# 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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### Help me understand Bayesian prior and posterior distributions

In a group of students, there are 2 out of 18 that are left-handed. Find the posterior distribution of left-handed students in the population assuming uninformative prior. Summarize the results. ...
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### Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution?

My understanding is that when using a Bayesian approach to estimate parameter values: The posterior distribution is the combination of the prior distribution and the likelihood distribution. We ...
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### What are posterior predictive checks and what makes them useful?

I understand what the posterior predictive distribution is, and I have been reading about posterior predictive checks, although it isn't clear to me what it does yet. What exactly is the posterior ...
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### Effective Sample Size for posterior inference from MCMC sampling

When obtaining MCMC samples to make inference on a particular parameter, what are good guides for the minimum number of effective samples that one should aim for? And, does this advice change as the ...
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### Computation of likelihood when $n$ is very large, so likelihood gets very small?

I am trying to compute this posterior distribution: $$(\theta|-)=\frac{\prod_{i=1}^{n}p_i^{y_i}(1-p_i)^{1-y_i}}{\sum_{\text{all}\,\theta,p_i|\theta}\prod_{i=1}^{n}p_i^{y_i}(1-p_i)^{1-y_i}}$$ The ...
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### How are artificially balanced datasets corrected for?

I came across the following in Pattern Recognition and Machine Learning by Christopher Bishop - A balanced data set in which we have selected equal numbers of examples from each of the classes would ...
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### Does the Bayesian posterior need to be a proper distribution?

I know that priors need not be proper and that the likelihood function does not integrate to 1 either. But does the posterior need to be a proper distribution? What are the implications if it is/is ...
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### How can an improper prior lead to a proper posterior distribution?

We know that in the case of a proper prior distribution, $P(\theta \mid X) = \dfrac{P(X \mid \theta)P(\theta)}{P(X)}$ $\propto P(X \mid \theta)P(\theta)$. The usual justification for this step is ...
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### Predictive distributions for data from common distributions

I have data that I assume comes from some distribution, such as normal. I don't care about estimating the parameters of this distribution. Instead, I'd like to know the distribution that future ...
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### Are the mean of samples taken from Metropolis-Hastings MCMC normally distributed?

I've come across the following theorem while studying MCMC. It seems to suggest that the sample mean taken from the MCMC – the posterior marginal expectation – should be normally distributed, using ...
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### Relation Between Bayesian Estimation and Maximum a posteriori estimation

Is maximum a posteriori estimation some kind of Bayesian Estimation? If yes, can you point out other Bayesian estimators? Edit: So I've come to know the following (don't know if they are correct): ...
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### Why does $P(\theta_1\mid D, \theta_2) \propto P(D \mid \theta_1, \theta_2)P(\theta_1)$ hold?

Suppose that in a Bayesian framework we have observed data $D$, using independent prior distributions on the parameters of the model, denoted by $\theta_1, \theta_2$. Then, the joint posterior ...
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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 ... 41k views

### What is the difference between posterior and posterior predictive distribution?

I understand what a Posterior is, but I'm not sure what the latter means? How are the 2 different? Kevin P Murphy indicated in his textbook, Machine Learning: a Probabilistic Perspective, that it is ...
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### Posterior distribution and MCMC [duplicate]

I have read something like 6 articles on Markov Chain Monte carlo methods, there are a couple of basic points I can't seem to wrap my head around. How can you "draw samples from the posterior ...
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### Example of maximum a posteriori estimation

I have been reading about maximum likelihood estimation and maximum a posteriori estimation and so far I have met concrete examples only with maximum likelihood estimation. I have found some abstract ...
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### How to interpret Bayesian (posterior predictive) p-value of 0.5?

In the following paper found here and reference below, the author suggests that "if the model is true or close to true, the posterior predictive p-value will almost certainly be very close to 0.5" . ...
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### MAP estimation as regularisation of MLE

Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this: It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
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### Gibbs sampling to produce posterior pdf

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all \$i = 1, 2, \cdots,...