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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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SD of a likelihood function: can it replace the Standard error of a sampling distribution

I was wondering if "standard deviation" of a "likelihood function" could ever represent the "Standard error" of a "sampling distribution"? I ask this, because when one follows a Bayesian approach ...
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Admissible and Inadmissible actions

Consider the following loss matrix. $\begin{array}{|c|c|c|c|} \hline & \alpha_1 & \alpha_2 & \alpha_3 \\ \hline \theta_1 & 1000& -300& 4000\\ \hline \theta_2 & -1000&...
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How to infer the distribution of a statistic (Bayesian inference?)

I have a list of approximately 30,000 venues in a major US city. These venues hold all kinds of events, sports, conferences, concerts etc. I want to know the distribution of the 'capacity' of these ...
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sequential Monte Carlo sampler, why the extended space and backward kernel?

Hello cross validated, I am currently studying sequential Monte Carlo samplers. My current understanding is as follows: We are interested in the marginal distribution of some sequence of joint ...
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Is my Bayesian approach (mixed effect probit regression) appropriate

I am hoping someone more experienced than I can let me know if I'm on the right track. I was asked to take a Bayesian approach to evaluate the null hypothesis from a collaborator's experiment. I’m ...
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log uniform prior vs gaussian prior to compute KL-divergence in Variational Inference

The optimal value of variational parameters can be found by maximization of the variational lower bound: In some papers, we see that they have proposed log uniform prior and tried to approximate the ...
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Does conjugate prior for natural exponential family needs jacobian to transform natural parameter back to original parameter?

From bayesian theory, we have that if $f(x|\eta) \propto \exp(\eta \cdot T(x)- A(\eta))$ - a natural exponential family, then the prior conjugate of $\eta$ is $\pi^*(\eta | \mu, \lambda) \propto \exp(\...
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Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$ P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no ...
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Calculating priors with large number of classes

I'm trying to classify users into ~1K different groups. I'm trying to build a MAP classifier and have estimated my prior and posterior distributions using large amounts of data. The issue that I've ...
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34 views

Confusion when Learning Parameters in BAYESIAN MODELS

I'm learning Bayesian Models but i still have some issues with the training of the parameters. These are my two questions : 1) Recall the Bayesian formula : $$p(\theta|X) = \frac{ p(X|\theta) \; p(\...
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Why does this improper prior = constant?

MacKay has an exercise on using Laplace's method for a Poisson model: $$ p(r \mid \lambda ) = \frac{e^{-\lambda} \lambda^r}{r!}, \qquad p(\lambda) = \frac{1}{\lambda} $$ And he asks the reader to ...
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Estimating sample size needed to reach estimate precision (Bayesian CIs)

I want to estimate the required sample size to reach a specified width (precision) of the Bayesian credible intervals of the parameter of interest by using the standard error (posterior standard ...
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Is it valid to make a fraction out of output of naive bayesian classifiers?

I'm working on a model of a twitter network that attempts to determine the likelihood of a tweet being retweeted. For every user in the network, I have a list of all of the tweets by all of the users ...
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1answer
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posterior distribution of f for Gaussian process model given existed observation data and input

In the chapter 2 of [Gaussian Process], equations (2.22-2.24) gives the predictive equations for Gaussian process regression, shown as follows. My question is how to derive f|X,y. It seems that the ...
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Bayesian A/B testing and error estimates

Someone from my work started using an AB calculator that they found online and asked my take on the approach which i assume is incorrect. The approach takes the conversion and visits for control and ...
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Prior distribution on a transform parameter [duplicate]

Let say that my likelihood has the following form: $[logit^{-1}\{\theta_1 + \theta_2\log(x_i)\}]^{y_i}[1-logit^{-1}\{\theta_1 + \theta_2\log(x_i)\}]^{n_i-y_i}$ where $\theta_1$ and $\theta_2$ are my ...
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bayesian analysis for learning hyper parameters of gaussian process model

For the Gaussian process, what are the approaches to learn the hyper-parameters? Are there any ways to apply Bayesian analysis over these hyper parameters based on new observations?
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68 views

What mathematically enables one to Bayesian update multiple times?

