# 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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### Implementing Predictive Posterior Distribution Using Stan

Background I had an example that sought to demonstrate the posterior predictive distribution in the context of a normal measurement model. The data that was used is as follows: ...
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### Density estimation/approximation from MCMC samples

I'm looking to accurately describe the density function of a multivariate posterior probability distribution based on samples from MCMC. As far as I know, in most cases this is done either with a ...
8 votes
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### Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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### Dealing with dependent data in a Bayesian model

Background: Consider a series of dependent data points, $$y_1,y_2,y_3,\cdots,y_N.$$ In cases where the dependence is well described by an exponentially decaying auto-correlation function, it is ...
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8 votes
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### What do the terms "nearly-optimal rate", "near-minimax rate", "minimax optimal rate" and "minimax rate" mean in the context of posterior consistency?

Definition: A sequence $\epsilon_n$ is a posterior contraction rate at the parameter $θ_0$ if $$\Pi_n(θ: d(θ, θ_0) ≥ M_n \epsilon_n| X^{(n)}) → 0$$ in $P^{(n)}_{θ_0}$-probability, for every $M_n → ∞$. ...
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### Including feature-dependent priors on output class, in bayesian logistic regression

When doing logistic regression with data $D_N = \{(x_i, y_i)\}_i^N$ with $x_i \in \mathbf{X}^N$ (each data point has N features) and $y_i \in \mathbf{Y}$ being assigned output classes, in a Bayesian ...
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7 votes
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### Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
7 votes
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### Parameter Estimation for Naive Bayes - Maximum a posteriori and Maximum Likelihood

I am wondering if I understand those terms correctly. To summarize my thoughts: In naive Bayes, our decision rule is basically the Maximum a posteriori (MAP) estimate of our hypothesis. We assign an ...
7 votes
1 answer
978 views

### Generate Posterior predictive distribution at every step in the MCMC chain for a hierarchical regression model

I'm trying to fit a Bayesian Hierarchical regression model with a random correlated coefficients using R ,I'm using data having 160 groups (schools) to fit a model of math score as a function of one ...
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6 votes
1 answer
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### Lipschitzness of posterior mean of Gaussian process?

Let $T$ be a compact set, and let $K \colon T \times T \to \mathbb{R}$ be a positive definite kernel. Consider the canonical pseudo-distance $$d_K(x,y) = \sqrt{K(x,x) + K(y,y) - 2 K(x,y)}.$$ Let $f$ ...
5 votes
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### How do I sample from the posterior distribution with gamma likelihood with unknown alpha and beta?

I realize that this Wikipedia page provides the proportional form of the conjugate prior to the gamma distribution with unknown $\alpha$ and $\beta$ parameters, as well as the posterior values of $p$, ...
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4 votes
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### Posterior distribution under Cauchy prior?

I have a (I hope) simple question! If I had a linear regression, $Y_t = \alpha + \beta X_t + \epsilon_t$ with $\epsilon_t \sim N(0,\sigma^2)$ and I assume a Cauchy prior for $\sigma$, is it ...
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### Implementation of Bayes posterior predictive check

I have a question concerning the implementation of a bayes posterior predictive check. Let us assume i have this model (implementation is in R and jags): ...
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### How do I perform an actual "posterior predictive check"?

This question is the follow-up of this previous question: Bayesian inference and testable implications. For concreteness, consider the following bayesian model. This model is not to be taken ...
3 votes
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373 views

### What is the posterior kernel lengthscale of a Gaussian process?

If I have access to multiple samples from a Gaussian process with known covariance kernel but unknown parameters (i.e. unknown lengthscale), it is straightforward to estimate the lengthscale using ...
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### Do most scientific discoveries commit the conditional probability fallacy?

Usually, in an experimental scientific paper where we have observations $X$ taken in laboratory conditions, we either reject some model $\mathcal{M}$ under the basis that $P(X \mid \mathcal{M}) \ll 1$,...
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2 votes
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### Multivariate bayesian parameter estimation

I am implementing an example for pymc3 in python and I want to understand the mathematical formulation of this code. ...