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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Setting parameters and functions for a single component Metropolis-Hastings algorithm

I'm trying to rebuild the results from an article by Dos Reis et al., called 'Capturing model risk and rating momentum in the estimation of probabilities of default and credit rating migrations'. ...
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66 views

Closed-form likelihood. Why to use Bayesian parameter estimation instead of maximum likelihood? [duplicate]

Given a model with a closed-form likelihood, what are reasons to use Bayesian parameter estimation instead of maximum likelihood? In other words, what are additional estimation results that you can ...
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The differences between Bayesian network and other Bayesian modeling approaches, such as Bayesian regression

What are the differences between Bayesian network and normal Bayesian regression, it seems that both of them aim to update the prior information via posterior analysis. What are the differences from ...
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Ideas how to report/emphasize a non-linear effect in abstract as text in case of no p-values?

I have a result that needs to be noted in abstract as this is important. I use Bayesian regression modelling and I have non-linear? effects as shown on plots below, what is a good way for reporting it ...
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How are artificially balanced datasets corrected for?

I came across the following in Pattern Recognition and Machine Learning by Christopher Bishop - A balanced data set in which we have selected equal numbers of examples from each of the classes would ...
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What is a “batch” of coefficients in Bayesian multilevel modeling?

I’m well acquainted with frequentist approaches for multilevel models (i.e. mixed/random effects models with random intercepts or slopes), and empirical Bayes estimation, but I’m trying to get ...
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1answer
50 views

Why is computing $\log p(x)$ difficult, but not the ELBO?

This question is in the context where we have some observed data $x$ and some latent variables $z$ which may be used to 'explain' the data. Let's say we have some likelihood model $p(x \vert z)$ and ...
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Relation of Bayes Theorem, Conditional Probability and Likelihood

Bayes Theorem is derived using conditional probability: $$P(A|B)\cdot P(B) = P(A) \cdot P(B|A)$$ Here terms are either probability or conditional probability. But later, in $$P(A|B) = P(A)\cdot \frac{...
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Running several MCMC chains after convergence?

I am running a MCMC Gibbs sampler for a computationally expensive model. It takes ~12 hours to obtain 1000 iterations of this MCMC sampler. I have tested the sampler, and I found that the chain seems ...
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42 views

Several trivial questions about applying Bayes’ rule

Question 1 I wonder can I calculate Bayesian rule like following: P(Raining|Peter uses an umbrella) Since raining or not is irrelevant if Peter uses an umbrella Can we still use Bayes rule? If yes how ...
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Understanding Bayes Rule Application in POMD Belief State Update

I am trying to wrap my head around Partially Observed Markov Decision Process (POMDP). However, I am unable to understand the application of the Bayes Rule in following equation (Step Nr. 2): Can ...
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What does it mean to have a “gaussian prior?”

When reading up on ridge regression, I saw it stated that it has a "gaussian prior." I realized that I don't know what the word prior means in this context and what it is applied to? I ...
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Use inference on the whole data as a prior to regularize estimates in a subgroup

Suppose that the probability of a certain condition in the data $D$ is $\theta$ which can be estimated by updating a Beta prior as $Pr(\theta|s, N) = Beta(\alpha + s, \beta + N - s)$ with $s$ being ...
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Beta Binomial Distribution Derivation- Bayesian

This is from Hoff's Book:First Course in Bayesian Statistics How is p(y|θ) in equation (1) is similar or different than that of equation (2)?. In equation (1) p(y|θ) is treated as joint ...
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Sanity check: Ensemble-based sequential state inference with different parameters (Ensemble Kalman Filter)

I have recently been invited to review a publication which employs the Ensemble Kalman Filter (EnKF) for the sequential inference of dynamic state variables $x_{1,...,t}=(x_1,...,x_t)$. In the study, ...
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Bayesian inference: all at once vs one at a time

Suppose that I have a prior on a parameter $\theta$ and update this prior in light of the realisation of $n$ random variables. It seems plausible that it is equivalent to update the prior $n$ times, ...
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Combination of bayesian models in pymc3

Simplified problem is: I noted down peoples’ physiological parameters (age, BMI, resting heart rate, etc.) and asked them to perform exercises. From that I measured their HRmax and how well they ...
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Gamma-Poisson conjugate prior, posterior exploding?

