# 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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Consider a Gaussian linear model with an $n \times 1$ outcome vector $y$ and an $n \times p$ matrix of centered predictors $X$: $y = \iota\alpha + X\beta + \varepsilon \quad \quad \varepsilon ... • 700 0 votes 1 answer 29 views ### Bayesian MCMC and Only Updating Some Variables at a Time I want to do Bayesian MCMC on a Gaussian Mixture Model. But, I want to update the means, weights, and covariance matrix for a single component separate from the others. Would there be the issue of ... 0 votes 0 answers 11 views ### Bayesian Mediation Analysis I have: 1 binary outcome (0, 1) 1 continuous quantitative mediator 1 continuous quantitative predictor I would like to compute Bayesian mediation analysis with Liu et al. (2023) method. The formula ... • 1 0 votes 1 answer 31 views ### When are Bayes estimators injective as a function of sufficient statistics? I know that Bayes estimators can be written only as a function of sufficient statistics. When are those functions injectives? That is, when can I say that, given a bayes estimator$\delta (\cdot)$and ... 0 votes 0 answers 13 views ### How to choose default uninformative prior in the R Package BAS I'm conducting a Bayesian multilevel logistic regression based on the Rpackage BAS. I'm a beginner in Bayesian statistics. But in bas.glm, I don't understand and I don't know how to specify my prior. ... • 1 0 votes 0 answers 12 views ### Mutual Information of nonadjacent nodes in Bayesian Network How do you compute the mutual information of two non-adjacent nodes in a Bayesian network? In this case, what would$I(D;A)$be? Would I need to take the conditional probabilities of all intemediate ... 0 votes 0 answers 29 views ### Marginal probability expressed as posterior expectation [closed] I'm reading Probabilistic Machine Learning: Advanced Topics by Kevin Murphy and fail to see how, on page 339, plugging$g(\boldsymbol{\theta}) = p(\theta_1=\theta_1^*\vert\boldsymbol{\theta}_{2:D})$... 0 votes 0 answers 25 views ### Bayesian Logistic Regression: default uninformative priors choice on JASP [closed] I'm currently trying to perform Bayesian logistic regression using JASP. For this, I need to choose a prior distribution. JASP offers the following options: AIC, BIC, EB-local, g-prior, CCH, Beta-... • 1 4 votes 2 answers 276 views ### What is the Gold Standard for Evaluating the Posterior of a Bayesian Regression Model? Let me explain my meaning & the context: I mean evaluating the correctness of the posterior (e.g. for approximate Bayesian inference methods). I care mostly about Bayesian deep learning, I'd like ... • 401 1 vote 1 answer 40 views ### How to Derive the Conditional Variance in a Bivariate Normal Distribution After Bayesian Updating? I'm working with a bivariate normal distribution of two variables,$\theta_1$and$\theta_2in a Bayesian framework, with an intial joint prior distribution defined as: \begin{pmatrix} \theta_1 \\ \... 0 votes 0 answers 24 views ### Proof of Strong consistency of Beta posterior distribution Suppose that we have random variable X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0}) with p_{0} true unknown probability in [0,1]. Now, I want to implement Bayesian machinery to ... • 2,286 0 votes 0 answers 44 views ### laplace law of succession paradox (I am a newcomer in bayesian probability... so please be kind if this is very naive!) Laplace law of succession: (r+1)/(n+2) for a binary next event Jaynes ("Probability Theory", p 571) ... 1 vote 0 answers 30 views ### What's the difference between Bayesian and frequentist curve fitting? I was reading Bishop's Pattern Recognition and Machine Learning (PRML) and I am not completely sure I understand Bayesian (polynomial) curve-fitting. This might be an elementary question, but I ... 1 vote 0 answers 45 views ### My MCMC Simulation I am new to MCMC Simulation and Bayesian Analysis, so I wonder if my simulation has converged. My posterior is highly correlated by nature, so I'm facing some difficulty to ensure a sufficient number ... • 41 1 vote 0 answers 11 views ### Assessing Random Search Cross Validation: Tuning in ElasticNet with Large Feature Sets I'm working on estimating an ElasticNet model for a large dataframe with over 100,000 variables, resulting in a well overidentified scenario. To tune my model, I've set up a grid of hyperparameters (... • 301 1 vote 1 answer 38 views ### Understanding Empirical Bayes I am trying to understand the basics of empirical bayes. I found myself struggling a lot to understand this so I tried to create a toy example involving the estimation of the success probabilities ... 