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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Inferring the bias of two types of coin

I would like to find the bias of a type of coin, when there is uncertainty about which kind of coin I am testing. The scenario is as follows, there are 2 mints in my neighbourhood that produce ...
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14 views

guidance for picking Likelihood in Bayesian analysis

I'm doing a Bayesian analysis for a time series response and wonder whether it is possible to get the Likelihood function without making distributional assumptions. I suppose my response is ...
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1answer
22 views

High dimensional sampling with low measurement noise

Assume that you have a model $$ Y = G(\Theta) + \varepsilon,$$ where $\Theta$ is a parameter vector with $\sim 8$ dimensions, $G$ is a highly nonlinear function of the parameters, $Y$ is observed ...
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53 views

How to score predictions in test set taking into account the full predictive posterior distribution?

I have three predictive models (regressions) which parameters are estimated by Markov Chain Monte Carlo. Predictions are made over a test set of size $N$. Since I compare the models under different ...
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53 views

How to do inference over two steps in a graphical model simultaneously?

Here is my problem: I have observed data $D$ about a physical object described by $M$. I would like to determine the posterior distribution of $M$ given $D$, or $p(M|D)$. Now I can't infer this ...
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20 views

How to fit a Gaussian Mixture Model to data with correlated errors?

I'm restating this question in the hope of getting more interest. The usual function for scoring a Gaussian mixture model assumes independent measurements. But what if we have correlated measurements? ...
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1answer
40 views

completing the square for Gaussian multivariate estimation

I have been trying to derive the posterior distribution in the case of weighted Bayesian regression in the case of multivariate normal distribution for a few days and have been stuck. I am not sure if ...
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1answer
36 views

Computing the conditional distribution for the mean of a Gaussian

I have the following distributional assumptions on some on my RV and model parameters: $$ y_i \sim N(\beta x_i, w_i^{-1}\Sigma_y) $$ There is a normal prior on the parameters $\beta$ as well: $$ ...
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16 views

Can I have a Bayesian approach to analyse my data set? [on hold]

I have a data set with tax incomes from different types of taxes(eg: Income tax, VAT, Goods & services).I want to forecast the each tax revenue for next 10 years using an statistical approach.Can ...
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6 views

computing expectations in variational updates

I have a complete log-likelihood expression as follows: $$ L = \sum_{i=1}^N \log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \sum_{i=1}^N \log P(w_i) $$ Now, I need to compute the expectation of these ...
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20 views

PyMC consistently under estimating results found in paper. Possibly not sampling enough?

I have been trying to build confidence in (my ability to correctly use) PyMC by working examples. Namely, I have been working on Chickering and Pearl 1997, and more specifically on their 'artificial' ...
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2answers
27 views

Suspiciously high Multivariate PSRF from gelman.diag()

I am using "Multivariate PSRF" statistics from gelman.diag() function to analyze my MCMC chains. Now I analyzed convergence 471 variables (parameters for each ...
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1answer
33 views

How to calculate a sample size for validating correct/incorrectness of records in a data table?

I have read through existing answers on CrossValidated (plus elsewhere online) and can't find what I'm looking for, but do please point me to existing sources if I've missed them. Let's say I have a ...
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0answers
66 views

Parameters vs latent variables

I have asked about this before and have really been struggling with identifying what makes a model parameter and what makes it a latent variable. So looking at various threads on this topic on this ...
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18 views

Combining several posteriors

Is there an accepted method of combining the posterior distributions from a model fit to several participants to obtain a posterior for the entire group of participants? The reason I am asking is ...
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2answers
42 views

How might Google go about estimating and updating traffic speeds?

This is, I guess, a specific example of a wider class of problem, one to which there must be a well-established solution, but which I, as a relative layman when it comes to statistics have thus far ...
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1answer
26 views

How to rank rankings? Two factors - freq and rank but freq has to has much more weight

My respondents has to evaluate number of items (say, 20). Formerly they had to just check three that they liked (like/dislike) and I simply counted the most checked. I 'improved' survey by replacing ...
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2answers
248 views

Example for a prior, that unlike Jeffreys, leads to a posterior that is not invariant

I am reposting an "answer" to a question that I had given some two weeks ago here: Why is the Jeffreys prior useful? It really was a question (and I did not have the right to post comments at the ...
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41 views

Bayes Test and decision regions

I'm given the following hypotheses: $$f(x|H_0) = e^{-x}u(x)$$ $$f(x|H_1) = 0.5\{u(x)-u(x-2)\}$$ and asked set up a Bayes Test for equally likely hypotheses and determine the decision regions. My ...
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31 views

How do I classify data with multiple dimensions using a gaussian classifier? [closed]

I've computed the equation inside the brackets (but not i): Features=dimensions (x,y)..R^n Ck being the covariance matrix, z being the input vector, u being the mean vector, N being the number of ...
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1answer
48 views

How is ABC more computationally efficient than exact Bayesian Computation for parameter estimation in dynamical systems (ODE) models?

