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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Confidence interval for Bayesian expected probability in categorical data

The context of the question is survey results analysis. I am focusing on categorical data : N respondents answer some questions and each question has $k_{q}$ choices. I want to compare what subgroups ...
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33 views

Probability of getting paid back over time

I'm trying to understand conditional probability and I thought of this problem, I'm not sure it might be related. I lent 500$ to a friend of mine, he told me that there are 20% chances that he gives ...
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measuring errors of bias, dispersion and outlier rate

I fit different models to a sample of data using Bayesian statistics. I have obtained for each data point in the sample a posterior probability distribution. Assuming I know the true answers for the ...
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13 views

Bayesian Variable Selection with NMIG

I have a Bayesian linear model like this: $Y_i = X_i*\beta + \epsilon_i$ . Just for completion: ($\epsilon_i \sim N(0,\sigma^2)$ $\beta \sim N_p(b_0,B_0)$, $\sigma^2 \sim Inv-Gamma (a,b)$) I would ...
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Why is the risk set convex, when we allow for randomized estimators

A randomized estimator $\delta^*(X)$ such that its loss function $L(\theta,\delta^*(x))=\int_\mathcal{D}L(\theta,a)\delta^*(x,a) \ da$, where $\delta^*(x, \cdot)$ is the estimator's density on the ...
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121 views

Is it always true that $P(X| Y \text{ OR } Z) \le P(X|Y)$?

Consider the following argument: If $(X| Y \ \text{OR} \ Z)$ is true, $(X| Y)$ must be true. For example, if $f(t)=10 $ when $ t=1 $ or $ \ t=0$ is true, then $f(t)=10 $ when $ t=1$ must be true ...
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1answer
17 views

Update gamma prior with new rate parameter instead of observations

From wikipedia: In Bayesian inference, the conjugate prior for the rate parameter $λ$ of the Poisson distribution is the gamma distribution. Let $$\lambda \sim \mathrm{Gamma}(\alpha, \beta)$...
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Bayesian Dirichlet equivalent (BDe), Bayesian Dirichlet equivalent uniform (BDeu) and Mutual Information Test (MIT)

To estimate structures of Bayesian networks, I am thinking about three score functions, BDe, BDeu and MIT. I have several questions. What are the differences between BDe and BDeu? Can I convert BDe ...
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57 views

Are we frequentists really just implicit/unwitting Bayesians?

For a given inference problem, we know that a Bayesian approach usually differ in both form and results from a fequentist approach. Frequentists (usually includes me) often point out that their ...
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1answer
14 views

Use of Metropolis-Hastings in Bayesian Inference

I am now studying the Metropolis-Hastings algorithm and I want to apply it in order to made a Bayesian Inference of a function $y=f(x)$ to a dataset $D=\{x_i,y_i\}$. Five parameters of the function ...
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8 views

Hierarchical regression model with just 2 populations

I have a dataset with the scores in Mathematics of students coming from 2 different schools. I'm trying to implement a hierarchical linear model to predict the student score given a set of covariates. ...
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A uniformly optimal statistical procedure? an exercise from The Bayesian Choice

The previous exercise is from the book 'The Bayesian Choice', page 87. What does the author mean by uniformly optimal stat. procedure? This exercise refers to a Decision theory chapter, in a section ...
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27 views

Using Bayes Rule to make inference about conditional probability using sample proportions

Suppose your target is to estimate $P(A|B)$, but it is impossible to do so directly. However, you have reasonable estimates of $P(B|A)$ $P(B)$ $P(A)$ from separate sources. Generally these are ...
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1answer
40 views

“Unable to resolve the following parameters:” jags error for Latent Class Model

I'm looking to fit a Bayesian latent class model in JAGS, but am running into an issue, which I'm seeking help to resolve. The model I'm trying to fit is described below (model details can be found ...
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10 views

Bayesian hierarchical multivariate linear model

Do you know any package in R performing Posterior inference for a Hierarchical multivariate Linear Model? Thanks
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14 views

Uninformative prior for a gamma distribution

I am seeking a vague prior for a Gamma distribution $G(\alpha, \beta)$, where both $\alpha$ and $\beta$ are unknown. My teacher suggest a prior $p(\alpha,\beta) = \exp\{-\beta\}$. However, it seems to ...
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2answers
199 views

Why computing P(x,D) is simpler than P(x|D) in exponential bayesian networks?

