All Questions
195 questions
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19
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Finding posterior of M/N given posterior of M. (Gamma Distribution)
I'm trying to solve a question about posterior distribution of Gamma scalar multiplication/division. I will give some context to the question here:
$Y_1,..., Y_n$ is modelled as IID Pois($\mu$) random ...
- 1
0
votes
1
answer
53
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Estimating the Posterior Probability of Playing Coin 1
Suppose someone has two coins with probability $p_1$ and $p_2$ of getting heads with $p_2\le p_1$. The person is throwing the coins and may switch from playing coin 1 to coin 2 with probability $\...
- 1
1
vote
1
answer
78
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Which proposal function should be used in this particular case of the Metropolis-Hastings algorithm?
As part of my research, I would like to apply the Metropolis-Hastings in order to sample from some posterior distribution. More precisely, the data comes from a multivariate normal distribution in the ...
- 11
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0
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44
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Bayesian: Formally comparing prior and posterior distributions
Consider Bayesian inference in the regression model:
$$ \begin{align*} y &= \beta_0+\beta_1x_1+\beta_2x_2 + \varepsilon \\ \varepsilon &\sim
\mathcal{N}(0,\sigma^2) \end{align*}$$
Suppose we ...
- 523
5
votes
1
answer
166
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Uniform prior and poisson likelihood, what posterior distribution will be produced?
If i have a uniform distribution over a fixed specified and a finite range, and a Poisson likelihood distribution, what posterior will be produced?
The likelihood has this form $$P(\pmb{X}|
\pmb{\...
- 75
1
vote
0
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40
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how can predictive distributions be considered as expectations?
I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
0
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0
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30
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How to obtain likelihood ($P(B/R)$ given the prior $P(R)$ and the posterior $P(R/B)$
I am working on a topic related to multiple-choice response. I would like to measure the efficiency of the information source (or a student’s information search) and I believe Bayesian statistics is ...
- 21
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14
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Turning a list of cost into categorical probability mass distribution
Background
Given a noisy dataset $D$, I have to solve a classification problem where the possible anserwer is $i\in\{1,\dots,N\}$. So far I can get pretty decent result with an algorithm that, based ...
- 473
1
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0
answers
38
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How to derive conditional destribution of MVN variable
I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147):
$$Y \sim MVN(X\beta, \sigma^2I)$$
$$\beta|\sigma^2 \...
- 161
1
vote
1
answer
78
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Full conditional posteriors
so up to now I dealt with posteriors in the form of:
$$p(\theta|x) \propto p(x|\theta) p(\theta)$$
No we started to model a linear regression with the bayesian approach:
$$Y \sim MVN(X\beta, \sigma^2I)...
- 161
2
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0
answers
62
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Credible intervals with parameter near boundary
When doing Bayesian inference on a parameter that is bounded, often we use priors that approach 0 as the parameter approaches the boundary. For example, when estimating $(\mu, \sigma^2)$ for normal ...
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1
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0
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35
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Understanding the Binomial likelihood notation
Let $X \sim Bin(n,\pi)$.
I don't understand why the binomial likelihood is then given by $f(x|\theta)=\binom{n}{x} \theta^x (1-\theta)^{n-x}$. Shouldn't it be $B(x|\pi,n)=P(X=k)=\binom{n}{k} \pi^k (1-\...
- 161
2
votes
1
answer
97
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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 ...
- 163
3
votes
1
answer
78
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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}(\...
3
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0
answers
53
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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}{\...
- 9,317
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235
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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
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0
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88
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Laplace approximation from a log-posterior in R
I would like to perform a Laplace approximation of a log-posterior.
The evolution of a cancer cell at given time $t_j$, $j = 1,\cdots,n$ for an experiment $i$ follows the following Poisson ...
- 111
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0
answers
147
views
Computing log-posterior for large variance priors
Let's say that some quantity is modelled by a time-dependent Poisson distribution,
$$
y(t) \sim \text{Pois}(\mu(t))
$$
where
$$
\mu(t) = \alpha_0 \exp(-\alpha_1 e^{-\alpha_2 t})
$$
and $\alpha_k > ...
- 111
2
votes
1
answer
46
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Some questions about the posterior distribution when the marginal distribution is zero
Let $\{f(\cdot|\theta): \theta \in \Theta \}$ be a family of pdfs and let $\pi: \Theta \to \mathbb{R}$ be a prior. According to Bayes' theorem (as stated in, e.g., Casella and Berger), the posterior ...
