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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54 views

Why do the non-informative a priori distributions give better results than the frequentist estimate?

For example, in the specific case of Markov-Switching GARCH models why is a non-informative prior distribution chosen for GARCH models with Bayesian estimation and why is this approach better than the ...
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42 views

Metropolis Hastings proposal for one parameter restricted to less than the other

Suppose I have parameters $\theta_0$ and $\theta_1$ with prior $$ p(\theta_0,\theta_1)=p(\theta_0|\theta_0<\theta_1)p(\theta_1),$$ that is, $\theta_0$ is less than $\theta_1$. The distributions ...
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43 views

Motivations for experiment design in statistical learning?

My interests in statistics centre around statistical learning, including Bayesian inference, inference in combinatorial spaces, Monte Carlo methods, Markov decision processes, modeling stochastic ...
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41 views

Combining multiple predictions with Bayes Theorem

I have multiple weather forecasters who each use their own unique, independent calculation for prediction of the weather for the next day. We are only concerned with rain predictions to know if we may ...
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1answer
61 views

Probability of a box containing a combination of color

Let's say, we have a box containing 3 balls in it, they can be either red or blue. Someone draw a ball 5 times with replacement and get 4 red and 1 blue (not necessarily in order). Do you know how to ...
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33 views

Two sample test for equality of 2 dimensional distributions

I have a large sample from a 2 dimensional continuous unknown distribution. From that sample I could compute any data structure I need to hold an approximation of the sample distribution. This will be ...
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13 views

how to use depended / non-random observations when trying to inference exponential parameter

consider this case: There is a price rate for a certain product that changes throw time, The price rate is changed every x minutes (unknown, not constant). This price has depended / non-random ...
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30 views

fit a model to data

I want to fit a model to a data set, however each point is actually a distribution (i.e. I have the samples for each distribution). In an ideal world, I would assume that the distributions are ...
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1answer
83 views

Variance of evidence lower bound(ELBO) loss function

When using Bayesian optimisation in a neural network our loss function is equal to: Here the first term is the KL divergence between the approximate and true posteriors. The second term is the ...
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1answer
50 views

Intercept in a Bayesian model with categorical predictors (with brms)

I have a Bayesian logistic model fitted in R with brms. The predicted variable is binomial, the predictors are categorical. The model uses bernoulli family and a ...
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24 views

Gibbs sampling for mixture with Dirichlet prior?

I want to sample from the distribution of a mixture distribution. The hierarchical model is $x_i\sim f$, where: $$f(x\mid \theta_1,\dots,\theta_p, w_1,\dots,\omega_p) = \sum_{j=1}w_p\varphi(x\mid\...
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1answer
59 views

Bayesian predictions from posterior parameter distributions

I have two physical models $f(\theta)$ and $g(\theta)$ (not probability distributions) parameterized on the same set of parameters $\theta$. I also have data $y$ with measurement noise $\epsilon$ ...
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25 views

Likelihood, posterior, prior interpretation and credibility/confidence_level with bayesian/frequentist approaches

This question was originally posted on physics exchange but one advised me to transfer it here. I try to understand the following article : testing general relativity from curvature and energy ...
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What kind of a priori distribution for the Markov Switching models?

Why in the Markov-Switching models is chosen as prior distribution for the probability of the transaction as follows: $$f(P) \propto \prod_{i=1}^K \left(\prod_{j=1}^K p_{i,j}\right) I \left\{0 < ...
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29 views

Modeling Bayesian inference using Dirichlet conjugate

I'm trying to formalize my research question and want to know whether the following set up makes any sense or not. Suppose there are two coins $a$ and $b$. Probability of tossing heads are given by $...
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1answer
38 views

Geometric distribution with a capped number of trials - finding expectation and prior predictive distribution

So I am modeling a random variable which follows a geometric distribution with probability $\theta$ except that the total number of trials is capped at some value $n$. I.e., the probability mass ...
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1answer
19 views

Bayesian repeated updates, likelihood functions with different nature

Let's say we have a prior probability of some diseases 'D'. Then we have some data and likelihood function of symptoms (S) P(S|D) and we update priors. Then we have age (A) likelihood function P(A|D) ...
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23 views

