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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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Best/Simplest ways to do Bayesian Inference? [on hold]

For my project, I need to perform Bayesian Inference. Right now, computational speed is not the most important factor, but accuracy is. I believe the problem should be more or less straight forward, ...
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29 views

Parameters in the prior and posterior distributions

In the answer of this question Joint posterior distribution of $(\mu,\sigma^2)$ in the Normal model (the expression below $(1)$ ), Why the parameters are $\frac{1}{2}(v_o+n+1,v_os_o^2+(n-1)s^2+n(\...
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17 views

Bayesian analysis: comparison of marginal probability distributions

Is it valid to compare mariginal probability distributions from separate Bayesian analyses to infer which scenario is most likely? Specifically, in phylogenetic (evolutionary) analysis, if I ...
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2answers
33 views

NUTS Drawing samples from slice sampler; how to keep bounds on log scale?

I'm currently working to adapt the No U-Turn Sampler from this paper for a model I'm working on. The No-U Turn sampler augments the typical hamiltonian system by incorporating a slice variable $u$ ...
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1answer
39 views

Understanding the parameters needed for a distribution in Bayes networks?

Since I have a discriminative mindset hardly can I intuit the so-called parameters needed to specify a distribution in a generative Bayesian Network. I'd like to borrow an example from this blog. If ...
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32 views

Bayesian hypothesis test: Type I and II errors

In a Bayesian hypothesis test between two alternatives A and B, what is the probability of making a type I and type II error? This question has been asked many times on this forum in various formats: ...
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15 views

Covariance specification in hierarchical model

I am currently working on hierarchical models and try to get my head around the following question: What influence has the prior choice of the covariance matrix in the 2nd stage, especially when ...
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0answers
28 views

Defining custom Bayesian priors in R (BayesFactor package)

I'm performing some Bayesian analyses in R using the BayesFactor package, and was wondering whether it is possible to specify priors for the alternative not centered on zero (the current defaults ...
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26 views

bayesian decision making - comparing expected loss

The problem is like this: Suppose that I am considering which country should I invest on, country A and country B, based on their GDP growth rate $\alpha$. There are two possible choices for each ...
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4answers
60 views

How to build a Bayesian Model to estimate the probability distribution of the parameters given the output?

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
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1answer
33 views

Bayesian posterior pmf for weighted dice with uniform prior

We want to find posterior probability mass function for dice tossing with uniform prior. We are interested in rolling of weighted dice. The outcome is 1,2,...,6. We assume that prior probability ...
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1answer
137 views

Posterior distribution of mixture models

In the context of mixture models in Bayesian inference, one can assume that the general form of the joint posterior for a mixture model of $k$ components is $$ \begin{equation} p( \boldsymbol{\...
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2answers
24 views

Denoting random variable $\theta$ with capital $\Theta$?

It is common practice to denote random variables with a capital letter $X$ and the realization with a small $x$. But how about in Bayesian statistics? The parameter $\theta$ is a RV, so shouldn't it ...
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1answer
23 views

Coherence and calibration

I am trying to find good definitions and examples for both these concepts regarding frequentist vs Bayesian statistics. Can anyone please shed light on them and explain them? Furthermore, why are ...
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0answers
16 views

Bias correction when using loo cross-validation to replace unreliable PSIS-LOO estimates

The PSIS-LOO information criterion (see this paper by Vehtari, Gelman, and Gabry) assigns a Pareto shape parameter $\hat k$ to each observation in the data, and these $\hat k$ values can be used to ...
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0answers
26 views

Measure of accuracy for a Bayesian model

I am reading Statistical Rethinking (Section 6.2.1.2). The topic of this section is measuring accuracy for a Bayesian model, i.e. accuracy of the model of predicting correctly an outcome. The ...
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0answers
38 views

Find the distribution function $F$ for $min_{1 \le i \le n}{X_i}$ [duplicate]

Given a random sample $X_1, X_2, ..., X_n$ where each $X_i$ has pdf: $$ f(x; \theta) = 3 \theta^3 x^{-4} $$ and $0 \lt \theta \le x \le \infty$. Show that the distribution function $F$ for $min_{1 \...
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0answers
19 views

Prior predictive distribution usage

I understand the mechanics and math behind prior predictive distributions, but I don't understand its practical uses. Theoretically and application wise, what is its purpose?
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What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions?