I am reading this tutorial on variational inference and wonder why the statement in the question title which is mentioned on page 3 is true.
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1answer
23 views

Multi parameter Metropolis-Hastings

I need to formulate a multi parameter Metropolis-Hastings algorithm. My question is related to how to define the condition to accept or not the candidate value. In my problem (it is a curve fitting)...
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23 views

Bayesian model and Jensen inequality [on hold]

We estimate a Bayesian model which has transforms in it $y \sim normal(\beta t(\theta, x), \sigma)$, where t() is a nonlinear transform, we then want to translate the many chains and iterations into a ...
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1answer
29 views

Making the model explicit in Bayes' rule

I'm reading a book about Bayesian statistics and at some stage it explain the Bayes' rule as follow: $$p(\theta|D) = \frac{p(D|\theta)\,p(\theta)}{p(D)}$$ Where $\theta$ is the model parameter and D ...
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2answers
70 views

How to determine a prior, if there is no relevant prior empirical information?

I am using Bayesian probability. In my case, I have an empirical prior probability to use in calculating the posterior probability. But this isn’t the subjective way of doing it, I believe. If I didn’...
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5answers
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Wikipedia entry on likelihood seems ambiguous

I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.) It starts from the Wikipedia page on likelihood. They say ...
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How to implement Bayesian Model Combination?

I'm interested in formal procedure mentioned in "Turning Bayesian Model Averaging Into Bayesian Model Combination" (Kristine Monteith 2011). I have a set of $N$ "best" AIC ranked models and I want to ...
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1answer
17 views

How to determine correct changepoints from Posterior Probabilities (bcp R package)?

I am using the bcp package in R to determine change points in a time series. The output that this package gives is a distribution of posterior probabilities. As far as I can understand, the peak ...
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1answer
20 views

Definition and calculation of the log pointwise predictive density

I want to calculate the log pointwise predictive density from an MCMC sample. Gelman et al (2014) define the lppd as $$ lppd = \sum_i^n \log \int( p(y_i| \theta) p_\text{post}(\theta) d\theta $$ My ...
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2answers
276 views

What does “mixing” mean in sampling?

I keep seeing this term "mixing": when people want to show their sampler works better, they say it "mixes" better. The term is a little counter-intuitive.
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1answer
59 views

Is correlation between parameters a problem when fitting a Bayesian model using MCMC?

Assuming some Bayesian model, for example: $$y \sim N(X\beta, \sigma)$$ where this model has: Response vector: $$ y = \pmatrix{y_{1} \\ y_{2} \\ \vdots \\ y_{n}} $$ Predictor matrix: $$ \...
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How to make shrinked forecast for the extreme value?

Let me use made-up example: John loves running. He decided to run in his local half-marathon for the first time in his life. He never measured exactly how fast he runs the distance, but while ...
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1answer
29 views

METROPOLIS-HASTINGS with likelihood

I am trying to set up a Metropolis-Hastings algorithm in Matlab in order to estimate the parameters ${\theta}$ (it is a vector of 5 elements) to fit a curve to a set of data $D={X_i,Y_i,\delta_i}$. $X$...
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6 views

Convergence of the iterative scheme while maximizing the model evidence

I understand all the derivations in Bishop 3.5 about evidence approximation, except the part where they ignore the term $m_N$, (for the time being), while differentiating w.r.t $\alpha$. This doesn't ...
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1answer
53 views

How to improve model by Bayesian Statistics/Inference? [closed]

I am puzzled about model improvement in implementing Bayesian statistics/inference. Normally, we will use a fixed model in bayesian statistics, e.g. normal distribution with parameter mean and sd. ...
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1answer
40 views

MCMC for Probit/Logit model with some 1's flipped to 0's

I would like help constructing a sampler for the following model, which is the latent variable interpretation of either logistic or probit glm (doesn't matter which one to me), with a small twist: ...
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16 views

Logistic regression in JAGS [migrated]

I'm new to Bayesian analysis. I have a hierarchical model with a binary response variable. There is only one predictor (categorical), which has 3 levels: HLL, LHL and LLL. I prepared my data file by ...
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1answer
40 views

