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Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Probability question using Bayes rule

I have a probability question here which I believe I need to apply Bayes rule to solve it. Here is the question: A specific enzyme, QQ, which is designed to quickly help cows gain weight, is ...
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Dirichlet Multinomial Posterior Predictive Distribution for Language Model

I have been trying to teach myself about Bayesian analysis, and whilst I have been through the theory several times, I am struggling to actually apply it. I have found some questions online to ...
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Twist to 3 prisoners problem applying Bayes rule

T, J and B work for a company but the chairman has decided to fire one person randomly chosen through 3 cards. The chairman decides to fire with unequal probabilities - T with probability of 15%, B ...
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Is there an HMC algorithm that estimates a model with noncontinuous parameters?

Is there an HMC algorithm that estimates a model with noncontinuous parameters? All of the intuition I have for how HMC surfs around in the phase space is based on examples for posterior distributions ...
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Can Expectation-Maximization algorithm estimate parameters other than mean and variance (from a model distribution)?

We know that we can use Expectation-Maximization algorithm to estimate parameters from a Gaussian mixture model, say $\mu$, $\sigma$, and $\phi$ (they are parameters of the Gaussian distributions)as ...
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164 views

Approximating expectation with Taylor series

I want to get the second-order Taylor approximation for an expectation. I have the following distribution, which is a Generalized Dirichlet distribution with parameters $\boldsymbol\alpha$ and $\...
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Are random variables sampled upon stopping rules exchangeable?

In this article from D. Berry https://www.jstor.org/stable/2684222?seq=1#page_scan_tab_contents the author uses an example to introduce some limits of p-values in frequentists analyses. He takes an ...
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Calculate probability based on part of features

I have a knowledge base, where each row represent a class and columns represent features of classes: ...
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2answers
46 views

importance sampling from posterior distribution in R

Today I read that Importance Sampling can be used to draw posterior distribution samples just like Rejection Sampling. However, my understanding of Importance Sampling is that its main purpose is to ...
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Question regarding joint posterior distribution [on hold]

I also have the mean and standard deviation, not sure if they are needed for this question. I am totally lost, please help. Thanks! This is the rest part of the question, sorry i didnt post it at ...
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1answer
47 views

Why is the product of two gaussian process $f1$ and $f2$ not a gaussian process?

In the book from Rasmussen/Williams on Gaussian Processes we have the following statement without proof (Page 95): "If f1 and f2 are Gaussian processes then the product f will not in general be a ...
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Bayesian vs Frequentist inference in the presence of noisy data

I'm wondering how Bayesian inference and Classical/Frequentist inference fair towards noisy data. I can't seem to find too much literature addressing this issue and it seems the conclusion is usually ...
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Which gradient to compute in a hierarchical model for M-H MCMC?

We have the following model: $$y_t=Mx_t+\epsilon_t$$ with $M$ being a matrix such that $M\sim F_{\lambda}$(assume it's a conjugate prior). The $\lambda$ does not appear in $M$, only in its ...
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What is the probability that the person has actually taken the alcohol? [on hold]

What is the probability that the person has actually taken the alcohol?
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Linear regression: right-hand tail of $\sigma$ marginal posterior

Suppose we're doing plain vanilla linear regression for $y$. The likelihood: $y_i \sim {\cal N}(\mu, \sigma^2)$, $i=1:N$. Priors: $\mu \sim {\cal N}(0,1)$, $\sigma \sim {\rm HalfNormal}(1)$. As ...
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Bayesian model initial values impact posterior values

I using Winbugs and having trouble getting the model to converge, but I think real question is understanding what is going on with the bayesian model and the initial values which is why I post on ...
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what is the connection between variational inference and re- parameterisation trick?

