and that we've run an MCMC algorithm to approximate this distribution. I know that there is a Markov chain ...
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26 views

Gibbs sampling for simple posterior distribution?

I have a likelihood function, $$ p(x) = \theta^{\sum x} (1- \theta)^{n-\sum x} $$ and prior distribution, $$ p(\theta) \propto \theta^{\alpha - 1} (1- \theta)^{\beta - 1}$$ then the posterior ...
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How to tune MCMC with unwieldy posterior [duplicate]

Let's say I have $n$ observations of a random variable, $X_1, \dotsm, X_n \sim \mathcal{N}(0, \sigma^2)$. I also assume $\sigma^2$ has a Gamma(1,1) prior distribution, $\pi(x) = \exp(-x)$. I'm now ...
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Need some help understanding the factorised posterior in semi-supervised generative modelling

I am having a bit of trouble with the derivation in Kingma's semi-supervised generative modelling paper for the M2-model. The M2 model assumes a probabilistic model where the data $x$ is generated by ...
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85 views

How can I derive mathematical posterior predictive distribution calculation steps for beta prior and binomial likelihood

I would like to know the mathematical calculation step by step processes with beta prior and binomial likelihood for posterior predictive distribution.
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Notation for conditional density

Are $p(\mu \mid \sigma)$ and $p(\mu ; \sigma)$ equivalent? I've seen the notation $p(b_i \mid T_i, \delta_i, y_i ; \theta)$ used to represent the posterior distribution for $b_i$. I am assuming that ...
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Shrinkage effects in a hierarchical model

I am working on the chimpanzees dataset from Richard McElreath's text, "Statistical Rethinking", edition 2. I have built 2 simple models, one a fixed effects model and the other a hierarchical model. ...
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What distributions are conjugate to themselves, besides the normal?

I know the normal distribution is conjugate to itself; are there others? Is there some sort of intuition behind why a given distribution would be conjugate to itself?
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Estimate random effects for a new individual with a linear mixed effects model

Consider repeated observations $\mathcal{Y} = (y_{i,j})_{i,j}$ obtained for $p$ individuals ($1 \leq i \leq p$), at different time points $t_{i,j}$ $(1 \leq j \leq n_i$). The "random slope and ...
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exact form for the marginal posterior

I have a question that I come across for practicing. Basically the question is this: Consider a random sample from the normal distribution with unknown mean and variance $Y_i \sim^{i.i.d.} N(\mu, \...
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Question about marginalization

This is related to a research project I'm working on, I hope someone can help me clear up this confusion... I have a 2d array of log-likelihood values that I obtain after sampling from my posterior ...
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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 ...
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mismatch in sampling between t distribution and normal-inverse-gamma distribution

I am looking at equivalence of sampling between t distribution and normal-inverse-gamma (NIG) distribution in python. The results don't match, and I want to see if there's a mistake in how I am ...
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Bayesian inference problem with three parameters

Assume that I observe a yearly time series for the number of certain events occurring per year. Also, assume that the data points come from a Poisson process with parameter $\lambda$. The dataset in ...
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103 views

Marginal probability in Gaussian Process

Let $\mathbf{a} \sim \mathcal{GP}(\mathbf{m},\mathbf{C})$ where $\mathbf{a} \in \mathbb{R}^T$ is modeled as Gaussian process with mean $\mathbf{m} \in \mathbb{R}^T$ and prior covariance $\mathbf{C} \...
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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 ...
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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 ...
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Find posterior distribution given prior distribution

Problem Be $X_1,...,X_n$ a random sample of $X$ ~ $Geometric(\theta)$, i.e., $f(x|\theta)=\theta(1-\theta)^x \forall x = 0,1,2,...$ Assuming a prior distribution for $\theta$ find the posterior ...
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Posterior Predictive Distribution for Uniform Likelihood and Pareto Prior

I'm trying to find the posterior predictive distribution for data $X_i, \dots X_n$ from a a $Uniform [0, \theta]$ distribution. The prior distribution for $\theta$ is a $Pareto[\alpha, \beta]$ ...
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Deriving full conditional of ordered probit model (Bayes)

I have a question regarding the following exercise: I am able to compute the complete (full) data likelihood function, the full conditionals of $y^{*}_i$ and $\beta$. However, I do not know how to ...
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Dynamically updating posterior density in R

I want to redefine my function in a loop by calling the function from last iteration. However I know this is basically a recursive way which I don't want. To give an example, see the following ...

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