Regret Minimization with Hidden Markov Processes

Consider a hidden Markov process with two states $\{0, 1\}$ represented with $Z_t$. The transition matrix is unknown, although we can assume it's strongly diagonal (i.e. slow-switching). At any time, ...
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2answers
24 views

Bayesian Model Comparison - Neutral Prior Information

Let's say I estimate two models, $M_{0}$ and $M_{1}$. The posterior odds ratio for for model $M_{0}$ against $M_{1}$ given the data, $y$, is, $\frac{Pr\left(M_{0}\mid y\right)}{Pr\left(M_{1}\mid y\...
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PyMC sampling is slow

I'm using pymc2 to estimate the parameters of a normal distribution. My data has shape 50000 x 6. Basically, I have 50K independent distributions and I want to obtain the parameters for each of them, ...
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4 views

How to correct for base rate when pooling different conditions?

My question is as follows: In each trial of my task there can either be a target or a distractor. For targets, the outcome can be true positive or false negative, whereas for the distractor the ...
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26 views

Partitions, and Random variable indexes in Dirichlet Process

I am going over this tutorial and am confused by the notations on pages 14 and 15. Here is my understanding for the notations on page 14: $G\sim DP(\alpha,G_0)$: Means $G$ is a draw from a DP, with ...
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How should I call the complement of a credible region?

Incredible interval/region? More explicitly, if I have a unimodal distribution with a 95% credible interval in [A,B], what would I call the complementary region ]A,B[? It is 100% credible region ...
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10 views

Upper bounds on mixing times in MCMC for bayesian analysis in practice

I'm familiar with how, for a general markov chain with some transition kernel, the spectral gap and the log-Sobolev constant both provide an upper bound on mixing time. I also have heard people ...
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1answer
25 views

Can a likelihood function be integrated to find the CDF and probabilities?

Likelihood analysis uses the likelihood function: $L(\Theta | data) = P(data | \Theta)$ to determine how likely it is that some value is the true population parameter ($\Theta$) compared to some ...
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1answer
26 views

Bayesian Probability - Hybrid approach to calculate components?

The task The following question is from a Basic Statistics course on Coursera: In a shop, people can take chewing gum from a dispenser on the right, or the left. The dispenser on the right has 7 ...
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2answers
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Probability rule involving conditional and marginal distributions

The context of my question is Baysian generative models. The text book I am reading states $p(\tilde{x} \, \vert \, D) = \int p(\tilde{x} \, \vert \, \theta) p(\theta \, \vert \, D) \, \mathrm{d}\...
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0answers
33 views

Posterior distribution under Cauchy prior?

I have a (I hope) simple question! If I had a linear regression, $Y_t = \alpha + \beta X_t + \epsilon_t$ with $\epsilon_t \sim N(0,\sigma^2)$ and I assume a Cauchy prior for $\sigma$, is it ...
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Bayesian Confirmatory Factor Analysis: goodness of fit indexes

I'm trying to fit a Bayesian Confirmatory Factor Analysis with R. R seems to have excellent packages like 'blavaan' and 'MCMCpackage' which enable users to fit bayesian CFA and I have fit some models. ...
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How do I statistically rephrase this question

I am analyzing a dataset containing observations from n number of attempts by players in a game. If I am building a regression model to predict the outcome of each attempt given 1 or more descriptors ...
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1answer
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Metropolis-Hastings Convergence in R

I'm trying to implement the algorithm Metropolis Hastings for the next FDP. \begin{equation} f_{X}(x)=(5+\exp(5/2))*\sqrt{2\pi}\left[\exp\left\{-\frac{(x-2)^{2}-2x}{2}\right\}+5\exp\left\{-\frac{x+2}{...
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Defining “p” value for glmm randomisation psuedo-p value

I am currently using a Bayesian approach to glmm (MCMCglmm in R) to estimate regression coefficients (see here where I ask whether randomisation of fixed effects is ...
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37 views

What are we optimizing in the hill climb algorithm for Bayesian? [closed]

With R you can create the hill climb algorithm. When and why are you using the hill climb algorithm in the Bayesian Model? Can it be used to check if there is dependencies between variables? What ...
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13 views

Threshold to segment count data following different distribution

I have a dataset D, wherein $ D = \{x_i, x_{i+1}, \ldots, x_n\} $. each data point in D is a discrete count data and since this is spatial data, one can expect dependence between contiguous data-...