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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79 views

Hypothesis testing via separate inference for each group and then combining

Suppose there are two groups, A and B, and we are interested in inferring a certain parameter for each one and also the difference between the two parameters. Here we can take a Bayesian perspective ...
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Calculate Conversion "Weight" based on Multiple Conversions

I'd like to estimate the value of each "conversion" starting from a free trial signup all the way to that user becoming a paid user. Let's say I have an online coding course website that ...
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Interpreting non linear brms output - estimates of posterior cooefficient and user supplied formula

I am a bit confused about how to approximate the equation from a nonlinear model constructed in brms, and was hoping someone could explain it to me. Say I have the below model: ...
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How did the posterior distribution get factorized in this manner? (bayes rule)

I am refering to this course on sampling https://www.youtube.com/watch?v=TNZk8lo4e-Q. At around minute 6, the lecturer shows on the slide the posterior probability factored as: $$p(\Theta |X,Y)=\frac{...
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effect of multiplying by Tikhonov regularization factor after an inverse?

I came across a repository which uses Tikhonov regularization to compute an inverse, but then in the inference step they multiply by the Tikhonov factor again... Compute $\Phi\Phi^T$ Compute the ...
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30 views

Understanding calculation of "expected log predictive density"

Suppose, I have a posterior distribution of parameters $\theta$, $P(\theta|y)=\frac{P(y|\theta)P(\theta)}{\int_\theta P(y|\theta)P(\theta)d\theta}$. Now, in order to assess the fit of the model, I ...
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22 views

What is the relationship between Bayesian Optimization and Sequential Model Based Optimization?

I often see these two terms being used almost interchangeably: Bayesian Optimization (BO) and Sequential Model Based Optimization (SMBO). Sometimes, SMBO is referred to as a formalisation/formalism of ...
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Can you use a prior on the population distribution when using BTYD?

In Counting Your Customers: Who-Are They and What Will They Do Next?, David C. Schmittlein, Donald G. Morrison, Richard Colombo, 1987. which defines the BTYD model, ...
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Marginalization over the nuisance variable

I was reading a paper in which they state $$ \text{P}(\mathbf{y}, \mathbf{f}, \mathbf{u}) = \text{P}(\mathbf{y}| \mathbf{f})\text{P}(\mathbf{f}| \mathbf{u})\text{P}(\mathbf{u})$$ With $\mathbf{f}$ ...
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MALA vs NUTS for sampling in MCMC fashion

Could someone please point out the pros and cons of Metropolis-adjusted Langevin algorithm and No-U-Turn Sampling algorithm wrt sampling from an intractable posterior? Which is better? Please try to ...
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Tied Bayesian Mixture of Gaussians

I am bit confused when it comes to modelling a Bayesian Gaussian mixture model that assumes a shared covariance/precision matrix for all Gaussian components. I followed the derivation in Bishop and ...
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1answer
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Computation of Posterior Mixture Weights

My question concerns the below example, where the author analyzes rainfall occurrences via a first order Markov chain. The transition probabilities are such that $p_{11} + p_{12} = 1$ and $p_{21} + p_{...
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Code availability for Bayesian inference based resolution of Simpson’s Paradox

I went though this blog. It describes how to compute the effect of an intervention involving a situation like Simpson’s Paradox. They are basically trying to answer a what-if question and they show ...
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1answer
42 views

How to use a sample from the posterior predictive distribution

Suppose I have a sample drawn from a posterior predictive distribution of a previously trained Bayesian Network (or any other Bayesian model). I.e., I have a vector $\tilde{\textbf{y}}_n = [\tilde{{y}}...
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40 views

How to calculate BIC from Gaussian Process Regression?

I'm currently using scikit-learn to fit a Gaussian Process Regression (GPR) to some 2-dimensional data. I want to compare this to another model using the Bayesian ...
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1answer
46 views

How can we attribute observations to observers in a hierarchical Bayesian model?

I am trying to make a hierarchical Bayesian model of latent variables based on many observations by noisy oracles. I want to leverage the information of which observations are from which oracles, as I ...
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1answer
58 views

Kullback-Leibler distance calculation for discrete distributions?

