Bayesian inference is a method of statistical inference which uses Bayes' theorem to find probability estimates of parameters or hypotheses.

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tutorial on sampling methods and MC

I'm looking for good tutorials that cover the various sampling methods: simple sampling, MCMC, Gibbs Sampling, and Metropolis Hastings Algorithm. I barely know what is an MCMC. I would like to learn ...
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Bayesian modeling with multivariate normal

Suppose you have an explanatory variable ${\bf{X}} = \left(X(s_{1}),\ldots,X(s_{n})\right)$ where $s$ represents a given coordinate. You also have a response variable ${\bf{Y}} = ...
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Is my Bayesian analysis correct?

This is my first time doing a Bayesian analysis, so I'm not sure whether what I did makes perfect sense. I'm trying to tell if two samples come from the same distribution, more specifically, if they ...
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15 views

Are pooled results from multiple imputation equivalent to a posterior mean?

I am fairly new to multiple imputation and trying to be sure I understand the approach. Say I have a data set with missing values, so I create 5 imputed data sets using multiple imputation by ...
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Bayesian Updating Process, 3 signals and 2 states of the world

Suppose Nature chooses a state $\omega = \{X,Y\}$ at $t=0$. Long-lived agents observe a signal $s_t$ at every period $t$, where $s_t = \{ x,y,z \}$. Agents all hold a common prior $\mu_0 \in (0,1)$ ...
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24 views

MCMC-Draws from different MCMC- chains

I have questions about the usage of draws from a MCMC. I estimate a hierarchical bayesian Multinomial Logit model (using bayesm in R). I am interested in the ratio of two coefficients 1 and 2, say ...
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java.lang.OutOfMemoryError using bartMachine package in R [migrated]

I ran a BART model with 11000 samples and 20 features(half of them are categorical variable). My mac has 8G ram. At first, I set memory to 5000 MB via function set_bart_machine_memory(5000). Then I ...
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Sequential weighted sampling

I need to figure out the total path (A to Z) followed by an agent through a squared-element grid. Each grid element has a probability density function $\Delta$ assigned to it that represents the ...
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52 views

P values depend on researchers' sampling intentions

John Kruschke has written widely on the misleading information provided by p values. I understand nearly everything he says, although there is one aspect I am a little confused about. Kruschke ...
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27 views

Posterior parameter distribution

I am considering the following non-linear state space model: $X_t=\frac{X_{t-1}}{2}+25\frac{X_{t-1}}{1+X_{t-1}^2}+8\cos{1.2t}+\epsilon_t, \epsilon_t\sim N(0,\sigma_x^2 ) $ ...
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Predictive posterior distribution with multivariate normal distribution

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = ...
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1answer
36 views

Simulating from a normal with “unknown” variance

Suppose I want to performing sampling from a normal distribution with an unspecified variance, and I want a way to sample so that I am in some sense "averaging out the possible values of the ...
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15 views

Posterior Predictive Checks

I understand what the posterior predictive distribution is, and I have been reading about posterior predictive checks, although it isn't clear to me what it does yet. What exactly is the posterior ...
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1answer
22 views

Empirical Bayes vs “non-informative” priors

I am familiar with the mechanics with both methods, but don't know what factors I should consider when choosing between these two approaches for adjusting a prior. I would imagine that, on a case by ...
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2answers
58 views

How does one interpret the distribution over parameters in bayesian estimation?

I am new to Bayesian estimation. The assumption that the parameters are random variables seems a little unsettling to me. For example when considering a model for data, what physical interpretation ...
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81 views

What is the mathematical difference between using a un-informative prior and a frequentist approach?

Un-informative priors are preferred in instances where bias is not acceptable (ie. courtrooms, etc.) However, it seems to me that it would just make sense to use a frequentist approach instead. Why ...
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121 views

MCMC packages in R

Is there an R package for MCMC that can accept my self-defined (log)likelihood function (can be done in MCMCpack) and lets the user define contraints to the ...
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1answer
38 views

PYMC Confusion: are observed nodes fixed or stochastic?

I've been trying to gain a better understanding of factor potentials in PYMC. In reading this article by Cam Davidson-Pilon on Yhat, I got confused about how observed nodes are understood by PYMC. ...
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20 views

Bayesian Spatial Prediction: What is f?

I'm working my way through Gelfand, A.E. & Li Zhu, B.P.C. (2001). On the change of support problem in spatio-temporal data. Biostatistics, 2:1, 31-45. I'm stuck at: This is absolutely the ...
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96 views

Deriving the posterior density for a lognormal likelihood and Jeffreys's prior

The likelihood function of a lognormal distribution is: $f(x; \mu, \sigma) \propto \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right ) $ and Jeffreys's ...
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55 views

Is rstan or my grid approximation incorrect: deciding between conflicting quantile estimates in Bayesian inference

I have a model to achieve Bayesian estimates the population size $N$ and probability of detection $\theta$ in a binomial distribution solely based on the observed number of observed objects $y$: $$ ...
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65 views

Bayesian inference of a clinical trial for clinicians

I am a clinician who is more adept than average at interpreting clinical trials in a frequentist manner. At this point, interpreting a trial as a frequentist has kind of become a procedure: check ...
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73 views

Inferring prior distribution

Suppose that we take a sample ($X_1, X_2, ... X_n$) from a distribution where we assume that $X_i $~$ Bin(n_i, p_i)$ and $n_i$ is known for every $i$. We also assume that $p_i$'s are independent and ...
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E-mail answered probability after n days of waiting for a reply - based on a sample of e-mails and replies

Here is the task: I have a sample of replies to my e-mail from my mail box. A sample is taken over a period of 90 days, 1000 e-mails and replies if any. (We only consider a pair of {my original ...
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27 views

multiple imputation for categorical data question

How do i design the use of Multiple Imputation based on Bayesian Inference when I am dealing with categorical data and my dataset does not contain complete prior observations at every combination ? ...
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MCMC of a mixture and the label switching problem

