Questions tagged [posterior]
In Bayesian statistics, the term 'posterior' refers to the probability distribution of a parameter conditioned on the observed data.
1,017
questions
3
votes
1
answer
219
views
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 ...
- 43
1
vote
0
answers
27
views
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 ...
0
votes
0
answers
11
views
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 ...
- 5,222
1
vote
1
answer
29
views
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 ...
- 63
1
vote
0
answers
19
views
Bayes factor for hypothesis
I am studying Bayesian hypothesis testing and I want to calculate the Bayes factor for
\begin{align*}
H_0: \lambda = 1 \hspace{0.2cm} vs \hspace{0.2cm} H_1:\lambda > 2
\end{align*}
with $p(\...
- 155
0
votes
0
answers
26
views
How to obtain likelihood ($P(B/R)$ given the prior $P(R)$ and the posterior $P(R/B)$
I am working on a topic related to multiple-choice response. I would like to measure the efficiency of the information source (or a student’s information search) and I believe Bayesian statistics is ...
- 21
2
votes
0
answers
39
views
Likelihood from posterior [closed]
This question is strange and perhaps silly but it would be very useful for my research. Is there any method to find the likelihood given a prior distribution and its corresponding posterior ...
- 21
1
vote
1
answer
47
views
How to choose what to integrate out or what to condition on for marginal distributions?
I am trying to work out the Bayesian posteriors of $\theta$, $\tau$ and the $\varepsilon$ in the following model:
$$y(t) = \phi(t,\tau)\theta+v(t),$$
where $\{v(t)\}$ is an iid sequence of random ...
- 11
0
votes
1
answer
85
views
Normal approximation for posterior distribution
I am reading the example 4.3.3 of "The Bayesian Choice" by Christian P. Robert and I was wondering if it is possible to obtain a normal approximation in this case to estimate the posterior. ...
- 155
0
votes
1
answer
30
views
Interaction: Posterior comparison brms, difference between as_draws, and posterior_predict. Is it correct to interpret posterior_predict instead?
I have an interaction effect in my model, and I want to extract the posterior of each of my parameter in order to compare them and make inference about them. I couldn't simply use the as_draws() ...
1
vote
0
answers
37
views
How to prove the posterior probability for multivariate case (i.e. dimension $d\ge 2$)?
Suppose there are $k$ groups, $\pi_1, \pi_2, \cdots, \pi_k$, with the probability density function for group $\pi_i$ being $f_i(\boldsymbol{x})$, where $\boldsymbol x\in R^d$. The prior probability ...
- 249
1
vote
0
answers
22
views
How to leverage the separable functions in MCMC sampling? [closed]
I'm considering the posterior of a parametric model via the Bayesian approach. More specificity, I have a parametric model $u(p_1,p_2, p_3) = u_1(p_1) \times u_2(p_2) \times u_3(p_3)$ and I want to ...
- 11
0
votes
0
answers
27
views
Overcoming posterior correlation for a model with random effects (for a Gibbs sampler)
I am trying to infer parameters for a model of case numbers of different infectious diseases in different locations over time. The model is
$$
\log \left(1 + y_{ijt}\right)\sim\mathsf{Normal}\left(\mu ...
- 1,017
5
votes
1
answer
55
views
Can we get probabilistic predictions evaluable by proper scoring rules from bayesian inference without evaluating the marginal likelihood?
Let's say we have a vector of inputs, $X=[x_0,\dots, x_{n-1}]$, and a vector of outputs, $Y=[y_0, \dots, y_{n-1}]$.
We would like to predict the distribution of a new output ,$\hat{y}$, given a new ...
- 451
1
vote
0
answers
28
views
Mutual Information decay
Consider $m$ channels indexed by $i$ with $1 \leq i \leq m$. The input alphabets are from the same finite set $\mathcal{X}$. Let $\pi$ denote a probability distribution on $\mathcal{X}$. Define the ...
- 155