I understand how variational inference work when approximating a posterior distribution especially in cases like variational autoencoder. Reparameterization trick is then used to keep the ...
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Is the sum $f+g$ of two gaussian processes with $f$~$GP(m_a, k_a)$ and $g$~$GP(m_b, k_b)$ a gaussian process? [closed]

I have problems with proving or disproving whether the sum of two functions from gaussian process is again a gaussian process. My toughts on proving it would be to sum both kernel functions and state ...
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1answer
37 views

Bayesian regression - prior dist for variables

In a multiple Bayesian linear regression model, do all variables (dependent and predictors) get prior distributions? If so, can one mix non-informative and substantive priors in the model? Thanks!
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Why is Bayesian Statistics becoming a more and more popular research topic? [closed]

Browsing through the research area of the top 100 US News statistics program, almost all of them are heavy in Bayesian statistics. However, if I go to lower tier school, most of them are still doing ...
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Posterior distribution of Bayesian Parameters

I am confused about how to get the posterior distribution of Bayesian Parameters. I have $t = w_1x + w_0 + \epsilon$ with $\epsilon = N(0, \sigma^2)$ How do I find $p(\textbf{w}|x_1, t_1, ...., x_N, ...
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1answer
163 views

Naive Monte Carlo, MCMC and their use in Bayesian Theory

So let's suppose I have a random variable X which follows a PDF fX(x) which is known. I can use the Naive Monte Carlo method (...
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Incorporating linear combination predictions into multiple regression

Let's say I have a predictor $p_1$ of the form: $\textbf{y} = f(\textbf{x})$ Let's suppose that I found another predictor $p_2$ of the form: $E[y_1 - y_0] = c$ (e.g. I have a predictor of linear ...
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1answer
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Modelling and interpreting brms output

I do apologize in advance for this might be very basic questions. I am an absolute newbie in Bayesian statistics and too, unfortunately, this is the very first time I am analysing data and I'm really ...
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Connections between logistic regression, information value and Kullback-Leibler

Suppose that we are interested in modeling a binary predictor $Y=0,1$ subject to $m$ predictors $x_1,...,x_m$. First, let us examine a simpler model of the impact of $x_j$ on the response $Y$. By the ...
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1answer
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Bayes logistic vs. standard logistic regression model interpretation

I performed a logistic regression using Stata's bayes: wrapper and obtain the following histogram from 10,000 posterior distribution samples of the log(odds) of my ...
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Updating the variance of a Normal Distribution Using Bayes [closed]

I have a prior belief for the mean $\mu_p$ and std $\sigma_p$ of a normally distributed variable $X$. (no dataset). So I have the mean of these parameters, but NOT their variance (I'll just presume ...
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numerical on decision boundary of bayes [closed]

please help me solve this question on decision boundary
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1answer
24 views

Conditional Posterior Distributions

So I'm trying to find the conditional posterior distributions of n given $\theta$ and x as well as $\theta$ given n and x. These are my priors (poisson and beta) \begin{equation*} \begin{aligned} &...
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1answer
14 views

Equivalence of prior alternatives

Suppose I have Bernoulli distribution with (p) parameter. Suppose I assume a uniform prior over this parameter. My question is, would this be equivalent to assuming a Beta prior, and then uniform ...
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Why volume preservation is important for Metropolis update? [duplicate]

I think my question is naive but I would like to ask why why volume preservation is important for MCMC and specifically Metropolis update.I'm reading the following paper https://arxiv.org/pdf/1206....
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2answers
33 views

The product rule of probability - Specific rewriting

So, the product rule of probability states $$ p(X,Y) = p(X|Y)*p(Y) $$ In general for any set of variables: $$ p(X_1, X_2, ..., X_N) = \prod_{n=1}^N p(X_n|X_1, X_2, ..., X_{n-1})$$ ...
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1answer
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An alternative to Beta-Binomial distribution?

I'm reading about Beta-Binomial. Assume a r.v. x~Bin(n, pi). (pi) in itself, is a r.v. drawn from Beta(alpha, beta) I was thinking, may be I can take the Expected value of pi (which is alpha/(alpha+...
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1answer
25 views

Finding MAP estimate

I think after all the reading I've done I still don't fully understand MAP estimation. I came across a problem that's leaving me dumbfounded. Suppose $A$ ~ $N(0,\sigma^2_1) $ and $\epsilon$ ~ $N(0,\...
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In directed acyclic graphs, is there a dependency in opposite directions?

