# 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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### Basic question about deriving MAP estimator

Say we have a random process $X(t, u)$ parametrized by $t$ and $u$ that generates data $x$. We also have a prior on $u$, $p(u)$. Am I correct in stating that the expression to find the maximum a ...
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### how can predictive distributions be considered as expectations?

I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
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### two-step gibbs sampling vs block gibbs sampling

While reading Bayesian-related technical articles, I can see algorithms such as two-step Gibbs sampling and block gibbs sampling ...
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### known variance in conjugate normal

$Posterior\ mean=\frac{1}{\frac{1}{\sigma_{0}^{2}} + \frac{n}{\sigma^{2}}}\left( \frac{\mu_{0}}{\sigma_{0}^{2}} + \frac{\sum_{i=1}^{n} x_i}{\sigma^2} \right)$ Using this updating equation with known ...
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### Posterior of Inverse Wishart distribution with a subset of data observed

Suppose: $$x_1\in \mathbb{R}^{p_1}\\ x_2\in \mathbb{R}^{p_2}$$ such that x \sim \mathcal{N}( \begin{bmatrix} x_1\\ x_2 \end{bmatrix}; \begin{bmatrix} \...
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### Mean and covariance kernel for the posterior GP of a Hidden Markov Model

In a hidden Markov model (HMM) we have a process $X_k$ that evolves according to: $$X_{k+1} = X_k + W_{k+1}, \quad W_{k+1} \sim N(0, \sigma_{W}^2),$$ where $\{W_k \}$ are IID and $X_0 = W_0$. We can ...
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### Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
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Assuming a Gaussian likelihood, $y \mid x, w \sim \mathcal{N}(w^\top x, \sigma^2)$, the variance of the least squares estimate $\hat{w} = \mathrm{argmax}_w p(y \mid X, w)$ is $\mathbb{V}[\hat{w} \mid ... • 43 0 votes 0 answers 11 views ### Why do we need the variance term in SWAG method? My question is about the SWA-Gaussian paper. I do not really understand why they need the 1/2 factor for the covariance matrix (as underlined in the picture). I understand that it is needed because ... 4 votes 1 answer 76 views ### Bayesian ROPE (region of practical equivalence) for simultaneous comparison of multiple parameters? I very much like the idea of the ROPE (region of practical equivalence) (e.g. see here), where you compute the posterior probability that a given parameter is in a previous range that counts as "... • 44.5k 2 votes 1 answer 71 views ### Using old posterior as new prior given new data [duplicate] Suppose I have some data, and use this data to create a posterior distribution. Now suppose I have some new data that I believe is from the same population as the data before. Can I now use my old ... 3 votes 1 answer 65 views ### Posterior probability for$\theta$with a discrete prior I'm trying to find a posterior probability for this model but I can't find the solution. Help would be appreciated! Prior distribution:$\theta$follows a discrete probability function:$\mathbb{P}(\...
In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: \widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \...