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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Question on rules of conditional probability

This is a somewhat trivial question, but I can't find anything anywhere to tell me how to solve it. I have the values of the following: ...
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Question on how to apply Bayes Theorem realistically

So I'm new to Bayes Theorem and am trying to understand it. For simplicity I'll refer to the usual example of testing for cancer. ...
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How do I set a prior on regression coefficients given partial information over them?

Let's say I have a dataset about student performance in mathematics: together with the scores (response variable, I can approximate continuity) I have a set of covariates regarding each student. For ...
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1answer
12 views

How to fit a Pareto distribution via Bayesian estimation (with a Pareto prior)?

I don't know Bayesian statistics very well, so I don't know if the question makes sense. Let me give an example. We assume that the income distribution of a country is a Pareto distribution (the ...
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error with using runiregGibbs in R: not compatible with requested type [on hold]

I want to run a hierarchical Bayes regression model using this runiregGibbs function. My data is like the following: ...
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26 views

State space model affected from future events?

I understand that a state-space model is a common model where the current observation $y_t$ depends on the current state $x_t$. Is there any common model where the current observation $y_t$ depends ...
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1answer
29 views

What are the prerequisites to start learning Bayesian analysis?

Or in other words, How mature-I mean my statistical knowledge- should I be to start learning and doing Bayesian analysis?
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27 views

how to use sparse Dirichlet prior in Dirichlet-Multinomial models?

Consider a Dirichlet-Multinomial model, with a (symmetric) Dirichlet prior on a probability distribution $\theta$, where: $\theta \sim Dir(\alpha_1,...,\alpha_K)$ and $X \sim Mult(\theta)$. When we ...
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What is the difference between Bayesian seasonal adjustment and other types?

I need to learn this for a new task I have been allocated but it has been a few years since I studied maths! I have some books on time series and have read a few papers on it and I understand time ...
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15 views

Conditional statements in Winbugs

I am using Winbugs for bayesian estimation. I have a categorical data that consisted of the values:1,2,3. Here is what I have: ...
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17 views

markov blanket for gibbs sampling in graphical model

below is a bayes net where the nodes are discrete. I want to Gibbs sample each of the $S_t$ nodes. for example, to Gibbs sample $S_1$ conditioned on rest of variables, it should be sufficient to ...
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26 views

Improving the results coming from an image recognition API

We are developing a software application that will automatically suggest tags (keywords) for images that are being uploaded into a database of already-tagged (by a human) images. We are using a 3rd ...
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1answer
38 views

Bespoke MCMC priors & likelihoods, & feeding a posterior joint pdf back in as the prior next time

We're looking at PyStan, PyMC3 and emcee. (switching to R could also be an option, if need be). We have a lot of bespoke priors and bespoke likelihood functions: they are bespoke in the sense that ...
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52 views

Which regression model should I build?

I'm trying to handle a dataset about student performance in 2 Portuguese Schools in the subject of Portuguese. Student grades go from 0 to 20, discrete. I have a set of 30 Regressors (all but 3 ...
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14 views

How do I compute the posterior distribution of the success probability with uniform priors?

This is a question from a homework assignment A clinical trial is conducted to compare the effectiveness of three drugs. 100 patients are randomly assigned to each drug (300 total patients), and Y_1 ...
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1answer
49 views

Deriving the Ridge Regression $\boldsymbol{\beta}\mid \mathbf{y}$ distribution

Apparently the estimate $\hat{\boldsymbol{\beta}}$ for ridge regression comes up as the mean or mode of the posterior distribution given by $f_{\boldsymbol{\beta}\mid \mathbf{y}}$. This is the ...
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1answer
23 views

What does the error “pre.period must span at least 3 time points” in the CausalImpact R package mean?

I've been encountering the error "pre.period must span at least 3 time points" when using the package. Can someone help me understand why the package requires me to have at least 3 time points and ...
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60 views

Do you have to adhere to the likelihood principle to be a Bayesian?