Suppose we have this graph: (a) ==> (b) ==> (c) Does this mean that P(a|b)=P(a) because the arrows indicate that b is dependent on a and not the other way? If not, then why do we use arrows?
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1answer
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Machine learning to estimate p(y>N | X)

To illustrate, let's say I have a mobile game and I want to predict the duration of each session $y$ when they start. Say I have a training set with multiple useful features $X$ from previous ...
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Controlling for spatial confounding in point-referenced data

I have a point-referenced data set with 2 binary outcomes. The data shows a strong correlation between these binary attributes - however, the geographic clustering is also qualitatively clear. I would ...
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How do I specify a moving average model in R-INLA?

I have a dynamic regression model specified as follows: $f_{c,t+1} \sim N(\eta_{c,t+1} + \phi\epsilon_t,\sigma^2_{f})$ $\epsilon_t=f_{c,t}-\eta_{c,t}$ $\eta_{c,t}=\beta_0+\beta_1x_{1,c,t}$ How ...
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1answer
61 views

How does mice::mice work?

The idea of multiple imputation seems to be based on the decomposition $$ p(\theta \mid y_{\text{obs}}) = \int p(\theta \mid y_{\text{obs}}, y_{\text{mis}})p( y_{\text{mis}} \mid y_{\text{obs}}) \text{...
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1answer
61 views

Learning prior distribution from data

Suppose I have a dataset. How can I learn the prior distributions of the parameters of a model from this data? I want to learn the prior from this data in order to use them in a Bayesian model. Sorry ...
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25 views

Discriminant functions vs. non-Discriminant [closed]

How can I prove which ones from the following functions are non-Discriminant and which are? is it all Bayesian rule? can I use the joint rule in proving? x is the data w1 is the class/label P(...
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Proving Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model

How does one prove that the Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model? This prior is a generalization of the Normal-Inverse-Wishart Distribution. By Matrix-...
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Upper limits from Bayesian inference

While setting an upper limit for the new physics interaction with Bayesian inference, how do people decide on whether to claim 90% or 95% upper limit? What are the reasons behind one or another choice?...
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Bayesian (In)Decision

Let $A_j$ be the action of person $j$, $A_k$ be the action of person $k$, and $p(A)$ be the probability of an action. Using Bayes Rule, $$p(A_j=x|A_k=y)=\frac{p(A_k=y|A_j=x)p(A_j=x)}{p(A_k=y)}$$ If $...
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What is the $p$ in Bernoulli distribution?

In the Bayesian theory of probability, probability is our expression of knowledge about a certain thing, not a property of that thing. However, I always see people treat $p$ as a parameter that needs ...
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1answer
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Coding a simple Stick-Breaking Process in Python

I've just red the great 2012 blog post of Edwin Chen about Dirichlet Process with companion code in R and Ruby. Then I'm trying to translate the Stick-Breaking Process from R to Python. I've got this ...
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1answer
49 views

Learning a Gaussian Process from function observations (not GP regression)

Suppose we have a set of observations, where each observation represents a function. For example, our set is $\{f_1, f_2, ..., f_n\}$ where each $f_i = \{(x_1, y_1), (x_2, y_2), ..., (x_{p_i}, y_{p_i})...
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How to prevent heteroskedastic models from overfitting?

I'm trying to fit neuroscience data using a Gaussian Process, but noticed that it behaves poisson-like (var = mean). Since classic GP models assume iid noise, I figured I could get a better fit by ...
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Regression Non-Normal distribution [on hold]

I'm trying to make regression models for this sample data. And distribution is this: The net hourly electrical energy output (EP) is the response variable, ...
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Defining a posterior for poisson distributed data

I am trying to make a formulation of a posterior for a problem with this structure. For a chemistry experiment we have completed two rounds. The first round lasted for 4 months. In this round two ...