Adjusting for probabilities of different updates in Metropolis Hastings

I have a problem with specifying the update probability in MH. Assume we have the following setup in Metropolis-Hastings. We want to target a (2N-1)-dimensional posterior of parameters $(\alpha_2, \...
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2answers
45 views

Interpreting mixture of Gaussians (Variational Inference)

I've recently stated reading about mixture models and variational inference in this excellent paper, but I'm having troubles dissecting the models described, and have a couple of questions. Please see ...
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1answer
36 views

Adjustment for multiple comparison in bayesian multivariate regression model (using brms)

I am investigating age and timepoint effects on different (correlated) EEG parameters in a repeated measurements structure. I chose to use the brms R package to fit a bayesian multivariate model with ...
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35 views

Exact posterior and marginal likelihood

I am working on an approximate method of Bayesian inference and I want to study its approximation properties by comparing my approximate posterior and marginal likelihood with its exact counterpart. I ...
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47 views

Deriving conditional probability of bivariate bernoulli by using Dirichlet

While I was working on my research project, I found it difficult to derive a conditional probability from Dirichlet dist. Consider two Bernoulli trials that are possibly correlated with each other. ...
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15 views

Multivariate bayesian inference: learnig about the mean of a variable by observing another variable

I want to derive a Bayesian learning procedure where I don't only learn from my own signal, but also from other signals which are correlated to mine. I thought it could simply work with Bayesian ...
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4answers
107 views

Bayesian Hypothesis Tests with continuous priors

I am new to the Bayesian world, and I'm trying to understand how hypotheses tests are performed here (as opposed to the frequentist framework). I am aware that likelihoods, priors and posteriors can ...
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3answers
69 views

Bayesian Inference: Feeding Posterior back in as Prior

I've just started reading about Bayesian Inference, and one thing I've wondered about is if it's possible to feed the posterior in as a new prior for a new model, using the same data. Or would that ...
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43 views

If $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ conjugate to $p(\theta)$?

If exponential family $f(x|\theta)$ is conjugate to $p(\theta)$ then is $f(x|r\theta)$ for $r>0$ conjugate to $p(\theta)$? If not, what can we do about it in terms of sampling to make use of ...
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27 views

Is there a conjugate prior for normal location family N(x|u,1) such that the mean is always positive?

The conjugate prior to normal location family is usually a normal distribution. However, I want to constrain the mean to be positive. Is there a conjugate prior to the normal location family $x\sim N(...
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9 views

Bayesian Comparison of Dependent Correlation Coefficients

I would like to compare the correlation coefficients from two different models of the same data points. One of them is the raw bivariate data (Value vs Time), and the other is given a biologically-...
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28 views

Why is a frequentist confidence interval equivalent to a credible interval with flat priors?

It's a commonly quoted result that frequentist confidence intervals are equivalent to a bayesian credible interval assuming a flat prior. Ignoring for now questions about invariance under ...
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1answer
40 views

Updating a probability with additional knowledge. Bayes Theorem

I am quite confused with using Bayes theorem for the following problem. And I am not sure it can be applied at all. I have a football website data with user views. Each view corresponds to a specific ...
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17 views

Why BSTS model gives very different forecasts depending on length of testing set?

When I run a BSTS model in R, I get a finished model from a data set, and when I use that model to predict, for example, like this: predict(model3,newdata=futuredata) -> pred_1 I get a set of ...
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2answers
139 views

Did Jaynes ever comment on Lindley’s paradox?