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions? The purpose of finding this information is to enlighten me on how should one decide on a prior for a positive-...
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0answers
25 views

Trouble with MLE [closed]

I have a random sample $X_1, X_2, ..., X_n$ with $X_i$ having a pdf $$ f(x;\theta) = 2\theta^2x^{-2} $$ I'd like to find the MLE of $\theta$. First, because this is a random sample, all $X_i$ are ...
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0answers
26 views

Getting main and interaction effects from Bayesian factorial ANOVA in Stan

I am using Rstan to build a factorial ANOVA model with two factors and their interaction. The sample dataset has 2 factors, A (levels A1 and A2) and B (levels B1, B2, B3) and 10 replicates for each ...
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0answers
15 views

Should weights be applied in generated quantities block in stan?

I want to do predictions via generated quantities block in stan. I have two questions: Should the weights be applied again in the generated quantities block in addition to the likelihood in the ...
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1answer
107 views
+50

Bayesian inference for non-Gaussian errors

Following from a previously unanswered question, regression tasks involving measurements with normally distributed noise apply Gaussian processes. But are there any recommended approaches for ...
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0answers
39 views

Sequential monte carlo : A simple example

I am attempting to understand how to implement the sequential monte carlo algorithm using this article. Here are the steps that the author proposes: Example problem: Say I have a self moving robot ...
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0answers
19 views

Bayesian- Can we use uninformative prior for time

I have a doubt about what type of prior to use for time. For example, I am trying to estimate the time in a port. I collected a bunch of data that measures time for a ship to stay in a port, but I do ...
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0answers
19 views

Likelihood function vs probability distribution function [duplicate]

I've been reading about Bayesian statistics and data analysis, and constantly see that $\text{posterior} \propto \text{prior} \ \times \text{likelihood}$. I'm familiar with fundamental probability and ...
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1answer
25 views

Recursive Bayes Learning

I'm trying to work through an example from Richard Dudas Pattern Classification on Recursive Bayes Learning. My main question is why do we choose the $max[D^n] $ in: $$max[D^n] \le \theta \le 10 $$ ...
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0answers
21 views

Bayesian- How to determine the distribution for likelihood function [closed]

I have a question about likelihood model. Given I have a set of data, how do i find out what type of distribution is suitable for my likelihood function? (eg. Poisson, exponential etc.)
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1answer
28 views

Probability of finding a lost item

I am trying to solve the following problem and was wondering if someone can verify my answers. Big Joe has lost an important document. There is a 70% probability it is at home, and a 30% chance it is ...
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2answers
43 views

Bayesian- If I only have prior distribution, is there a way to calculate posterior distribution? [closed]

I have a question about Bayesian inference. In my research, I have only a normal prior distribution with known parameters (mean and std.). I do not have a likelihood function, but I need to calculate ...
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1answer
140 views

Joint posterior distribution of $(\mu,\sigma^2)$ in the Normal model

Find the joint posterior of $(\mu, \sigma^2)$ given Normal data. I've found the joint prior of $\mu$ and $\sigma^2$ (where $\displaystyle\sigma^2\sim\chi^{-2}(v_o,v_os_o^2)$ and $\mu|\sigma^2\sim N(\...
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0answers
56 views

Trouble replicating simple example of Bayesian inference

On pages 20-21 of John Kruschke's Doing Bayesian Data Analysis book (2nd ed.), there is an introductory illustration of Bayesian inference. We know that balls can have four sizes: 1, 2, 3 and 4, but ...
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44 views

Recommended point estimate for non-normal distribution?

