Bayesian inference is a method of statistical inference that relies on turning the model parameters into random variables and applying Bayes' theorem to deduce probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Are all generative Models based on Bayes?

Reading about deep learning I encounter various different kinds of hierarchical networks, many of which are generative. 1) Are all of the generative networks based on Bayes? 2) If not, how do they ...
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RJAGS bayesian approach of mixed effects model

Why my posterior result always shows that the sigma and sigma.c estimates to be around 50? It should not be that large as I know from another approach of analysis and also summary of the data. Is it ...
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Marginal likelihood: why integration is used

From wiki: Given a set of independent identically distributed data points $\mathbb{X}=(x_1,\ldots,x_n)$, where $x_i \sim p(x_i|\theta)$ according to some probability distribution parameterized ...
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+100

Verifying and/or changing priors assumptions on Bayesian ANOVA

I am performing a Bayesian analysis of around 1500 data, divided into 2 factors, one that I am interested x1, and the id-variable for the paired/within-subject x2. x1 has 15 levels, and x2 around 100 ...
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choosing prior parameters for variational mixture of Gaussians

I am implementing a vanilla variational mixture of multivariate Gaussians, as per Chapter 10 of Pattern Recognition and Machine Learning (Bishop, 2007). The Bayesian approach requires to specify ...
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44 views

Power of Uniform Distribution?

In the Bayesian analysis, $\mathtt{rjags}$ in particular, it is very frequent to see the code: sigma ~ dunif(0, 100) sigma.1 <- pow(sigma, -2) But, what does ...
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Estimate polynomial model [on hold]

For a polynomial regression of temperature data I want to estimate a polynomial model of order 8. It is asked for a suitable prior that mitigates the problem of 1) overfitting the data 2) higher ...
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Bayesian question [on hold]

A team of scientists have found that experimental mice sometimes (1 in every 10,000 mice) spontaneously show signs of a disease (that looks very similar to a human disease). The diagnostic sign is a ...
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Point null hypothesis in Bayesian statistics

Let $X\sim N(\theta,1)$ and consider 5 independent observations $X=(4.9,5.6,5.1,4.6,3.6)$. The prior probability that $\theta=4.01$ is $0.5$. The remain values of $\theta$ are given a prior ...
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How to calculate the perplexity of the twitter-LDA?

I am trying to evaluate different topic models. When it comes to twitter-LDA, I am not sure how to evaluate the perplexity. I regard the background assignment as another topics. Thus there are totally ...
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Bayesian linear regression - Defining hyperparameters

I am currently working on a linear regression task from a textbook and I have some questions. In the following you will find the task as well as some thoughts of mine to possible results. Short ...
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First order model vs n-order models

Plenty of different research models showed that n-order models give better results than first order models. For example, for location this is work that shows this ...
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Computation of the marginal likelihood from MCMC samples

This is a recurring question (see this post, this post and this post), but I have a different spin. Suppose I have a bunch of samples from a generic MCMC sampler. For each sample $\theta$, I know the ...
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Bayesian treatment of outliers

In a supervised learning problem, I have a training dataset $D$ comprised of samples $x$ and their corresponding labels $\omega$. From this data, I attempt to learn the true distributions ...
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34 views

How to combine two beta-binomial distributions

Say I have the following situation. I have two weighted coins: Coin 1: In the past I've seen this coin flipped 10 times, 8 of which it came up heads. So I can model the probability of $n$ heads out ...
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3answers
63 views

“Unidentified” hierarchical model in brms/stan - where to go from here?