This question is spurred from the question: When (if ever) is a frequentist approach substantively better than a Bayesian? As I posted in my solution to that question, in my opinion, if you are a ...
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The likelihood of response variables in variational Bayesian probit regression

I read the paper Explaining Variational Approximations (J.T. Ormerod & M.P. Wand) and there is a part where they explain variational probit regression with auxiliary variable since the posterior ...
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2answers
51 views

priors for Gamma shape and scale parameters

I have a random variable $X$ that is Gamma distributed with unknown parameters $\alpha$ and $\beta$: $$ X\sim \text{Gamma}(\alpha, \beta) $$ I now want to estimate $\alpha$ and $\beta$ from samples ...
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28 views

Ones trick in BUGS gives node inconsistent with parents error [closed]

Edit: This issue doesn't come up if I use OpenBUGS. But I can't use it for my bigger problem as it seems "very slow" compared to JAGS at least on my machine. I am using JAGS as my BUGS flavor to run ...
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practical implementation detail of Bayesian Optimization

I'm giving Bayesian Optimization a go, following Snoek, Larochelle, and Adams [http://arxiv.org/pdf/1206.2944.pdf], using GPML [http://www.gaussianprocess.org/gpml/code/matlab/doc/]. I've implemented ...
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Advantages of Particle Swarm Optimization over Bayesian Optimization for hyperparameter tuning?

There's substantial contemporary research on Bayesian Optimization (1) for tuning ML hyperparameters. The driving motivation here is that a minimal number of data points are required to make informed ...
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11answers
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When (if ever) is a frequentist approach substantively better than a Bayesian?

Background: I do not have an formal training in Bayesian statistics (though I am very interested in learning more), but I know enough--I think--to get the gist of why many feel as though they are ...
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1answer
31 views

Relation Between Bayesian Estimation and Maximum a posteriori estimation

Is maximum a posteriori estimation some kind of Bayesian Estimation? If yes, can you point out other Bayesian estimators? Edit: So I've come to know the following (don't know if they are correct): ...
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1answer
35 views

How to combine two measurements of the same quantity with different confidences in order to obtain a single value and confidence

Back in the lab at university, we were taught to measure the quantity of interest some number of times (call this N), and then calculate the standard error. The underlying assumption here is that you ...
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29 views

Find Bayes rule/action under given prior

I am able to solve for Bayes actions/rules with no data and am able to follow problems with simple data. However, I'm not sure how to solve a question where the data, $X$, is conditional on the state ...
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1answer
35 views

Probability the next draw from a distribution is greater than some number given a previous draw

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think ...
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3answers
56 views

Bayesian inconsistency

I have a small knowledge of Bayesian analysis which I want to apply to invert some instrumentation data which has a complex nonlinear response. However this simple example confused me so before I go ...
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1answer
41 views

How to choose t-distribution degrees of freedom in “robust” Bayesian linear models

It is well known that in both frequentist and Bayesian linear models, outliers can greatly influence the parameter estimates. Consider the simple example where one outcome variable, $y$, is predicted ...
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1answer
25 views

Levels of “hyperparameterization” in Hierarchical Modeling

Suppose we have observations $y$ that we wish to model as having being randomly sampled from a distribution with parameter $\theta$. General Bayesian approach assumes a prior distribution over ...
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1answer
27 views

Multivariate posterior probability

This is a 2-dimensional pattern recognition system that I am working on. It is known that the distribution between the two classes are $1/2$ and $1/2$ respectively for class $\omega_1$ and class ...
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47 views

Cross validation or EM for selecting strength of the prior?

