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 intervals for machine learning predictors

Assume I have a regression problem. I fit models on a train data set and tune their hyperparameters using CV. I then run the models on the test set. What is the best way to calculate confidence ...
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Graphical Model Question — Stochastic Block Model Priors

In the generic stochastic block model (binary edge data, no degree correction, etc.), if an uninformed prior is used for the Bernoulli coefficients i.e. Beta with $(a,b) = (1,1)$, will the model ...
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Evaluating integral to obtain marginal PDF related to Tikhonov Regularization

I am attempting to derive the marginal PDF for an application of the Gibbs Sampler. My joint PDF contains: $P(b,x) = \frac{1}{\sigma^{n}}\exp \left( -\frac{1}{2\sigma^2}\left\lVert ...
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2answers
56 views

Priors in Bayesian MCMC

I am trying to understand how the choice of priors affects a Bayesian model estimated using MCMC. At a basic level I understand that the product of the prior and the likelihood are proportional to the ...
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10 views

What are good values for autocorrelation, Gelman, and cross-correlation in rjags?

I don't want to post my whole code since it is long, so I will only post part of it: ...
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1answer
28 views

Estimating bias in surveys

I run into the following problem on a job interview, and am still wondering what is a principled way to solve it. I think the problem is general enough that will hopefully have enough educational ...
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1answer
25 views

Prediction based on bayesian model

I have created a bayesian model that estimates 6 parameters using rjags from R. Now i want to do some predictions based on new data in R. Can anyone help me with an example. ...
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19 views

Bayesian Mixture Model Gibbs Sampler for two linear relationships

I am attempting to use a Gibbs Sampler to model a mixture of two groups, where the group membership is defined by a linear relationship conditional on x. Both groups have the same slope and intercept, ...
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0answers
13 views

Jeffrey's Prior for normal distribution with mean = 0

How would I go about calculating Jeffrey's Prior for a normal distribution with mean = 0, So far I get: But then don't know where to go next. Any help much appreciated
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0answers
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What are some advanced algorithms in bayesian networks? [on hold]

What are some advanced algorithms in bayesian networks? I am familiar with the conventional algorithms of network construction and inference in bayesian networks. What are some algorithms that provide ...
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2answers
154 views

How does one use Bayes theorem with a continuous prior?

If my prior is modelled as a continuous probability distribution, say, a beta distribution skewed to reflect my bias towards certain models, how can I calculate the posterior probability? The ...
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17 views

Likelihood of hypothesis in live data

Bayes rule is $P(H|E)=\frac{P(H)P(E|H)}{P(E)}$ I have a prior distribution from categorical data prior={'a':0.2,'b':0.6,'c':0.1,'d':0.1} Which forms my ...
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Posterior predictive for Gamma distribution with unknown scale and shape

I have a question that needs clarification. The posterior predictive distribution can be described as the distribution that a new i.i.d. data point $\tilde{x}$ would have, given a set of $N$ existing ...
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1answer
16 views

Log likelihood for inverse gamma

For a gamma distribution, the answer to this question shows that you can just use the log of the gamma distribution density function. Is the same true for inverse gamma? It is the same as the log of ...
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+50

When should I be worried about the Jeffreys-Lindley paradox in Bayesian model choice?

I am considering a large (but finite) space of models of varying complexity which I explore using RJMCMC. The prior on the parameter vector for each model is fairly informative. In what cases (if ...
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2answers
32 views

Marginal likelihood vs. prior predictive probability

In the Bayesian framework, to me, it seems that the marginal likelihood and the prior predictive distribution/probability are equal. Is that the case? Or maybe this just holds for single data points? ...
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3answers
44 views

Bayesian Risk and Subjectivity

I am studying the differences in bayesian and frequentist approaches to point estimation. I understand that there are objective and subjective approaches to Bayesian and some people don't like the ...
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0answers
56 views