- 121
2
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0
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28
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Distribution families whose likelihoods integrate to $+\infty$ for some sample values
I've recently started learning about Bayesian statistics, and I came across this very nice answer by Xi'an https://stats.stackexchange.com/a/129908/268693, which [in my slight paraphrasing] says the ...
- 121
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1
answer
3k
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How to calculate the posterior distribution with a normal likelihood function and a prior that involves sigma
In the problem, the data X follows a normal distribution, or $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{1}{2}(\frac{x-\mu}{\sigma})^2)$. Let's say I know the value of $\sigma^2$ and ...
- 297
1
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1
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80
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Is it ok to widen a prior during an MCMC which did not converge yet?
I am calibrating parameters of a process model. The runtime of the model is high and the calibration already ran for more than two weeks with many cores on a HPC.
After almost 150k iterations I ...
- 21
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vote
2
answers
88
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Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?
Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?
Suppose you assume a probability distribution.
You ...
user318514
0
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0
answers
69
views
How should I deduce the conjugate prior and corresponding posterior for a geometric distribution
The given pmf is for a geometric distribution and is $f(x_i|\theta) = (1-\theta)^{x_i - 1}\theta; ~x_i = 1, 2 ,\cdots, $ and the 1-parameter exponential family I have obtained is; $$f(x|\theta) = \exp ...
user365863
0
votes
1
answer
77
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Posterior distribution when the domain of the likelihood depends on the parameter
I am trying to calculate a posterior density given distribution and a prior. And I am a bit confused about how I should act as the domain of the distribution depends on the parameter.
I am talking ...
0
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0
answers
46
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How to update a prior probability distribution of hurricane occurrence based on absence of hurricanes to date?
For a forecasting tournament, I am trying to forecast the number of Atlantic basin hurricanes in the 2022 hurricane season. I have reason to believe that my prior distribution looks as follows:
At ...
1
vote
0
answers
49
views
Is there an implicit independence assumption in Bayesian inference between X and parameters?
I often see things like
$$ p(w|X,y) \propto p(y|X,w) p(w)$$
where $w\in\mathbb R^p$ denotes some parameters, $y\in \mathbb R^n$ denotes some observed outcome values, and $X\in \mathbb R^{n\times d}$ ...
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2
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1
answer
241
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For multivariate normal posterior with improper prior, why posterior is proper only if $n\geq d$
This is related to Gelman's BDA chapter 3 section 5's noninformative prior density for $\mu$.
Let $\Sigma$ be fixed positive definite symmetric matrix of size $d$ by $d$. Let $y_1,\dots, y_n$ be iid ...
- 1,465
2
votes
2
answers
169
views
Find a likelihood to calculate a posterior probability
I am having trouble understanding a basic Bayesian inference exercise:
Suppose we are interested in inferring the proportion $\theta$ of
individuals in a given population suffering from a certain ...
- 121
1
vote
1
answer
347
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How can I find the posterior distribution for gammadistributed data and prior?
I am 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 Gamma distribution with ...
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2
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1
answer
203
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How to find the marginal prior distribution?
Suppose that $\beta$ has the following prior
$$
\beta|\zeta \sim f(\beta,\zeta)
$$
Then I know that the marginal prior distribution of $\beta$ is given by
$$
\int f(\beta,\zeta) d\zeta
$$
However, ...
- 273
1
vote
1
answer
492
views
Kl Divergence between factorized Gaussian and standard normal
Given two distributions, one a parameterized gaussian and the other a standard normal gaussian:
$q(x) \sim \mathcal{N}(\mu,\sigma)$
$p(x) \sim \mathcal{N}(0,I)$
We want to compute the KL Divergence $...
2
votes
1
answer
137
views
Partially specified Bayesian prior?
In bayesian linear regression for example, we may specify a model as:
$$y_i \sim N(\beta_0 + \beta_1 x_i, \epsilon^2) \\\\
\beta_0 \sim N(0, \tau_0^2) \\\\
\beta_1 \sim N(0, \tau_1^2) \\\\
\epsilon \...
- 343
0
votes
0
answers
136
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Exponential Posteriori with a Uniform Prior
I'm studyng for a final exam and found this problem from another generation, but I don't know how I should continue... I will be gratefull for any help, thanks you.
Let be $X|\theta\sim U(0,\theta)$ ...
- 71
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0
answers
370
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Is a draw from the posterior always the same as a draw from the prior?
I'm reading Bayesian data analysis. On page 155, the authors state:
Each of the [...] parameters were assigned independent Beta(2, 2) prior distributions. ... If the model were true, we would expect ...