Often when I'm looking at bayesian analyses, the influence of the prior is chosen via cross validation. For example, suppose $X$ and $Y$ represent some real valued data that I want perform a bayesian ...
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Full conditionals for the parameters of a Bayesian regression with dependent components

Let $\mathbf{y}_i=\{y_{ij}\}_{j=1}^p$, $i=1,\dots ,n $ be a $p-variate$ vector and $$ y_{i,j} = \alpha_{j}+\beta_{j}x_{ij}+\epsilon_{ij}, $$ where $x_{ij}$s are known constants and ...
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1answer
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With complete data and a factored prior, the posterior also factors

In the second paragraph of Section 11.3 in Machine Learning A Probabilistic Perspective, the author concisely summarizes Section 10.4.2 by saying that for the standard bayesian model ...
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1answer
65 views

How to determine posterior distribution of the parameter in a binomial

Assuming that I performed n iid tests, and the total number of test is n which is a fixed value, and the observaton of 1 which corresponding to successful results is X observations yeild with ...
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Why does the sim function in Gelman's arm package simulate sigma from inverse chi square?

In getMethod(arm::sim, "lm"), the source code shows that $\sigma$ is simulated from inverse chi square: ...
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20 views

Is there any way to convert from a posterior probability to p-value, or the opposite?

I have results of a study from associations of a variant with a phenotype in the form of posterior probabilities but I was wondering if there is any way to convert these to p-values, even making ...
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1answer
19 views

How do I show that the mean of the posterior density minimizes this squared error loss function?

This exercise comes from Koop's Bayesian Econometrics. Given $\theta$, the parameter(s) of a model (in this case $\theta$ is a scalar), $\tilde{\theta}$, the point estimate of $\theta$, and constants ...
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23 views

Bayes factor from posterior odds

I tried to answer my own question Comparing two Bayesian models under disjoint prior supports using MCMC. Here is my intent. I am not confident in what I wrote so prefer to post it as a question : Is ...
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+50

Is Independent jeffreys prior different from independent reference prior?

I have a model involving two scalar parameters $\theta_1$ and $\theta_2$ and derive the Jeffreys prior for $\theta_1$ and $\theta_2$ independently (so for, e.g. $\pi(\theta_1)$, setting in the ...
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1answer
46 views

Can BIC be Used for Hypothesis Testing

Define the Bayesian information criterion as $$ \mathrm{BIC} = {-2 \cdot \ln{\hat L} + k \cdot (\ln(n) - \ln(2 \pi))} $$ (I do not drop the constant, $ - \ln(2 \pi)$, to avoid issues when equating to ...
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1answer
58 views

Calculating the probability a predicted point is 0

I have a deterministic function $f(x)$ and have evaluated some points $x_1,...,x_n$. So essentially I have pairs of data $(x_1,f(x_1)),...,(x_n,f(x_n))$. I am modeling the function $f(x)$ using a ...
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38 views

model fitting of data to multiple distributions

I have a set of numbers $ X = \{x_1, x_2,\ldots,x_n\}$ and I am interested in finding the most fitting combination of these numbers to multiple exponential distributions. Using predefined rules, I ...
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2answers
54 views

Monte Carlo Simulation of Complex Dynamical System

Assume that $\vec{z}(t)$, the state at time $t$ of a particle in a two-dimensional space, can be fully described by its position and velocity: $\vec{z}(t) = [r_x(t)\ r_y(t)\ v_x(t)\ v_y(t)]$. ...
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1answer
271 views

Concrete examples of a frequentist approach that is superior to a Bayesian one [closed]

Can you help me understand the frequentist point of view in the bayesian vs frequentist debate? I have read a lot and all the sources I found are either filled with complex equations or written from a ...
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1answer
41 views

Aside from the exponential family, where else can conjugate priors come from?

Do all conjugate priors have to come from the exponential family? If not, what other families are known to have/produce conjugate priors?
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What should an uninformative prior be for the slope when doing linear regression?

When performing bayesian linear regression, one needs to assign a prior for the slope $a$ and intercept $b$. Since $b$ is a location parameter it makes sense to assign an uniform prior; however, it ...
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Bayesian with multiple variables

I have a set of transitions from type of points A, B, C and D. For example A -> B -> C. And I am trying to predict next location by knowing two previous ...
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1answer
86 views

Does this Monte Carlo Technique Have a Name?

I sketched this algorithm out the other night. I am sure it has a name, I just do not know what it is yet. It would be helpful if someone could point me in the right direction for research. I ...