Questions tagged [posterior]
In Bayesian statistics, the term 'posterior' refers to the probability distribution of a parameter conditioned on the observed data.
975
questions
1
vote
1
answer
288
views
How can I find the posterior distribution for gammadistributed data and prior?
I'm working on a project where I believe Bayesian statistics should be useful. However, my knowledge about bayesian statistics are very scarce. Suppose I got data following a Gammadistribution with a ...
CommunityBot
- 1
5
votes
1
answer
2k
views
Determining overdispersion of count variable in bayesian model (brms)
I am trying to determine whether my response count data are too overdispersed for a (brms) Bayesian poisson model. I constructed a poisson-generated response variable with low and high levels of noise/...
- 221
0
votes
1
answer
577
views
Confusion about the use of the MLE & the posterior in parameter estimation for logistic regression
In classification one usually computes
$$
C = \operatorname*{argmax}_k p(C=k\mid X)
$$
where $p(C=k\mid X)$ is the posterior distribution.
In a simple logistic regression setting with $C \in \{0, 1\}...
CommunityBot
- 1
6
votes
3
answers
207
views
How to interpret the population parameters of a Bayesian Hierarchical model?
This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.
This is incredibly ...
1
vote
1
answer
183
views
Posterior Distribution in a Bayesian Multivariate Normal Model
I am currently working on a Bayesian inference problem and would appreciate some help on computing the posterior distribution of a hyperparameter within a specific multivariate normal model. Below, I ...
- 5,340
7
votes
2
answers
7k
views
Why use mean of posterior distribution instead of probability?
I'm reading the Think Bayes (pdf link) by Allen B. Downey, and on this example I don't understand well the purpose of Mean in the chapter 3.2 The locomotive problem....
- 4,903
1
vote
0
answers
74
views
Estimating expected value with respect to posterior
I have a neural network and I need to calculate the following:
$$\mathbb{E}_{P(\theta|D)}[f(\theta)]=\frac{\sum_\theta P(D|\theta)P(\theta)f(\theta)}{\sum_\theta P(D|\theta)P(\theta)}$$
Where $f$, ...
- 253
2
votes
2
answers
62
views
prior and posterior predictive distributions, Bayes Theory
Consider the binomial sampling model with a Beta prior on $\theta$ and the prior predictive
distribution. Let $n$ be the binomial sample size.
\begin{align}
p(y^{new}) &= \int_{\theta}f(y^{new}|\...
- 31.8k
1
vote
2
answers
1k
views
In this Bayesian network, where does this posterior probability come from?
I'm reading Building Intelligent Interactive Tutors (Woolf, 2009) on student models for ITSs. On page 261, the author presents an example for a simple Bayesian network ($S \rightarrow E$), where $S$ ...
CommunityBot
- 1
1
vote
1
answer
291
views
Dispersion measure - probability density function
I am wondering whether someone has a tip on this potentially very basic question. i have done some grid-based bayesian analysis and ended up with a non-standard discrete posterior density function ...
CommunityBot
- 1
0
votes
0
answers
7
views
Confusions modeling times of related events
I am currently interested in modelling the time of related events, and I am currently confused as to how to incorporate all different sources of information in a single model.
Consider a toy example ...
- 133
2
votes
1
answer
17
views
Show that the multinomial distribution with $k$ categories and Dirichlet distribution are conjugate prior
Problem: Show that the following distributions are conjugate priors for the corresponding densities..
The multinomial distribution with $k$ categories and
$$ p_{X|\theta_1 , \dots, \theta_k} (x_1, \...
- 71.9k
0
votes
1
answer
313
views
How to restrict positive posterior?
Is there a way to restrict posterior, say, to only positive values or an interval of values? Let's say I want to estimate a linear model, y = a + Xb, using Bayesian techniques. I specify priors for a ...
CommunityBot
- 1
34
votes
3
answers
17k
views
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 ...
- 103k
1
vote
0
answers
27
views
Bayesian hypothesis testing using posterior samples of estimated parameter
I'm modeling recruitment curves using a Hierarchical Bayesian model. There is a key parameter in my recruitment curve, let's call it $P$. I have two groups (A and B) of participants of respective size ...
- 11
5
votes
1
answer
125
views
Updating a Beta prior based on observations from a product of two Independent Bernoulli variables
I'm working on a problem involving Bayesian updating with a Beta prior, but the data I observe comes from a slightly complex source.
Let $X \sim \text{Bernoulli}(p)$ and $Y \sim \text{Bernoulli}(q)$, ...
- 118k
0
votes
0
answers
9
views
Simulate posterior from a linear mixed model with categorical and continuous variables and their interactions
Currently, I am working on a data set containing isotopic ratios. To understand differences in the ratios I am fitting a linear mixed model in R using the lme4 package. I am then using the arm package ...
