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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How to do bayesian model comparison for control and treatment

I have a model for a biological experiment which has a typical bayesian structure, that is $\lambda \rightarrow X$. Now let's assume for control the parameters are $\lambda_1 \rightarrow X_1$ and $\...
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Escobar and West Sampler for Dirichlet Process Parameters

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process. The issue I have is at the end of their section 6, ...
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27 views

Prove Causation [on hold]

Could anyone share me some good material to start to read How one can prove causation or the steps to prove it using statistical test via code e.g python or R? Thanks in advanced.
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10 views

Bayesian mixed ANOVA

I am analysing temperature series measured by three different sensors. I am interested to test if the temperature data series are credibly different. This is a sketch of my sample design: My ...
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37 views

Time Series Data Mining and Correlation guidance [on hold]

Introduction Editing this post to clarify what I am asking. This is a bit of a vague question to begin with. I am not asking if this particular statistics method with these variables will cause this ...
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Sampling concentration parameter of DP via Slice sampling?

Is there a published work which shows how sampling the Dirichlet Process's concentration parameter can be done via Slice sampling?
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Calculating the probability of a successful marriage lasting N years

In reading the US HHS report on marriage trends, I was interested in calculating the probability that a marriage for a specific person with different qualities (male/female, education, race, age) ...
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10 views

What is the junction tree of the following graph?

I got so confused in the remaining steps (from triangulated graph to junction tree) after turning my Bayesian network to a triangulated graph. Would anyone please help?
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17 views

Matrix Calculations and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
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8 views

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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37 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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19 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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127 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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22 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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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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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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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
43 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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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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15 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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207 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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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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Bayesian model and Jensen inequality [closed]

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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31 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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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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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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40 views

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
18 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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21 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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278 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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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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55 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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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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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
45 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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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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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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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 ...