I've been looking for simple code that can model ad clicks per day. Notionally, gamma-poisson would be a good conjugate prior. However, I'm finding that for slightly large daily click rate values, the ...
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How to use scipy stats gamma pdf to update the posterior distribution?

I'm trying to "get my bearings" performing bayesian analysis, specifically I'm exploring the Gamma-Poisson conjugate prior. The definition of the PDF is below If the prior takes the form of ...
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Posterior predictive check on non parametric posterior (not a stan model) in R

I am trying to figure out how to do a posterior predictive check on samples from a posterior distribution (a list of samples) obtained using Metropolis-Hastings in R. I have 6 parameters. When I look ...
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18 views

Prediction Interval for log-tranformed data

I am doing a linear regression on log-transformed data and I use the bayesian approach to model the predictive distribution and construct my 90% prediction Interval. The problem with this approach is ...
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How to model repeated measurements with the same outcome in a Bayesian framework?

Can't think of a more accurate title, so I'll illustrate the problem with an example. I want to record temperature using cheap noisy sensors. I also have recordings from a gold-standard reference ...
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Why does Quadratic (Normal/Laplace) Approximation fail on multilevel models?

In Statistical Rethinking, 2nd Edition, section 13.1, Richard McElreath says: Why doesn’t simple quadratic approximation, using for example quap, work with multilevel models? When a prior is itself a ...
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Metropolis-Hastings exercise with Cauchy and normal distributions [self-study]

I have the following exercise to solve and would appreciate some help. Consider linear regression model $y = X\beta + \varepsilon$, where $y = (y_1,...,y_T)'$, $X = (x_1,...,x_T)$, $x_t$ is a single ...
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Bayesian inference with an arbitrary prior

A classical problem in Bayesian inference arises when we wish to learn about (say) the fraction $\theta$ of balls in an urn that are white; and do so by sampling from the urn with replacement. In such ...
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24 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 ...
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Direct relationship between prior and posterior parameter

I've been trying to learn JAGS and Bayesian modeling more generally and I'm running into something I can't quite explain. I've noticed that when fitting simple mean and variances to normally ...
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1answer
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Are priors in Bayesian inference similar to levels in mixed-effects models?

I often see a frequentist multi-level model (MLM) structure is defined like so (made up parameters): $$ \theta_i \sim \mathcal{N}(10, 2.5) $$ $$y_{i,j} \sim \mathcal{N}(\theta_i, 0.5) $$ But this is ...
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Multi-object counting under uncertainty

I am interested in learning about different ways to model ones beliefs about hearing related sounds under uncertainty. Specifically, how might I model the following: There is a finite set of possible ...
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Exchangeability and joint distribution

The definition of an exchangeabilty for a finite sequence says that, if we have random variables $X_1,\ldots,X_n$, then for each permutation $\pi: \{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$, the joint ...
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Interpretation of coefficients in mixed-effects model with circular response?

I have a dataset from an experiment where wild ants were surveyed continuously for 24 hours under a number of temperature treatments (chambers). Whenever an ant was observed, the species of the ant ...
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Setting variance of an informative prior

I am creating a Bayesian Poisson Regression model and I have access to a dataset and a previous corresponding model. I want to use the previous model to create a prior that I will combine with the ...
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22 views

Account for autocorrelation without autoregressive model

I have a time series model that has seasonality (and therefore it has autocorrelated errors fitting an OLS model) - how can I account for autocorrelation without a complicated autoregressive model ...
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1answer
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Problem with bayesian implementation of a Time-lagged Linear Model in PyMC3

I am trying to build a GLM of a time-series y(t) with 2 predictor time series x1(t) and x2(t), where t is in days. But the second time-series influences y(t) with an unknown lag of l days. I was ...
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How to explain the difference between confidence and credible interval?