I'm going through the MIT OCW notes on probability and statistics and came across an example in some class notes where the posterior is updated after two events: $x_1 = 1$ and $x_2=1$, which ...
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How to determine hypothesis and evidence in Bayes theorem

I am new to Bayes theorem, and bit confused how to identify evidence and hypothesis in the theorem.For e.g.I have this problem - in a company 10% of laptops fail within their warranty period. 5% of ...
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How do we reuse and store the output of an MCMC?

For many Bayesian models, the posterior distribution is intractable... a solution is then to sample points from this unknow distribution with a Markov Chain Monte Carlo (MCMC). But at the end, how do ...
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Does intended model use affect Bayesian parameter estimation?

Bayesian parameter estimation results in a posterior distribution for model parameters. The user may or may not be interested equally much in all properties of the distribution. Perhaps the user ...
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Bayes Minimax Estimation

Let $S\sim B(n,\theta), l(\theta, a) = (\theta - a)^2,\delta=\bar{X} = S/n,$ and $$\delta^*(S)=\left(S+\frac{1}{2}\sqrt{n}\right)/\left(n+\sqrt{n}\right)$$ where $B(n,\theta)$ is Bernoulli ...
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Explanation of the Posterior Derivation of the Gaussian Distribution

I'm reading through my notes and I don't quite understand this bit: I understand how the likelihood was calculated but no more than that.Can anyone explain the steps and exactly how they go from one ...
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Trouble understanding sample spaces in Bayes Theorem

I wanted to check my understanding of what's going on in the problem below, which employs Bayes' Theorem. I would like to understand what's going on with respect to the sample space $\Omega$. It ...
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Bayesian Inference, Posteriors Priors and Likelihoods [duplicate]

So at the moment I'm reading through my notes on Bayesian Inference and I'm really just not understanding anything. If anyone has any good websites that explain this topic well then I'd be really ...
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notation of derivative probability function

I am reading article about bayesian predictive function. In the article it denote posterior distribution $\pi_n(d\theta) = \frac {\prod^n_{i=1}f(y_i|\theta) \pi(d\theta)}{\int \prod^n_{i=1}f(y_i|\...
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this is not Bayesian, does it have a name

Suppose that I have prior belief B. Then I get an information D from the source S. I update B with D. However, my update magnitude depends on my trust T(S) in S. Also, the marginal update diminishes ...
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Formal Bayesian justification of conditional modelling

I'm having some trouble following the logic of this passage from Chapter 14 in Bayesian Data Analysis, A. Gelman: The numerical 'data' in a regression problem includes both $X$ and $y$. Thus, a ...
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33 views

Calculation of Bayeain rule as classifier for mixture Gaussian model

Here is a paper which used a bayesian classification based on Gaussian mixture model I read many article saying that we can fit a gaussian mixture model to a data and then, based on the estimated ...
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Independence test with priors

Say you are selling a product, and you know from experience that the green version of the product sells better than the blue version. But you have two types of customer A and B, and you want to know ...
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1answer
54 views

MLE for logistic regression, formal derivation [duplicate]

I am currently working through Bishops' Pattern Recognition and Machine Learning where the following issue came up. It is closely related to the unanswered post below, but I wanted to propose a more ...
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30 views

Constraints on choice of marginal distribution and likelihood

For some time I have been reading into Bishop's Pattern Recognition and Machine Learning. Coming back to some earlier chapters the following got me confused and I am interested where, formally I go ...
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How to calculate the predictions using Bayes model avergaing?

suppose I have $K$ models $M_1, \dots, M_K$ (in fact I want to use NNs) and some observed data $D =\{(x^1, y^1), \dots, (x^n, y^n)\}$. Then by definition, the Bayes model average predictor is $\hat{y}...
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26 views

manual implementation of Gaussian naive bayesian returns posterior larger than 1

I try to implement Gaussian naive bayesian manually in R. I test my model on iris data set. I would like to build a predictive model. That is, I would like to ...
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1D Bayesian Inference clarification

I'd like some help making sure I understand a 1D Bayesian inference problem. Say I have a data vector which is an array of the number of flu cases reported weekly in California for the past 10 years. ...
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1answer
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How to interpret estimates and correlation of random effects (intercepts and slope) in a mixed-effects model in a Bayesian framework(brms)?