I am evaluating an intervention in which participants are grouped in teams and each participant fills in a survey before and after the intervention. As such, the data presents a classic multilevel ...
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64 views

Find the posterior distribution of $\pi$

An observation $x$ is taken from a negative binomial distribution $X \sim \text{Negative-Binomial}(k,\pi)$. The parameter, $\pi$, is allocated a beta prior $\pi \sim (\alpha,\beta)$. My attempt: ...
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Bayes factor and hypothesis test in Bayesian inference

Let $$\pi_0=P(\theta\in\Theta_0)=\int_{\Theta_0}\pi(\theta)d\theta$$ $$\pi_1=P(\theta\in\Theta_1)=\int_{\Theta_1}\pi(\theta)d\theta$$ $$a_0=P(\theta\in \Theta_0|x)$$ $$a_1=P(\theta\in \Theta_1|x)$$ ...
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Predictions from BSTS model (in R) are failing completely

After reading this blog post about Bayesian structural time series models, I wanted to look at implementing this in the context of a problem I'd previously used ARIMA for. I have some data with some ...
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Comparing results from reference coding and orthogonal coding in a linear model?

The problem: I'm trying to fit a zero-inflated negative binomial model to count data (catches of larval fish). I have three factors, and an offset variable, which is the volume of water filtered by ...
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Bayesian Optimization for a Stochastic Target that changes over time

Let's say there is a single slot machine that: costs zero to play can only be played once per day has a payout that is conditionally normal and is a function of the date and time. I want to use ...
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34 views

How to do calculate both causal and diagnostic inferences simultaneosly in bayesian networks?

Consider a simple Bayesian network as given below. Question: How to find $P(S|C,W)$? It is fairly straight forward to compute the causal inference $ P(W|S) = P(W|S,R)\cdot P(R) + ...
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What's an intuitive explanation for why MAP is variant under parameterization?

I understand why MAP is variant under parameterization mathematically, but I don't really understand it intuitively. To help me out, my professor gave me an example where reparameterizing MAP ...
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Categorical mixture model in PyMC2

I am currently trying to implement a simple categorical mixture model in PyMC2. However, I am not able to get it to run after trying some possible solutions. Here is my current attempt: ...
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Marginal prior $p(\mu)$ of mean of a normal distribution when both mean and variance are unknown

I read that if the data is normally distributed with mean $\mu$ and variance $\sigma^2$ (both unknown) then to have the joint posterior distribution $p(\mu, \sigma^2 | y)$ in closed form, one has to ...
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Hierarchical Bayesian model - issues with JAGS/BUGS switching between lognormal and normal

I'm trying to construct a hierarchical model using JAGS, but I'm running into issues converting between normal/lognormal distributions and the more I stare at my problem, the more confused I get. ...
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Bayesian model comparison in high school

I teach physics to high-school students, and I would like my students to conduct a rudimentary Bayesian model comparison for data from their experiments. I figured out a way for them to do so (see ...
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Evaluation of Bayes GARCH estimation

I am using the bayesGARCH package to estimate Bayesian GARCH models and I was wondering how to evaluate them in terms of precision of forecast or at least the quality of the model. I have encountered ...
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Classification on sequential data

Context: I am working on a classification project where I recommend items to customers based on their past purchase history. Question: How will "time leakage" affect training? Example: Let's say ...
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What is the meaning of admissibility within a class, that every decision rule in a class is admissible in that class?

Suppose that I have that $X$ is a Poisson random variable with mean $\lambda$. Suppose a decision rule is to estimate $\lambda$ by using $\delta(Y) = aY$. Now, let $K$ be the class of all decision ...
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HMM prior on stationary probability

I am trying to model a sensor that when mis-calibrated tends to vibrate alot (or atleast high varying readings). I used a HMM to model these vibrations. It is known that the sensor was calibrated ...
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45 views

Bayesian Time Series model in R

Similar to the scenario described in this paper, I need to forecast a seasonal time series with only a few periods. I am working with about 2 years of daily revenue data, and I want to forecast the ...
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Question on Bayesian Curve fitting (Pattern Recognition and Machine Learning by Bishop) [closed]

Let $y = \displaystyle\sum_{i=0}^Nw_ix^i$ be a polynomial fit curve. In this question, we are looking at this curve from a probabilistic perspective, as Bishop says, towards a full Bayesian treatment. ...
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Why is the posterior distribution in Bayesian Inference often intractable?