Bayesian Analysis of Box-Cox Transformation

This problem is problem 5 in Chapter 7 of Bayesian Data Analysis, 3rd edition. Consider the Box-Cox transformation: $y_i^{(\lambda)} \sim \mathcal{N}(\mu, \sigma^2)$ where $y_i^{(\lambda)} = ...
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1answer
46 views

Example Bayesian resolution of the Two Envelopes Problem [closed]

What is a concrete example of a Bayesian resolution to the Two Envelopes Problem?
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Bayesian Probabilistic Matrix Factorization (BPMF) with PyMC3: PositiveDefiniteError using `NUTS` [migrated]

I've implemented the Bayesian Probabilistic Matrix Factorization algorithm using pymc3 in Python. I also implemented it's precursor, Probabilistic Matrix ...
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32 views

Is this how a Bayesian bootstrap works?

I am a bit new to the whole nonparametric and Bayesian idea, so tell me if this is correct: to estimate, say, the mean of a dataset's population we do the following: We define a function $f(x)$ that ...
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1answer
22 views

JAGS Error: Invalid Parent Values on last observation

I am using R2jags to fit a model in R using JAGS. Here is my code: ...
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0answers
21 views

Identify distribution change [closed]

I have a categorical variable Product that can have one of $4$ possible values, ${x_1,x_2,x_3,x_4}$ The current distribution is ...
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2answers
58 views

Kernel of a Normal Distribution

From Wikipedia , The kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the ...
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sampling from distribution [duplicate]

In Monte Carlo Markov chain (Gibbs or Metropolis-hastings) samples are drawn from posterior distribution. In layman terms, how sampling is done from a distribution?
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8 views

HELP: Bayesian Multi-level model with seasonality

I am trying to define a Bayesian Multi-level model which has seasonality in BUGs. I have defined the model (below).I have attached a graphical representation of what im trying to model. eventually ...
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1answer
95 views

Bayesian meta analysis: implementation in BUGS/JAGS/STAN

I would like to conduct a meta analysis in order to collate the information from a number of studies. The parameter of interest is a probability $\theta$. In each of the studies, the observed data ...
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1answer
43 views

Neural network & Bayesian in this machine learning algorithm

I am new to machine learning etc and found this comprehensive algorithm: http://scikit-learn.org/stable/tutorial/machine_learning_map/ . However, I am not able to make out any reference to neural ...
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1answer
67 views

PyMC3 Implementation of Probabilistic Matrix Factorization (PMF): MAP produces all 0s

I've started working with pymc3 over the past few days, and after getting a feel for the basics, I've tried implementing the Probabilistic Matrix Factorization model. For validation, I use a subset ...
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0answers
22 views

Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline. With that assumption, the mean and trend ...
2
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0answers
22 views

EM algorithm: With prior vs. not prior

I have a working EM algorithm without prior. I am asking for some advice on how to add prior on latent variables. Define: $t_i \in \{ +1, -1 \} $: variables of interest to be predicted $p_j \in ...
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1answer
31 views
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1answer
36 views
+50

Multipliers on Top of Binomial Rate Estimates

I was wondering if anyone has come across a similar question to the following. I have data of the form $s_{x,y}, t_{x,y}$ (successes and trials) for varying groups with $x \in X, y \in Y$. I also ...
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1answer
52 views

Construct the likelihood with asymmetric uncertainties

I want to study the correlation between 2 parameters, this is done by fitting a straight line. I have uncertainties on both parameters. I want to solve my problem using the Bayesian approach, i.e. I ...
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0answers
27 views

Bayesian Monte Carlo modeling and selecting priors [duplicate]

Could anyone recommend some not-too-mathy introductory texts to Bayesian regression and Monte Carlo modeling? I am neither a statistician nor an econometrician. The frequentist perspective makes ...
4
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1answer
51 views

Evaluate posterior predictive distribution in Bayesian linear regression

I'm confused on how to evaluate the posterior predictive distribution for Bayesian linear regression, past the basic case described here on page 3, and copied below. $$ p(\tilde y \mid y) = \int ...
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1answer
230 views

Can a Multinomial(1/n, …, 1/n) be characterized as a discretized Dirichlet(1, .., 1)?