I do not understand how to interpret random slopes from the output of brms Among others, I read this post on the output from ...
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Problems estimating a “Bayesian version of FIML”

I am anticipating that my question exposes some basic ignorance about how mcmc works, but here we go: In an attempt to deal with missing data I am trying to simultaneously estimate a regression model ...
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1answer
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What is the real-life benefit and application of Bayesian regression [closed]

Question What is the real-life example of the benefit and application of the benefit of Bayesian regression? Having read the items and it looks having the range of inference (possible values and ...
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How to calculate forecast given variation by day

I'm trying to work through a problem and I'm wondering if I'm interpreting it correctly. Let's say we predict the price of stock (today worth $50) to vary by N~(0,1) every day, and you are looking to ...
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Why are the additional set of parameters in discriminative models necessary(in Minka's 2005 paper)?

In a short paper titled Discriminative models, not discriminative training by Tom Minka, it says that the discriminative training might work better because it has two sets of independent parameters ...
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1answer
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Conclusions from combination of P value and Effect size

Limitations of P values are being increasingly highlighted in recent literature (e.g. here and here). To evaluate results of unpaired t-test, following is an approach combining P values and effect ...
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1answer
29 views

Bayesian approach: ignoring the denominator leads to the conditional density equaling the joint density? [duplicate]

I know there are a lot of questions here about ignoring the denominator in a Bayesian approach, but I don't think mine is a duplicate of any of them. I am reading the book "Pattern recognition and ...
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Bayesian method to determine effect size

I have data on heights of 2 groups of persons. Usually Cohen's-d is used to determine effect size. Is there any Bayesian equivalent of Cohen's-d which reflects effect size of the difference between ...
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1answer
38 views

What happens if the observations are connected in a hidden Markov model (HMM)?

Suppose that we have an HMM with hidden variables $X_t$ and observed variables $Y_t$. Why do we always assume $p(Y_t|X_t)$? What happens if we have $p(Y_t|X_t, Y_{t-1})$? Is it because that wouldn't ...
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2answers
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What is the likelihood function of having heads 8 times out of 10 toss

Question What is the likelihood function for the event where 8 heads observed after 10 coin tossing? Is below in Python/Scipy using (scipy.stats.binom) correct? ...
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Why are the cut-offs used for Bayes factors and p-values so different?

I am trying to understand Bayes Factor (BF). I believe they are like likelihood ratio of 2 hypotheses. So if BF is 5, it means H1 is 5 times more likely than H0. And value of 3-10 indicates moderate ...
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How do Bayesian hierarchical models adaptively learn the prior?

It seems the main difference between a hierarchical and a non hierarchical model is that the hierarchical model learns the prior. That is it adaptively comes up with a regularizing prior to be applied ...
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Using different models within one bayesian optimization?

I'm using GPyOpt for Bayesian Optimization with Gaussian Proccesses. I have a dataset and want to know which of my models (LSTM, GRU, VANILLA RNN) works best. In Pytorch it is really simple to ...
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BNLearn: How to merge the estimating parameters of a Gaussian Bayesian network with its conditional structure?

I define the structure of a gaussian baesian network usind " iamb" function and then estimated the coeficients of the nodes using "bn.fit". ...
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Bayesian update of a confidence interval

How does one update a confidence interval using Bayes rule? Say, for example, an experiment shows that the mean lies in [A, B] with 95% confidence. Later, a colleague says they ran a similar ...