I have the following model $$N \sim Pois(\lambda) \\ n \sim Bin(N,p)$$ for which I calculate the posterior for the parameter $N$ as $$\pi(N|n,p,\lambda) = \frac{f(n|N,p)\pi(N|\lambda)}{f(n|p,\lambda)}...
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1answer
52 views

Is it practical to distinguish aleatoric uncertainty from epistemic uncertainty?

I know the difference between the two, but don't know if it is practical to tell them apart technically. Say, I have trained a deep neural network and I use some techniques to get the posterior ...
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15 views

Comparing two groups with measurements over multiple time points with Bayesian analysis

I have data from a surgical drug study with multiple measurements over time. In both groups A (treatment) and B (control), a biomarker indicating inflammatory response is measured at the following ...
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29 views

How to compute the mean of two conjugate distributions in an analytic posterior distribution?

I do not know why in the following picture the mean of this posterior is $\mu_n= (X^TX+\Lambda_0)^{-1}(X^TX\hat{\beta}+\Lambda_0\mu_0)$
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1answer
98 views

Frequentist vs bayesian and P(data | H0) vs P(H0 | data) giving same result

In hypothesis testing using a frequentist approach, we usually compute a p-value = $P(data\ or\ more\ extreme | H0)$. Moving to a bayesian approach, we are then able to compute different things, such ...
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9 views

dlmForecast error : "dlmForecast only works with constant models"

I have a dataset with intervention dummy variable to be incorporated inside the measurement equation (let's call Lambda) I picture my measurement and state are as below : measurement : Lambda + Et ...
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1answer
67 views

Likelihood in Bayesian inference: p(x|theta, I) = p(x| I)?

In page 164 of the book “Probability theory: the logic of science” the author says that: $$ p(D|\theta I) = \prod_{i=1}^{n} p(x_i|\theta I) = \theta^r(1-\theta)^{n-r} $$ $ \theta $, in this equation, ...
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1answer
38 views

Bayesian combination of expert opinion

In a population of $N$, $K$ experts pick $M_{k\in\{1, ..., K\}}$ individuals that will have a certain attribute. Note that $M$ can be different across experts (e.g., one expert can pick 5 individuals, ...
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1answer
43 views

Finding a,b parameters if Highest Posterior Density is known

I know that a beta distribution with unknown parameters a,b has a 95% HPD of [0.25, 0.75]. What is the correct approach to solve for a,b?
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6 views

Logit-link logistic regression: Calculating effect size between a continuous predictor's max and min value

Let's assume I have a logistic regression logit-link model as follows. binary_y ~ year I mostly work with bayesian regression models. Calculating the effect size ...
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31 views

Prior knowledge in context of nullhypothesis testing

I am currently working through the book Doing Bayesian Data Analysis - John Kruschke and have trouble to reason about a text paragraph [Page 315] Suppose that we are not flipping a coin, but we are ...
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1answer
130 views

How to understand maximum likelihood estimation from an objective Bayesian paradigm?

I am trying to understand maximum likelihood estimation from an objective Bayesian/Jaynesian paradigm. My current understanding is that: There is a parametric family of functions f(x; theta) indexed ...
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1answer
35 views

Interpreting false positives of Placebo group with full information

I'd like to make sure that I am interpreting a study correctly; I'm getting tripped up by Bayes Rule. Formulas and data are listed below my question. Here's the background: There is a study where ...
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27 views

why Gamma inverse is the conjugate prior of normal distribution?

I am trying to understand Bayesian regression. Then in Wikipedia enter link description here, it is written that by using the following relation we get the Gamma inverse as follow for the conjugate ...
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1answer
48 views

Binomial distribution - estimating confidence interval without mean?

This question is probably easy but I couldn't find the answer, nor remember my lectures in statistic. I have an (infinite) bag of red (A) and blue (B) chips, i.e. $P(A) = p = 1 - P(B)$ I want to ...
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1answer
36 views

Which kinds of priors could be applied in Bayesian linear regression?