So this question is slightly messy, but I'll include colourful graphs to make up for that! First the Background then the Question(s). Background Say you have a $n$-dimensional multinomial ...
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1answer
31 views

D-separation in a Bayesian Network [closed]

The above question asks to see if Radio is D-Separated from Petrol given certain evidence. For evidence (i), why would this mean D-Separation? If Battery is true, we have a inactive triple. If ...
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25 views

Questions about Linear regression vs bayesian way of parameter estimation

I have several questions about the difference between linear regression and bayesian way of parameter estimation using MCMC. I know their difference in mathematical stuffs and I can implement them ...
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Bayesian Inference for Poisson process 2

How do I calculate Bayesian posterior distribution of Poisson likelihood function with Pareto prior distribution?
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0answers
22 views

Expected ratio of probabilities--is there a term for it?

I recently came across the following quantity when I played around with some information theoretic quantities and Bayesian learning. Given three probability distributions $q(z), p(z)$ and $p(z|x)$. ...
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1answer
37 views

Why do we use Gamma($\epsilon, \epsilon$) as non-informative prior for precision and Normal prior for betas in Linear Regression

Suppose my regression model is $$Y_i = \beta_0 + \beta_1X_{i1} + \epsilon_i $$ In most books I am seeing that the prior used for precision $\tau = 1/\sigma^2 $ is $Gamma(\epsilon, \epsilon)$. However ...
2
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1answer
54 views

How to assess if a model is good in multinomial logistic regression?

I have some ordinal response $y$ that I modeled using both ordinal logistic regression and multinomial logistic regression (to avoid the proportional odds assumption), using two continuous variables ...
2
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1answer
40 views

Translating a bayesian model from MCMCpack (hregress) to JAGS

I am trying to convert a hierarchical model that currently works with the R package MCMCpack, into a JAGS version. I have experience working with regular models in JAGS, but not with hierarchical ...
2
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1answer
50 views

Estimating mean of Normal with unknown variance and then predict the future observation

I am trying to estimate population mean of 9 observations when the variance is unknown. I marginalized the posterior and understand that the t- distribution would give me the distribution of ...
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32 views

Bayes estimator, poisson distribution with exponential prior

If X ~ Poisson($\theta$) and $\pi(\theta)$~$exp(1)$(prior). Find the Bayes estimator for $P_\theta(X=0)$ with respect to quadratic loss $f(\theta|x)=\frac{e^{-n\theta}\theta^{\sum x_i}}{\prod x_i}$ ...
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19 views

Bayesian process with noise?

Does anyone know of any research considering Dirichlet processes (or other Bayesian nonparametric models), where sample points have a known gaussian noise attached to their input? I.e. I have a set ...
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11 views

How to derive type-2 maximum likelihood method for RVM regression

According to PRML book(7.85,7.86,exercise 7.12), the marginal likelihood for RVM regression is $$ \ln p(y|X,\alpha,\beta)=−1/2\{N\ln2π + \ln|C| + y^TC^{−1}y\} $$ $$ A=diag\{\alpha_1,..,\alpha_D\} $$ ...
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BIC for Bayesian ANOVA

I am doing a Bayesian ANOVA as follows: BIC0 = -2 * logLik0 + k0 * log(N) # null hypothesis BIC BIC1 = -2 * logLik1 + k1 * log(N) # alternate hypothesis BIC ...
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1answer
23 views

“Multiple definitions of node p[1]” Error using WinBUGS [closed]

I have written the following code in WinBUGS and every time I try to compile the data after loading in the data I get the same error which is "Multiple definitions of node ...