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 ... 4 votes 3 answers 107 views ### How to fit a Bayesian model to a mixture of Beta and One-Zero inflated data? I have very noisy data, which I believe is created through interactions of multiple physical processes. In the mapping Y = f(X), Y is a ratio [0, 1] and X \ge 0. While Y is a function of X,... • 101 0 votes 0 answers 15 views ### Modeling demographic patterns in voting using Bayes's rule If a town is 60% orange and 40% not orange and just voted 50% for Party A and 50% for Party B, and we have a prior \theta (maybe from a recent regional election) that gives us P(A|O,\theta) , P(A|... • 1 1 vote 0 answers 12 views ### Recycling MCMC samples for another data set from the same distribution Suppose I'm given \theta_0 and I want to sample data from a density f(Y|\theta_0) and then sample from the posterior of \theta|Y (given, obviously, some prior). I want to do this lots of times, ... • 41.6k 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 ... 1 vote 1 answer 111 views ### What does sample space look like for 3 dice? Learning Bayes statistics from Allen Downey's Think Bayes There are three dice, 6-sided, 8-sided and 12-sided. A randomly chosen dice is rolled and the outcome is "1". What's the probability ... • 43 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 3 votes 1 answer 102 views ### Applying Bayesian probability to a generalized Monty Hall problem I posted this question about the Monty Hall problem and Monty's knowledge of the probability distribution several months ago. I got some good answers and this one in particular helped me gain some ... 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 0 votes 0 answers 12 views ### Experimental Design: Selecting value of n given desired width of credible interval Note that this is a cross post from here. I realize this is probably a better space Suppose I have n IID Bernoulli trials with k successes. Assume that as a prior we are assuming that P(\theta) ... • 113 0 votes 0 answers 26 views ### Measured value represents range in timeseries I have datasets that consist of measured points (measuring m), across a landscape/area (x and y) and time (t). The issue comes in that the measurements are actually made over a period of time, a month ... 1 vote 1 answer 46 views ### When running a Bayesian mixed effects regression, if a random effect estimate has 95% CIs that include zero, should it be disregarded? Consider a Bayesian mixed effects regression. I am interested in the correlation between two of the random slopes. However, the 95% CIs for the correlation value include 0. Should I disregard the ... • 2,651 0 votes 0 answers 47 views ### Error in Bayesian Derivation of Covariance Matrix in Least Squares I know variants of this question have been asked a million times, but rather than just asking "how do I derive the covariance matrix" I ask you to check the error in my calculations, because ... • 372 0 votes 1 answer 30 views ### Bayesian updating with affine transformation of random variable I want to estimate a parameter \theta, and I have a prior \pi(\theta). I receive the realization of a random variable Y, which has some likelihood f_Y (y \mid \theta). My posterior then ... 0 votes 0 answers 12 views ### Mapping two Dirichlet Distributions into a comparative Dirichlet Assume I observe some draws from 2 choice options, and want to infer the probabilities of various outcomes, e.g. non-negative integers up to a limit L. I could simply use 2 Dirichlet distributions to ... 