# Tag Info

Accepted

### Posterior distribution and MCMC

If this was not a clear conflict of interest, I would suggest you invest more time on the topic of MCMC algorithm and read a whole book rather than a few (6?) articles that can only provide a partial ...
• 106k

### Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution?

This question has likely been considered already on this forum. When you state that you "have the posterior distribution", what exactly do you mean? "Having" an available$-$in the ...
• 106k

### Alternatives to Bayesian statistics when distributions are unknown

If the distribution of the data is unknown, the Bayesian way of handling this uncertainty is to put a prior on it. There exists a huge literature on Bayesian non-parametrics, including the ...
• 106k
Accepted

### Why is the normalisation constant in Bayesian not a marginal distribution

$p(D)$ is a constant with respect to the variable $\theta$, not with respect to the variable $D$. Think of $D$ as being some data given in the problem and $\theta$ as the parameter to be estimated ...
• 2,085

### Effective Sample Size for posterior inference from MCMC sampling

The question you are asking is different from "convergence diagnostics". Lets say you have run all convergence diagnostics(choose your favorite(s)), and now are ready to start sampling from the ...
• 15.6k

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

They refer to distributions of two different things. The posterior distribution refers to the distribution of the parameter, while the predictive posterior distribution (PPD) refers to the ...
• 461
Accepted

### What is/are the implicit priors in frequentist statistics?

In frequentist decision theory, there exist complete class results that characterise admissible procedures as Bayes procedures or as limits of Bayes procedures. For instance, Stein necessary and ...
• 106k
Accepted

### Alternatives to Bayesian statistics when distributions are unknown

From what you’re saying is that you want something Bayesian, but you can’t define the likelihood. For such cases there is approximate Bayesian computation (see abc), where in place of likelihood you ...
• 138k
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### Why does thinning work in Bayesian inference?

Thinning has nothing to do with Bayesian inference, but everything to do with computer-based pseudo-random simulation. The whole point in generating a Markov chain $(\theta_t)$ via MCMC algorithms is ...
• 106k

### What is/are the implicit priors in frequentist statistics?

@Xi'an's answer is more complete. But since you also asked for a pithy take-away, here's one. (The concepts I mention are not exactly the same as the admissibility setting above.) Frequentists often (...
• 3,814

### Multivariate normal posterior

With the distributions on our random vectors: $\mathbf x_i | \mathbf \mu \sim N(\mu , \mathbf \Sigma)$ $\mathbf \mu \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$ By Bayes's rule the posterior ...
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### Posterior distribution and MCMC

How can you "draw samples from the posterior distribution" without first knowing the properties of said distribution? In Bayesian analysis we usually know that the posterior distribution is ...
• 125k
Accepted

### Are MCMC based methods appropriate when Maximum a-posteriori estimation is available?

No need to use MCMC in this case: Markov Chain Monte-Carlo (MCMC) is a method used to generate values from a distribution. It produces a Markov chain of auto-correlated values with stationary ...
• 125k
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### Posterior Predictive Distribution as Expectation of Likelihood

$\newcommand{\y}{\mathbf y}$We have $$E_{\theta|\y}\left[f(\theta)\right] = \int f(\theta) p(\theta | \y)\,\text d\theta$$ just by definition of expectation (and you could cite the LOTUS as well), ...
• 20.2k
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• 91k

### Are MCMC based methods appropriate when Maximum a-posteriori estimation is available?

It is unclear to me what you call an analytical posterior $\pi(\theta)$ and hence why this analyticity should preclude one from using MCMC. Even for a posterior distribution that is available in ...
• 106k
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### Under what circumstances can an improper prior be used in bayesian analysis?

The textbook definition is that an improper prior is a prior probability density that is not integrable, i.e. it does not have a finite integral. A CDF cannot be higher than 1 hence the name "...
• 1,267
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### Proving that a function is always increasing

By the law of iterative expectations and conditional independence, for any $n \in \mathbb{N}$ and $u_i \in \{0, 1\}$, $i = 1, \ldots, n$, we have \begin{align} & P(X_1 = u_1, \ldots X_n = u_n) = ...
• 18.8k
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### Bayesian inference: numerically sampling from the posterior predictive

If you can simulate values from $P(x_{\text{new}}|\theta)$, you can simply use your $N$ samples from posterior predictive and generate $x_{\text{new},i}$ for each posterior sample from this model to ...
• 484

### Use of prior and posterior predictive distributions?

Some uses of the posterior predictive: Simulating future data based on your model assumptions and data observed to this point. This is useful for predictions, forecasting, etc. Model checking via ...
• 8,814
When you have an independent prior on $X$ and $Y$, then the posterior might not factor into $X$ and $Y$ pieces just because the likelihood doesn't factor into $X$ and $Y$ pieces. It's easy to see ...