I have a problem understanding why Bayesian Inference leads to intractable problems. The problem is often explained like this: What I don't understand is why this integral has to be evaluated in ...
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Beta approaching Binomial

If we have a Beta likelihood and a binomial prior, we get a beta posterior. Can someone please explain why this approaches a binomial as $n\rightarrow\infty$. I plotted it and this appears to be the ...
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Calculation of hierarchical linear model with Wishart prior

Calculate a hierarchical random coefficients model with betas[0:3] distributed normally with the priors mu_beta ~ Normal, ...
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Bayesian Priors Update: Difference in Mean detection

Suppose I have measures of the life span of mice. I know the true expectancy in the beginning of the experiment - 1000 of days and true variance. At some (unknown) point mice begun to be fed by a new ...
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When is the probability of a variable equivalent to a function of the variable i.e. when does p(x)=f(x)?

What allows us to conclude that that p(z)= h(z) as shown in the yellow highlights in the below solution? If p(z) didn't equal h(z) then proportionality would still fail to show conditional ...
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If I have a nested multi-level model, how can I find the conditional expectation easily of the middle variable?

Suppose I have the following model: $$ y_i | x_i, V_1 \stackrel{ind}\sim N(x_i, V_2) $$ $$ x_i| V_1 \stackrel{iid}\sim N(0, V_1) $$ $$ V_1 \sim Unif(-V_2, \infty) $$ where the data is $y = (y_1, ...
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Can I use the Bhattacharyya distance as an acceptance criterion for Approximate Bayesian Computation?

I am researching the spread of a disease through a population and want to capture the behavior of this disease with a model. I already have a model and patient data. The data is a value per patient ...
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2answers
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Understanding Wishart Definition

I'm trying to understand the definition of the wishart distribution. In wiki, $X_{(i)}{=}(x_i^1,\dots,x_i^p)\sim N_p(0,V).$ What do they mean by this? Each component is drawn from a univariate ...
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Bayesian minimum mean square error estimator

In Fundamentals of Statistical Signal Processing, Estimation Theory, by Steven M. Kay the author shows on p. 312-313 that the estimator $p(A\mid x)$ minimises the Bayesian mean square error when you ...
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Sampling of a Gaussian posterior

Sorry for my simple doubt, but I'm quite newbie and don't clearly get how to sample the posterior Bayesian distribution. My likelihood and prior are normal and I know how to calculate the posterior. ...
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Implication of using independent priors for means of joint normally distributed random variables

I am using Bayesian methodology to estimate parameters of joint distribution(Multivariate normal) of random variables $(y_1, y_2) \sim N(\mu, \Sigma)$. I implemented the code for finding the posterior ...
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Fit a straight line in 3D with 3D uncertainties

So I'm trying to extend the recipe given here in chapter 7 to 3 dimensions. I have x,y,z data points each with their own uncertainties and I'm trying to fit a straight line. So I'm extending the the ...
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Understanding default Bayesian posterior probability shape parameters in multi-armed bandit

Multi-Armed Bandit indicates a best "arm" based on a simple test of proportions and Bayesian posterior probability shape parameters, alpha and beta (some implementations may take a different approach, ...
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Calculate log-odds posterior distribution

Given a gamma-posterior distribution $p(\theta|y)$ I want to compute the posterior distribution for the log-odds: $log\frac{\theta}{1-\theta}$ I tried to solve it with the change of variables ...
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Naive question about Bayesian multivariate logistic regression

I am stuck on what's probably a trivial question. I am reading a paper about multivariate logistic regression, and they say: ...
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Integrating to get the posterior distribution for the precision

Our model is the following: $x_i|\mu,\tau \sim N(\mu,\tau)$, where tau is the precision. $\mu|\tau \sim N(\mu_0,n_0\tau)$, $\tau\sim Ga(\alpha,\beta)$. In the picture below, how is the ...
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Calculate posterior distribution (gamma-prior, poisson-likelihood)

I want to calculate the posterior distribution given a gamma-prior and a poisson likelihood. The task is from a textbook and I just have the solutions (without a walkthrough). Please find all given ...