- 13
2
votes
2
answers
698
views
Squared Error Loss for Bayesian estimator of Normal distribution
I'm following Brad Efron and Trevor Hastie book "Computer Age Statistical Inference" (link). In chapter 7, they begin to debate the James-Stein estimator by calculating the Bayes rule, or ...
CommunityBot
- 1
1
vote
1
answer
272
views
How to average several posteriors distributions from a Monte Carlo Simulation
Say you produce several posteriors distributions from different runs of the same model under different seeds. That is to say you have something like the following:
...
CommunityBot
- 1
0
votes
0
answers
32
views
prior distribution for iid gaussian, with a known variance
I have been reading Pattern Recognition and Machine Learning by Bishop, and I have a question regarding the prior distribution of an iid Gaussian with known variance.
The relationship $\dfrac{n}{\...
- 8,787
1
vote
2
answers
70
views
Posterior of binomial and mixed prior
I'm currently studying posterior distribution with likelihood $y|\theta \sim B(n,\theta)$ and mixture of prior distribution $\theta \sim \pi Beta(\alpha_1, \beta_1) + (1-\pi)Beta(\alpha_2, \beta_2)$. ...
- 2,566
0
votes
0
answers
10
views
Is the spectrum of a signal circularly symmetric if the signal itself is circularly symmetric?
Let’s consider a signal that is circularly symmetric complex Gaussian process (proper):
$$ s \sim \mathcal{CGP}(0, C, 0) $$, and,
the covariance has the following form:
$$ \mathbf{C} = \mathbf{C}_{rr} ...
- 185
1
vote
1
answer
64
views
Why do T prior and likelihood make a bimodal posterior?
In this post, the author shows that when a likelihood and prior are both T-distributed with $2$ degrees of freedom, the posterior is bimodal. The given reason is that
The two modes persist - the ...
- 34.7k
0
votes
1
answer
40
views
Bayesian Analysis of Coin Toss with Three Outcomes: Incorporating a Fixed Probability of a 'Side flip' event
I'm working on a Bayesian analysis of a coin-toss scenario and have a conceptual question to clarify my understanding.
Background
Given a uniform prior on the probability that a coin lands tails over $...
- 2,308
3
votes
1
answer
4k
views
Beta-Binomial conjugate proof
Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely:
Is the integral being evaluated or not?
How does the entire expression reduce to a ...
- 1
0
votes
0
answers
47
views
Calculate posterior distribution and full conditional of a HMM
Set up a Bayesian analysis of an hidden Markov model and calculate the posterior distribution and the full conditionals, given this assumptions:
The state space of the hidden process has size m
$Z_t|...
0
votes
0
answers
48
views
How do we obtain the posterior of a beta binomial mixture of continuous and a discrete density?
In section 3.6 of Jim Albert's 2009 book "Bayesian Computation with R" he describes a test of whether a coin is fair using a mixture of priors. The coin tossing follows a binomial ...
- 74.7k
3
votes
1
answer
246
views
Is it acceptable to take the mean of a bunch of median values?
I use a Bayesian latent variable model to construct a time series cross-sectional measure of corruption for all countries in the world from 1960 to 2010. For each country-year observation, I obtain a ...
- 2,370
19
votes
2
answers
7k
views
Why does thinning work in Bayesian inference?
In Bayesian inference, one needs to determine the posterior distribution of the parameters from the prior distribution and the likelihood of the data. As this computation might not be possible ...
- 103k
1
vote
1
answer
483
views
Highest Posterior Density Interval (HPDI) using kernel
I'm trying to compute the 95% HPDI of a posterior from 10000 draws from a distribution.
I've been instructed to use density() in ...
CommunityBot
- 1
0
votes
0
answers
7
views
Correction variance estimation from the posterior in a Bayesian framework
My question is quite basic, I have posterior distributions for some parameters derived from an arbitrary Bayesian framework. Since I know that the posterior variance under-estimates the true variance, ...
- 523
3
votes
1
answer
35
views
Estimating posterior of proportion of positives in population from per-observation probabilities
I have a sample from some population of 0s and 1s and need to estimate the posterior of the proportion of 1s in this population. But the catch is: for each observation in the sample I only have ...
- 86.9k
2
votes
0
answers
24
views
How do I evaluate correlation of model parameters using MCMC posterior samples from a rstan fit?
Is there a better way to do so than simply by taking posterior parameter estimates and calculating the Spearman or Pearson correlation between them? Anything specific to having posterior samples from ...
- 21
2
votes
1
answer
72
views
Is it correct to use the posterior distribution from a Bayesian model in other analysis?