4 votes 3 answers 92 views ### Posterior expectation of normal distribution with "truncated" observation Consider the following problem of estimating an unknown parameter from normal samples: Suppose that \theta \sim N(0, \tau_\theta^{-1}), where \tau_\theta \ge 0 is the prior precision. Consider two ... • 215 -1 votes 0 answers 36 views ### What can we say about the variance of the posterior mean? In Bayesian inference, there's one famous theorem, Bernstein–von Mises theorem (see the Wikipedia or this lecture notes, page 35), states that in front of sufficiently large samples, that is ... • 185 1 vote 1 answer 20 views ### Extract credible/confidence interval of a threshold in a Bayesian posterior draws distribution [closed] I have a Bayesian model created through bayer package in R on which I need to calculate confidence/credible intervals for a ... • 229 0 votes 0 answers 18 views ### Adjust winrates based on number of games played using Bayesian estimator I am sorting players of an online video games based on their winrate, but I'm trying to use the number of games they scored that winrate on. The idea behind it is that even though 9 wins for 1 losse ... 1 vote 1 answer 27 views ### Understanding Discrepancies Between Frequentist and Bayesian Parametric Weibull Models in Accelerated Failure Time Analysis Currently in the process of identifying a Bayesian Weibull Survival AFT model that is equivalent to the survreg() Weibull AFT model results from the Survival package in R. I came across this ... • 51 0 votes 0 answers 45 views ### Help with completing a derivation of usefulness of cross-validation This question is raised as a result of my attempt to answer this other question of mine. Let's refer to all our prior knowledge, both explicit and implicit, as X_\text{true}. Almost always, we are ... • 197 0 votes 0 answers 23 views ### How to Compute the Posterior Distribution of Covariance matrix in a Matrix Normal Model with Inverse Wishart Prior I am working on a time series model involving Kalman filters and smoothing to estimate state variables Y_i. The part of model is structured as follows: Y_1, \ldots, Y_n are iid. Y_i \sim \mathcal{... 0 votes 0 answers 28 views ### Conjugate prior for a beta distribution [duplicate] What is the conjugate prior of the beta distribution? All I can find is the wikipedia page on conjugate prior. Is this correct? And does anyone know where it came from? Like a specific paper? Thanks 0 votes 0 answers 11 views ### Pros and Cons Using Bayesian Metrics or Time Series Approach? I have two years' worth of sales data for a range of products, including: Sales: Total sales revenue Usage Time: Total usage time in hours Reviews: Numeric ratings and textual reviews from users I'm ... • 1 0 votes 0 answers 27 views ### Is the mean of the Bayesian regression posterior a probability distribution itself? I know that to find the mean and covariance matrix of the posterior distribution of the regression coefficients, we have to equate the exponents of the product of the likelihood and the prior and the ... 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 ... 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 ... 5 votes 1 answer 64 views ### Do they use some Bayesian-frequentist amalgamate in astrophysics? Below is a figure (highlighting is mine) from Madhusudhan et al. (2023). It caught my attention in a recent video by Becky Smethurst, where she explains some more context (but that's not necessary for ... • 1,350 3 votes 1 answer 768 views ### Why a truly uninformative prior does not exist? [duplicate] It is said that there is no such thing as a truly uninformative prior. For example, here. Q: Has it been proven that a truly uninformative prior does not exist, or is it merely the case that such a ... • 755 0 votes 0 answers 32 views ### What is the difference between estimating parameters via MLE versus minimizing deviations from expectation? What is the difference between estimating parameters using MLE (or MAP with uniform priors):\theta^* = \arg \max_\theta p(X|\theta)$and estimating them according to which setting would engender ... • 145 0 votes 0 answers 39 views ### Bayesian Fubini Tonelli I am working on a bayesian framework where I place a Gaussian Process on my function$f\sim GP$and have data$D^n=\{(X_i,Z_i,W_i)\}^n$. I then have the posterior measure$\mu(f|D^n)$. The posterior ... • 90 0 votes 0 answers 15 views ### Predicting future states in hidden Markov models -- use the Viterbi algorithm? The Viterbi algorithm is used to decode hidden states in hidden Markov models (HMMs) by working out which sequence of states is most likely. To do this, it first identifies which state$j \in \{1, ...,...
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Background Let $y_k$ be a noisy measurement at time $k$ and let $\{p_{k-1}(i)\}_{i=1}^n$ be (a discrete) prior probability distribution. Using Bayes rule, one can update the prior in function of $y_k$ ...