- 213
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votes
1
answer
2k
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Calculating the Prior and Posterior Mean
I was recently asked to calculate the prior mean and posterior mean of the proportion of defective items in a production line assuming a uniform prior for this proportion. The question was stating ...
- 141
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0
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551
views
To calculate Bayes estimator for $\theta $ using squared error loss function
The life of an electric bulb, X (in hours), follows the exponential distribution with mean life $ 1/ \theta $ , where $ \theta(>0) $ is an unknown parameter. The prior distribution of $ \theta $ ...
- 387
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0
answers
327
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Prior after variable transformation
I am sampling the variables ($\alpha, \beta, \gamma$) with a Markov chain where I put the priors to be flat, e.g. $\pi(\alpha) = U(\alpha)$ and similarly for $\beta, \gamma$.
Now I am actually ...
user336446
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0
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53
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Using a different (but related) hypothesis for the prior in MAP
Say we have a general set of data $\mathcal{D} = \{\mathbf{x}_i, \mathbf{y}_i \}_{i \in N}$ of covariates $\mathbf{x}$ and observations $\mathbf{y}$. Our problem is in fitting a known model $\mathbf{y}...
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3
votes
2
answers
677
views
Posterior of one observation transform into posterior of several observations
Suppose $\mu$ has prior distribution $\mathcal{N}(M, A)$ and $x |\mu \sim \mathcal{N}(\mu, 1)$
After one observation, the posterior is $$\mu|x \sim \mathcal{N}(M + B(x-M), B), \tag{1}$$ where $B \...
- 471
1
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0
answers
300
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Postetior from Jeffrey prior of Normal distribtion
Context
I am given a sample from normal distribution $v_i \sim N(\gamma \cdot u_i, \sigma^2)$, $i =1,..., n$.
I need to obtain the posterior distribution using Jeffreys prior for $\gamma$.
My solution
...
- 261
2
votes
0
answers
192
views
When does this prior dominate likelihood?
This is a simple Bayesian inference problem, where we are trying to infer some weight parameter $w$. Our posterior distribution is
$$ P\propto \exp\left(-\frac{1}{\sigma^2} w^Tw\right) \exp\left(-f(w)\...
- 281
1
vote
0
answers
21
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Prior/Posterior of the Proxy
I am trying to understand a sentence from Caldara and Herbst (2019), who develop a baysian proxy SVAR model. The paragraph is:
"In case of weak identification, the prior plays an important role ...
- 11
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0
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106
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What is the posterior distribution $p(\textbf{f} | \textbf{y})$ for a Gaussian Process regression?
What is the posterior distribution $p(\textbf{f} | \textbf{y})$ for a Gaussian Process regression?
Suppose that $p(y_n |x_n, f) = N(f(x_n), \sigma^2)$ with prior on $\textbf{f} = [f(x_1), \ldots f(x_n)...
- 211
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0
answers
38
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Converting posteriors to likelihoods by removing prior
I have a set of MCMC chains (i.e., unnormalized posteriors) for a parameter I modeled for a sample of objects. I have a model that requires that I condition on the likelihoods of this parameter. My ...
- 101
0
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1
answer
90
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How is the likelihood different from the posterior? [duplicate]
I come from an applied mathematics background and have never looked at statistics, but I started studying Machine Learning recently. One thing that I am struggling to understand is: what is the ...
- 125
3
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1
answer
109
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How to calculate the posterior distribution from the density
I'm stuck on a answer from an old exam.
The task is to use a Poisson distribution and a Gamma distribution as prior to calculate the posterior density:
$$
p(\lambda|x) \propto L(\lambda)p(\lambda)\...
- 113
4
votes
1
answer
28
views
How to jointly model $N$ groups where data in each group is i.i.d. Normal and infer the posterior distribution?
I am given the following data of income scores of individuals from $N$ groups:
$$(\textbf{x}_1, \textbf{x}_2 \ldots \textbf{x}_N),$$
where $$\textbf{x}_j = (x_j^1, x_j^2 \ldots x_j^{N_j}),\quad j = 1, ...
- 151
1
vote
2
answers
130
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Bayesian Statistics: Properly updating the Prior for new analysis
I have three tables of information about $A$ and $B$ (gray cells, black font), their row and column marginal totals (black cells white font), and the grand total (white cell black font). The first two ...
- 447
2
votes
1
answer
1k
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Gaussian processes: The uncertainty is reduced close to the observations?
I am currently studying the textbook Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams. Chapter 1 Introduction says the following:
In this section we ...
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