I have written a Bayesian model in JAGS that I use to calculate the growth rates of several plant populations as well as their variance while taking into account the observation error during the ...
2
votes
1
answer
63
views
Some Problems in Auxiliary Particle Filter
recently I am studying PF. And I am stuck in APF for a few days, though I derived many times. Here is my question:
I followed the framework of this paper. The APF is defined in Algorithm 1:
The ...
- 20.2k
0
votes
1
answer
67
views
Why the Pitman estimator is given by the sample mean of X and Y?
Let $(X,Y)$ be bivariate normally distributed with $E[X] = E[Y] = \theta$, $Var[X] = Var[Y] = 1$
and $cov[X, Y] = \rho, |\rho| < 1$, where $\rho$ and $\theta$ are unknown. Find the minimum risk ...
- 103k
0
votes
0
answers
392
views
How to determine uncertainty of data from a bayesian posterior distribution
I am a bit confused as to how we determine the uncertainty of a set of data from Bayesian Analysis.
In my specific case, I am asked the following:
Assume $f(x,x_0)$ as the correct model for the ...
- 17.7k
0
votes
0
answers
7
views
Maxdiff Approach - Claims comparison , how to compare product claims of two different surveys Maxdiff gives Preference shares or count based analysis
My business objective: I want to create a MAX diff approach where I will have multiple surveys my output will be Claims and its Posterior Probability , count of best and worst selection.
For more ...
6
votes
4
answers
1k
views
Is it OK to choose the MH proposal as the prior in a posterior simulation problem?
Is it OK to choose the proposal distribution as the prior in a Metropolis algorithm?
Perhaps it's a simple question and to me, it's totally fine but as I always see people choose different ...
- 103k
8
votes
0
answers
289
views
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 ...
- 955
4
votes
1
answer
180
views
Is Inverse-Wishart a conjugate prior for Wishart likelihood?
Suppose I have a noisy observation $Z$ of a covariance matrix $F$, given a prior on $F: p(F)$, I would like to find the posterior of $p(F|Z)$, does the following specification forms conjugacy?:
$$ F \...
- 51
1
vote
0
answers
77
views
How to validate a Bayesian model using posterior predictive check
Prediction ability of a model is usually being used for model validation/evaluation. But in a high noise to signal ratio setting, and when you are not caring about prediction but inference, then ...
- 74.7k
14
votes
3
answers
4k
views
Derivation of Normal-Wishart posterior
I am working on the derivation of a Normal-Wishart posterior but I'm stuck at one of the parameters (the posterior of the scale matrix, see at the bottom).
Just for context and completeness, here is ...
0
votes
0
answers
85
views
How to derive a regularized machine learning objective function with the maximum a posteriori for random features?
My question is at the end of the post. I tried to give as much information as I can to clarify my understanding and to point out as precisely as possible where I am stuck.
Independent variables or ...
1
vote
0
answers
29
views
Bayesian VAR: Derivation of predictive distribution for reduced form VAR
I have a standard reduced form VAR of type without intercept: $y_t = A_{1}y_{t−1} + \ldots + A_py_{t−p} + e_t$, $e_t \sim N(0,Σ)$.
I need to derive the predictive posterior distribution $p(y_{T+h}|y_{...
- 955
7
votes
1
answer
104
views
Reproducing a didactic example of Lindley (1993)
Lindley (1993) discusses the following mixed discrete and continuous prior for the tea tasting lady experiment, where $\pi$ is probability of a correct classification: $p(\pi=0.5) = 0.8$ (discrete ...
- 37.3k
0
votes
0
answers
102
views
Bayesian Gaussian mixture - is my prior correct?
I'd like to sample from the Bayesian Posterior of a Gaussian mixture model, but I am not sure about the correct Bayesian formulation of the latter. Is the following correct?
I consider the 1-...
- 1
1
vote
0
answers
45
views
Could one use mixtures of Gaussians to turn MCMC posterior samples into a new prior?
Theoretically in Bayesian inference one could use one experiment's posterior as another experiment's prior, such that knowledge of the parameters accumulates from $p(\theta) \rightarrow p(\theta|\...
- 955
0
votes
0
answers
24
views
Where does the uncertainty of the "true" $p_{*}(y|x)$ come from?
You'll often see the goal of a statistical estimation problem as being to fit a model such that it $\approx p_{*}(y|x)$ where $p_{*}(y|x)$ is the "true distribution of the data".
My question ...
- 355
1
vote
2
answers
44
views
Computing a posterior distribution
I need to compute the posterior distribution of a parameter $\theta$ conditionally on signal $t$. $\theta$ is uniformly distributed in $[0,1]$, while $t=\theta+\eta$ where $\eta$ represents a noise, ...
- 8,806