I have a rather non-normal marginalized posterior for some parameters, resulting from a Bayesian MCMC. Example: I know that the actual distribution is what truly represents the parameter, but I need ...
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1answer
30 views

Gaussian Processes, basic question about how the prior is computed

I'm approaching the topic of GP, and I have a question regarding how functions are sampled. On my textbook is stated that to represent a distribution over a function (the prior): we only need to ...
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1answer
13 views

Calculating Bayes Factor from Z score, n, and No

I'm completely stuck on how to get this answer from a course below. I guessed the answer, but I'm lost on how they get to it. I did the following in R ...
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1answer
17 views

Prior/degree of belief/degree of lack-of-information/algorithms/complexity

For a long time I had a bit of difficulty understanding what "degree of belief" means. Recently I had some thoughts about it and I wonder if they make any sense, or is there some literature about ...
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0answers
35 views

Bayesian Inference and MSE. Need help to understand solution

I have problems with understanding solution of Problem 4.c (MSE) here. I couldn't get exact number. My solution is following (numbers for $X_M$ in table above): I start with calculating $E(X-X_M)^2 ...
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1answer
24 views

Example of maximum a posteriori that does not match the mean of a marginalized posterior

Given a N-parameter likelihood and prior, I can obtain the marginalized posterior for each parameter through Bayesian MCMC. I can also estimate the maximum a posteriori (MAP) of the N-parameter ...
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1answer
40 views

Specifying frequency parameter in the absence of occurrences

Let's say I have a process where the occurrences are independent, proportional to time. I made $n$ observations for which I only observed no occurrences. My goal is to define a frequency parameter and ...
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1answer
33 views

The distribution of a posterior predictive p-value under certain assumptions

I am wondering if anyone can check my understanding of the following passage concerning posterior predictive p-values in the textbook "Bayesian Data Analysis 3rd Edition" on page 151: In the ...
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0answers
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Algorithms for combining Bayesian networks? [closed]

Are there any algorithms for combining multiple Bayesian networks? For example, let's say I have 5 variables A, B, C, D and E, and I build 5 Bayesian networks on different random subsets of these, let'...
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0answers
26 views

Frequentist vs. Bayesian bias-variance decomposition

Iv'e read the answer to this related question and still have some issues. Suppose that given some data $X$, we want an estimator $\hat{\theta}$ for some parameter $\theta$. A common approach is to ...
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0answers
18 views

Resources for prerequisites to Probablistic Machine Learning Models

I am a self-learner and have done several machine learning courses but diving into Bayesian or Probabilistic Graphical Models I feel like my prior knowledge is inadequate. I have done some Probability ...
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1answer
72 views

Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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0answers
7 views

Bayes and DFMEA

In design-engineering technique of FMEA the severity of a failure mode, its rate of occurrence, and its detectability are given ordered rank scores from 1 to 10, and then these values are multiplied ...
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0answers
25 views

Ranking/Sorting Star Ratings - Bayesian Credible Interval

I recently started analyzing episode polling data from a website that uses a 1-10 rating system. I've been reading about ranking star rating systems using Bayesian Credible Interviews as explained by ...
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1answer
98 views

On a Bayesian hypothesis testing

Let $X_1,...,X_n$ be a random sample and $\lambda >0$ be a parameter, with $X_i |\lambda \sim Poisson (\lambda)$ and $\lambda \sim Gamma(\alpha, \beta) (\lambda)=\dfrac {1}{\Gamma(\alpha) \beta^\...
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0answers
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In Gelman's 8 school example, why is the standard error of the individual estimate assumed known?

Context: In Gelman's 8-school example (Bayesian Data Analysis, 3rd edition, Ch 5.5) there are eight parallel experiments in 8 schools testing the effect of coaching. Each experiment yields an ...
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1answer
46 views

Why is variational Bayesian mixture model an alternative to MCMC? What are the similarities?

Why do people say that a variational Bayesian mixture model could be an alternative to MCMC for clustering? For example see the details here: https://en.wikipedia.org/wiki/Variational_Bayesian_method. ...
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Modeling dichotomous outcomes with multiple nominal predictors

I am trying to model dichotomous outcomes with multiple nominal predictors. Predictors have multiple levels, like individuals within a group, so this should be modeled hierrachically. I am using R and ...