The key difference between Bayesian statistical inference and frequentist statistical methods concerns the nature of the unknown parameters that you are trying to estimate. In the frequentist ...
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Bayesian priors and probability distributions

Book "Bayesian Statistics the Fun Way: Understanding Statistics and Probability with Star Wars, Lego, and Rubber Ducks", chapter 9 "Bayesian priors and working with probability ...
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Importance sampling and Metropolis MC

I am evaluating numerically integral $$I(\theta) = \int_{-\infty}^{+\infty} dx_1 dx_2 dx_3 dx_4 \int_0^{+\infty} dy_1 dy_2 dy_3 dy_4 \prod_{k=0}^4\left[w_n(x_k)w_e(y_k)\right]F(x_1, x_2, x_3, x_4, y_1,...
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Bayesian evaluation if partitioning is justified for a dataset

I'd like to comparare whether partitioning of a dataset is justified. The data is categorical with two levels and the fitted parameter is the prevalence of positives for a certain condition in each ...
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1answer
31 views

How to choose a non-informative or weakly informative hyper priors for my hierarchical bayesian model?

I am learning Bayes on "Applied Bayesian Statistics" by MK Cowles. The chapter about "Bayesian Hierarchical Models" mentioned an example that we estimate a softball player’s ...
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Escape unsuccessful accept-reject step in MCMC

I have an MCMC procedure that samples latent variables $h_1, \dots, h_T$. It is based on Shephard and Pitt (1997), https://doi.org/10.1093/biomet/84.3.653. Let $f$ be the true conditional posterior ...
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Example of message passing algorithm

I've been reading about approximate message passing (AMP) but still don't know how it facilitate computing marginal densities. Can anyone show me an example with number how to calculate a marginal PDF ...
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time series: What is the performance difference between ARIMA models and Bayesian Structural Time Series models

I have been looking at ARIMA/SARIMA models and some of the Bayesian Structural Time-Series models lately. The formulation of the two models does not seem that different but the fitting method of ...
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113 views

Bayesian interpretation of logistic ridge regression

Most textbooks (also this blog) cover the fact that ridge regression, $$ \hat y = \hat \beta X; \\ \hat \beta = \underset{\beta}{\text{argmin}}\ \ \frac{(y-\beta X)^T(y-\beta X)}{\sigma^2} + \lambda \...
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1answer
28 views

integration of Gaussian with prior mean

I want to calculate the following Integration $$\int \mathcal{N}\left(\mathbf{x} \mid \boldsymbol{\mu}, \boldsymbol{\Lambda}^{-1}\right) \cdot \mathcal{N}\left(\boldsymbol{\mu} \mid \mathbf{m},\left(\...
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Likelihood function for switchpoint analysis--in this case example from PyMC3

I was watching a video on PyMC3 for fitting Bayesian models, and an example they gave was of "switchpoint" analysis for coal mining disasters. A picture of the data is below, and the goal is ...
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42 views

Manually calculating `false negative risk` (using Likelihood ratio and Bayesian analysis)

The question is with reference to this paper: https://arxiv.org/pdf/1802.04888.pdf and https://royalsocietypublishing.org/doi/full/10.1098/rsos.171085 It give clearly how to calculate ...
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1answer
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Understanding reparameterization trick and training process in variational autoencoders

I am trying to understand variational autoencoders, particularly the sampling component and the reparameterization trick. I understand that instead of using a fixed determinstic latent representation ...
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1answer
19 views

Bayesian estimation of a proportion

Suppose I am interested in learning about the proportion $p$ of the population with a certain property (e.g. the proportion who are over 6ft tall). I observe $n$ binary data points, $X_1$, ..., $X_n$ (...
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Should I report the results with random effects or not (brms, re_formula = NULL vs NA)?

I am struggling to understand should I report the results with random effects or not (re_formula = NULL or NA). I have a population-based data. Patients are nested in differently sized counties and I ...
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Interpretation of sampling distribution as the main distinction between Bayesian and classical statistics (Leamer)

In Hendry et al. (1990) p. 187-188, Edward Leamer says: To me the essential difference between the Bayesian and a classical point of view is not that the parameters are treated as random variables, ...

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