I generated some data according to a mixture of two lognormals: $f(x) = p \cdot \mathcal LN(\mu_1, \sigma) + (1-p) \cdot \mathcal LN (\mu_2, \sigma)$. given $p$ and $\sigma$, this is my code to find ...
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Bayesian estimation of $N$ of a binomial distribution

This question is a technical follow-up of this question. I have trouble understanding and replicating the model presented in Raftery (1988): Inference for the binomial $N$ parameter: a hierarchical ...
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94 views

Estimating $n$ and $p$ for Binomial distribution, repeated counting of partly hidden population

A brief motivation: $n$ critters live in an aquarium, where sadly they often hide in, under or behind things. When the aquarium is observed, each critter is only seen with probability $p$ ...
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103 views

what to chose for prior and proposal function for MCMC of a mixture

I generated some data according to a mixture of two lognormals: $f(x) = p \cdot \mathcal LN(\mu_1, \sigma) + (1-p) \cdot \mathcal LN (\mu_2, \sigma)$ Now I want to use MCMC to find the parameters $p, ...
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23 views

How can we combine learnings from multiple experiments in a single causal model?

I would like to use a causal network modelling to model the interaction of several variables and the effects of interventions. I have measurements for all priors of the model, that is without any ...
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52 views

Posterior distribution of proportional difference of two binomial variables

Can somebody point me in the right direction for a treatment of the following problem? I imagine this should be a fairly common problem in medical statistics... Given two binomial random variables ...
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Fisher information metric for hierarchical Bayesian model is negative-definite?

I'm strungling at the computation of Fisher information matrix for the hierarchical Bayesian model. For simplicity, consider theta following hierarchical Bayesian model: $$X|\sigma \sim ...
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Posterior predictive checking in Hierarchical logistic model

I'm fitting a Bayesian Hierarchical logistic model with a random intercept, having data at two levels: subjects and hospital, ...
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1answer
112 views

Do Bayesians believe in Fixed Effect Models? [duplicate]

Given the Bayesian paradigm, is there actually such a thing as a fixed effects model since Bayesians treat all parameters as random variables?
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Finding the most “uniform” or “least concentrated” density function, subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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SVM versus Bayesian regression example(s)?

I am trying to track down examples where some basic problems have been tackled via both classical Machine Learning algorithms and more formal statistical methods. In particular, I'm interested in ...
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Evaluating the $H_0$ of a contigency table

I have a 2 way contigency table with variables $A=\{a_1,a_2\}$ and $B=\{b_1,b_2\}$. I have the observed cell frequencies $O_1=a_1b_1$, $O_2=a_1b_2$, $O_3=a_2b_1$ and $O_4=a_2b_2$. My null hypothesis ...
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General theoretical properties of empirical Bayes estimates

I was wondering if someone could provide reference (if such exists) for the theoretical properties of empirical Bayes(EB) point estimates, in the sense of what can we say about their risk under ...
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Estimating parameters and latent variables in a split Poisson distribution using R and JAGS

I am trying to estimate parameters and latent variables in a split Poisson model that describes observable and unobservable counts in time assuming the split probability is $\pi$. An observable event ...
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Understanding a measure of convergence of MCMC simulations

I am trying to better understand better the Gelman/Rubin measure of convergence of MCMCs. The method starts off by defining two quantities: $B$ and $W$. $B$ is said to be the between chain variance ...
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In Bayesian hypothesis testing, do the prior model probabilities have to be equal?

The posterior odds is the product of the Bayes factor and prior odds: $\frac{p(M_1|data)}{p(M_2|data)}=\frac{p(data|M_1)}{p(data|M_2)}\times\frac{p(M_1)}{p(M_2)}$. I was under the impression that ...
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1answer
41 views

Bayes' factor vs. Bayes' Discriminant Rule

When we are comparing two models against some data, will we obtain the same (set of) posterior odds for the models both when we use the Bayes' factor and when we use the discriminant rule? If not, ...
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HPD interval for the mean

Suppose we have iid observation with the following model $ Y_t \sim \mathcal{N}(\mu,1/\mu) , t=1,2,..T$ The question is " Assuming a flat prior on $(0 ,\infty )$ find a 95% HPD interval for $\mu"$ ...
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62 views

about the definition of bayesian network

In this PDF http://people.csail.mit.edu/yks/documents/classes/mlbook/pdf/chapter2.pdf page 5 says: Given a set of functions $f(x_i,pa(x_i))$ non-negative and sum to 1, we define a joint probability ...
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Bayesian fitting - multiplying two probabilities with differing orders of magnitude

I am fitting a model to data using Bayesian inference. This is my first time of using this method. My posterior is $P = P_{prior} + P_{photometric} + P_{spectroscopic}$. Value of $P$ is negative and ...
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PyMC – Calculate Evidence (Bayes' Theorem denominator)

I have one simple model and one more complex model (both are PDEs that describe displacement, when different kind of forces and thermal effects take place). From those, given some data, I calculate ...
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multivariate dirichlet for multiple imputation

I dealing with 3 covariates {x1, x2, x3} all three are discrete and contain missing data. ...
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How to update probabilities where datasets and analysis methods differ

Any help would be appreciated, as I'm not sure how to proceed. I have a dataset with binary data and have been able to fit a logistic regression model to it (presence of disease, for example, as a ...
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1answer
131 views

How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
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Problem with Finding Likelihood: Bayesian

I am really unfamiliar with Bayesian methods particularly parameter estimation. Suppose I have a test to find a parameter, theta which is the number of packaged bag for retail sale that could contain ...