I would like to know can we use any kind of prior distribution in Bayesian linear regression and still convergent to a close mean solution to the Least square solution? Or it should be just a ...
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28 views

Trying to find a distribution for time dependent drought

I would like to seek your support on a modelling issue for which I could not find relevant past postings or published literature resolve it. I am running a cost benefit model to assess the impact of ...
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1answer
49 views

Bayes theorem with multiple draws

Setting I have a question on the "Cookie Problem Revisited" exercise from Allen Downey's Think Bayes 2e. The Bayes theorem is defined as: $$ P(H | E) = \frac{P(H) \ P(E | H)}{P(E)} $$ where ...
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16 views

How to calculate alpha and beta parameters from an known mean and variance in normal-inverse gamma distribution

How can I calculate the $\alpha$ and $\beta$ parameters for a normal-inverse gamma distribution if I know the mean and variance?
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1answer
48 views

What is the importance of non-informative prior in Bayesian Inference? [duplicate]

By the name, noninformative prior, the prior distribution doesn't contain any information about the parameter. Then why we use this thing to estimate the parameter by the Bayesian approach?
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46 views

Bayesian update for a Gaussian distribution with unknown mean and variance

I am trying to implement the Bayesian Online Changepoint: https://arxiv.org/pdf/0710.3742.pdf in python. And I also have the step to update the sufficient statistics (variance and mean). I have a ...
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0answers
32 views

Bayesian Prior - a distribution conditioned on a set of measure zero? (Definition of a bayesian statistical model)

I am trying to write down the exact definition of a bayesian statistical model in a similar way as the definition for a statistical model. So far I have the definition: A statistical model is a pair $(...
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14 views

Posterior distribution shape for Gaussian Likelihood and non-linear model

I have an easy question I can't seem to find the answer to. I'm trying to fit some data to a non-linear model: $$d_{L}\left(x; H_0, \Omega_{m}, w\right)=\dfrac{c}{H_0}(1+x) \int_{o}^{x} \frac{\mathrm{...
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24 views

Statistical test on Bayes estimator

I have the following problem. Given a drug test, with $N$ people, $k$ have failed the test. And we assume a priori that the distribution of the test failing is a $\beta(p=1,q=6)$. Then the bayes ...
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36 views

The probability of life on another planet given an observation

Suppose you have pointed a telescope at a planet outside of our solar system, and observed some property of that planet (e.g. atmospheric composition). And suppose you want to know the probability of ...
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Conditional probabilities: Overdetermined system (conditioning a probability with redundant/extra information)

Question: Does $p(A|B,C)=p(A|B)$ if we presume that $A=B$ (or $A=f(B)$ more generally)? Background: I've been wrestling with overdetermined systems for some time. Presume the following relation $$A = ...
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1answer
62 views

Correlated belief update: Is this understanding of Bayesian posterior wrong

I am reading this paper Knowledge-Gradient Policy for Correlated Normal Beliefs for Rank and Selection Problem. The idea is as follows: We have $M$ distinct alternatives and samples from alternative $...
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46 views

Variable selection in Bayesian hierarchical models with R-INLA

I'm working with Bayesian hierarchichal regressions fitted with R-INLA. I would like to simplify my model by reducing the number of covariates. According to my understanding, Bayesian variable ...
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30 views

Bayesian linear regression with given distributions X, y instead of pairs {(X1, y1),..(X100, y100)}

I'm wondering if is it possible to model data by knowing only distribution of features (X) and targets (y). Thus, instead of paired variables {(X1, y1), (X2, y2), .., (Xn, yn)} I know only mean value ...
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25 views

How to apply MCMC to bayes when likelihood is not easy to compute

Let $z$ be observations and $w$ be the parameter that we want to infer. Assuming that we know the prior $p(x)$, by using Bayes law, we have $p(x|z) = p(z|x)p(x)/p(z)$ where $Z$ is the marginal ...
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2answers
70 views

Why is the beta distribution so flat when a, b=1?

If the beta distribution is a prior of a Bernoulli distribution (i.e. a rate of success for a binary outcome), then it is completely counterintuitive to me that the beta distribution should be ...
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33 views

Predictive posterior update for unknown mean and variance

I am trying to implement the Bayesian Online Changepoint: https://arxiv.org/pdf/0710.3742.pdf in python. And I also have the step to update the sufficient statistics (variance and mean). The work is ...
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1answer
30 views

How to calculate Log Predictive Probability Desnity?

I am trying to follow example in the book Statistical Rethinking (Page 210, code 7.13 and 7.14) for calculating LPPD of a simple model. The given example code in the book is: ...
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59 views

Constructing a hypothesis test for "ask the audience" lifeline in "Who Wants to Be a Millionaire?"

I wish to construct a hypothesis test that assesses whether the differences between the highest voted answer and the other answers shown when a contestant selects the "ask the audience" ...

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