I wondered whether ET Jaynes ever wrote or expressed an opinion about Lindley’s famous statistical paradox? I would be curious about his take on it, and imagine he must have done since he wrote ...
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26 views

Modeling the Probability of Falling Asleep

Problem The goal of this exercise is to estimate $p = \text{Pr}(\text{fall asleep})$, the probability of falling asleep imminently, as a mental exercise for those nights when sleep doesn't seem to be ...
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15 views

Contrast coding for 2x3 interaction coefficients

I'm using a Bayesian Poisson ANOVA to estimate the strength of interaction between a 2-level and 3-level variable. I'd like to report a single contrast if possible, but I'm not sure how to design the ...
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1answer
220 views

Correct or not? Mixed Bayes' Rule - Noisy Communication

In this problem, we study a simple noisy communication channel. Suppose that $X$ is a binary signal that takes value $−1$ and $1$ with equal probability. This signal $X$ is sent through a ...
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1answer
61 views

Mechanics for combining likelihood and prior in non-trivial case

Bayes rule is simple enough on its face: $$ pr(B|A) = \frac{pr(A|B)pr(B)}{pr(A)} $$ If these things are known scalar probabilities, the answer is simple to compute. But I'm failing to understand ...
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1answer
44 views

Bayesian inference for a conditional probability

I'm simplifying my research question and want to know whether the question can be properly modeled or not. Suppose we have two coins $X_1,X_2$ and assume that the outcomes are possibly correlated. ...
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3answers
74 views

Why sometimes evidence is said to be too complex to compute and other times is negelected cause it is fixed?

When motivating something like variational inference they say the denominator in Bayes rule is too complex to compute. Other times I see that the denominator is neglected and equality is replaced by ...
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15 views

Complex Linear Regression - Problem with Multicollinearity and Bayesian approach

I want to test out complex linear regressions with a simple model y = a*x + b. The normal equation is not invertible because of multicollinearity. The first two columns are equal to 1 because the two ...
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1answer
68 views

MAP of Gaussian Process Classification in Tensorflow Probability

I'm attempting to implement Gaussian Process Classification learning in tensorflow-probability, but my estimator turns out to be very biased toward zero. As opposed ...
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9 views

Online learning from a Bayesian Perspective in a State-Space Model

I'm trying to learn how to do online learning from a Bayesian Perspective. My main interest is to use it for a State-Space model. However, any explanation/reference in a different context, which may ...
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1answer
23 views

Calculating Conditional Probability without knowing individual probabilities

I have a case where I know the probability that three annotators will agree with each other pairwise, and I'm trying to find the probability that all three will agree on a yes or no question given ...
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18 views

Relationship between posterior yielded from entire data vs posterior from subset of the data

I have a question about the relationship between a posterior for $\theta$ yielded from a pooled dataset of size $n$ and a posterior for $\theta$ yielded from a subset of the data, size $n_1$, and an ...
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343 views

When does a confidence interval “make sense” but the corresponding credible interval does not?

It is often the case that a confidence interval with 95% coverage is very similar to a credible interval that contains 95% of the posterior density. This happens when the prior is uniform or near ...
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75 views

Two class bayesian decision theory

I'm new to decision theory, but in the many "intro to bayesian decision theory" tutorials, the two-category classification example is usually given. It boils down to deciding action 1 if it's risk is ...
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1answer
60 views

Inverse Wishart Prior for linear model

I know some bayesian methods employ an inverse wishart distribution for the prior distribution of the covariance matrix in a linear regression. I.e. for the model: $$Y=X\beta+\epsilon$$ Where $\...
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33 views

Bayesian estimation of mixed effects models covariance matrix

For a mixed model of the form: $$Y = X\beta + Z u + \epsilon$$ I know it is usually assumed in the parametric approach that: $u \sim N(0, D)$ and $\epsilon \sim N(0, \sigma^2I)$ Where $D$ is a ...
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89 views

Incorporating Prior Information Into Time Series Prediction

Suppose I have data on my child C's height measured every week. Presumably there is a positive trend, due to growth, and some noise due to measurement errors, and maybe even seasonality (winter boots ...
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Methods to utilise training data with unknown complex links

I am basically asking the question, what types of data analysis techniques may be suitable for taking individual n-dimensional training data, then using real world measurements of n-1 of these ...
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1answer
38 views

How to solve MLE when noise variance is made learnable rather than fixed?

I saw some machine learning code assuming that variance of gaussian noise is a learnable parameter in linear regression problem. I'm wondering how is this solved theoretically? Below you see typcial ...