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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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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23 views

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

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 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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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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14 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?
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32 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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24 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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5 views

Node dimension does not match in WinBUGS [on hold]

I'm using WinBUGS14, though I am not adept at it. I have worked through several problems before, however, this error seems to be tripping me up. Node dimension does not match This EXACT code has ...
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1answer
34 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
41 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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36 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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24 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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22 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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42 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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13 views

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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53 views

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
61 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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20 views

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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19 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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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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19 views

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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44 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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56 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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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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193 views

Concrete examples when frequentist approach is superior to the 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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40 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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142 views

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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12 views

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
81 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 ...
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26 views

When is it appropriate to use linear regression vs. Naive Bayes?

Based on my readings, it appears as though linear regression lends itself to cases where both X and Y are numerical and you have a large sample size, whereas Bayes is better for categorical variables ...
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Calculating elasticities from spatialprobit Bayesian coefficients

I have run a model in R using the spatialprobit package and would like to calculate elasticities with respect to some of my coefficients of interest. I am a bit ...
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28 views

Why is Bayes Classifier the ideal classifier?

It is considered the ideal case in which the probability structure underlying the categories is known perfectly. Why is that with Bayes classifier we achieve the best performance that can be achieved ...
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34 views

Bayesian time series - more weight on recent observations?

Note: I'm a stats novice, please let me know if any of these terms are unclear or misused and I'll update the question! I'd like to predict future values of a time series. More precisely, I'm ...
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37 views

Bayesian GARCH: forecasting volatility and returns

I am interested in forecasting conditional volatility, $h_t$, and returns, $y_t$, in a Bayesian GARCH framework. I am using the bayesGARCH package by Ardia in R ...
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21 views

Is it possible to find the distribution parameters of a Normal hierarchical model given specific values for the hyperparameters?

I would like to start by making it clear that I am by no means a statistician. I am an engineer who, after two days of futile research, has realized that I am entirely out of my depth. Please excuse ...
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14 views

regression for left censored data with JAGS

I don't have much knowledge about how to use JAGS to do bayesian regression, I have seen several examples but my data is left censored and I am not sure how to construct the likelihood function, if ...
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292 views

Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
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88 views

How can I use bayesian reasoning to explain a Scrum team if they initial estimates were right or wrong and if the project is currently delayed or not?

Imagine a software development team estimates they are going to be able to complete 80 user stories in 5 sprints using Scrum: Sprint 1: 16 stories Sprint 2: 16 stories Sprint 3: 16 stories Sprint ...
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31 views

Calculating the marginal likelihood with multiple observations in a multivariate Normal-Normal model

Given $f, y_1, \ldots, y_n \in \mathcal{R}^d$ and $V$ fixed: $f \sim N(f; 0, V)$ $y_i | f \sim N(y_i; f, \sigma^2I_d)$ for $i = 1, \ldots, n$ [so they're iid] Find the marginal likelihood: $p(y_1, ...
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Power of a binomial based test

I have a Bernoulli trial, which by definition can be either success of failure, and I have a Binomial random variable of the number of successes, where I don't know p (I wish to estimate it). I have a ...
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17 views

Reporting MCMC autocorrelations of variables in a hierarchical model

I implemented a a Gibbs Sampler for a hierarchical model with priors and hyperpriors that has around 16 variables. When it comes to autocorrelation plots, I have seen in some papers that they do not ...
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14 views

How to write unnormalized posterior when prior is a mixture of continuous and discrete

Suppose I want to do bayesian inference on the regression problem $\beta$ for Y = X$\beta$ + $\epsilon$ for $\epsilon_i \sim N(0,\sigma^2)$. The complication is that the prior for each component ...
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15 views

Using bayes network to determine state

There are two states S1 and S2. There are three output states: O1, O2 and O3 that corresponds to low, medium and high respectively. I have a range specified in which: below 5 is low, 5 to 15 is ...
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Bayesian interpretation of the repeated sampling principle

My question is philosophical rather than practical, and I will try to explain it through an example: Consider a Kaggle competition. All these contests have a similar structure: A "train dataset" is ...
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Bayesian GARCH(1,1) Forecasting

I am using the following bayesGARCH here package in R. I am interested in forecasting $h_t$, the model setup is given bellow. $r_t$ = $\varepsilon_t(\frac{v-2}{v}\omega_th_t)^{1/2}$ $\quad$ with ...
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27 views

Is MAP and MLE the same if MAP uses uniform priors?

would MAP = maximum a posterior and MLE = maximum likelihood estimation be the same if the priors were uniform? since maximizing p(x|y) would be basically the same as p(x|y)c where c is some ...
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Bayesian A/B testing - Sample Size

I've been reading up on Bayesian A/B testing - in particular, the Math behind VWO's Smart Stats. I understand that the way they compute and decide which test is better is by comparing the threshold of ...