Approximate Bayesian Computation has been suggested as an approach to parameter estimation for computationally intensive simulations, most commonly in population genetics, but also in dynamical ...
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1answer
18 views

One-Step ahead predictive likelihood for time series forecasting

I am still new to Bayesian forecasting, so I am hoping to get some clarification on a simple concept (by the sounds of it). Suppose that we are interested in forecasting some time series one-step ...
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29 views

MCMC: The posteriors are too narrow

I have this very simple MCMC there I have ...
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30 views

Probability Interval for F(x) with Parameter Estimates from Bayesian Analyses

Problem: I estimated the shape $\alpha$ and scale $\lambda$ parameter of the Weibull distribution using Bayesian methods. That gave me a marginal posterior distributions for both parameters. ...
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2answers
84 views

Can my Bayesian prior reflect what the data should say rather than what it could say?

Can my Bayesian prior reflect what the data should say rather than what it could say? For example, assume I collect data where $Y_i$ is whether or not student $i$ passed the test and $X_i$ is whether ...
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33 views

Is is possible to determine conditional conjugacy in this case?

I'm working on a problem where I have to extract sufficient statistics for parameter estimation in a state-space model. Usually these come from the quantities used for conjugate updates. I'm OK with a ...
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1answer
34 views

Show posterior is proper for this poisson linear model

This question is 3.12 in Andrew Gelman's Bayesian Data Analysis 3rd edition. Let $y_i|\alpha,\beta \overset{iid}{\sim} \text{Poisson}$ with mean $\alpha+\beta t_i$. Find a prior distribution that ...
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31 views

implementating the bayesian linear prediction with NIG prior

In Bayesian linear regression when the covariance of weights is unknown; one can set Normal-Inverse-Gamma prior. Based on "Machine Learning: a Probabilistic Perspective", Page 235, \begin{equation} ...
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15 views

Factorization of probability distribution and its Bayesian Network

My question is if we have a distribution $P$ that can be factorized into cond. distributions, can we model it with Bayesian Networks? I mean, $P(X_1,X_2,...,X_n) = \prod_{i=1}^n P(X_i|Cond(X_i))$ ...
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1answer
47 views

Bayesian regression with independent variable drawn from distribution

I'm trying to set up a bayesian regression of the form $y_i \sim f(\beta_0 + \beta_1 x_i)$ but rather than $x_i$ fixed, they themselves are drawn from a distribution of (known) mean $x_i \sim ...
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4answers
1k views

What is a good book about the philosophy behind Bayesian thinking?

What is a good book about Bayesian philosophy, contrasting subjectivists against objectivists, explaining the view of probability as state of knowledge in Bayesian statistics, etc.? Maybe Savage's ...
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23 views

Marginalizing over a Chinese Restaurant Process prior

I am reading a paper by Kemp et al. and there is a part about marginalising over a Chinese Restaurant Process and I am quite clueless about how could one marginalise over such a prior! The details of ...
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1answer
60 views

Kalman filter with control inputs in python?

i am trying to fit a simple kalman filter with input controls (in this case step input) in python. i am using filterpy (http://filterpy.readthedocs.org/). my code is: ...
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32 views

MCMC convergence: why Heidelberg's test says normal samples are non-stationary?

I am learning about and playing with Heidelberg's convergence test to automatically stop a MCMC sampling. I would have said that if I sample, for instance, from a normal distribution, the test ...
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13 views

How to calculate the posterior probabilty of Gaussian Mixture Component

If the mean vector and the Covariance matrix of a Gaussian Mixture model are known, how could I calculate the posterior probability of each of the Gaussian Component in the mixture.
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1answer
35 views

Why is it called “mode” in MAP estimation?

When estimating parameters with MAP, why is it written that we are estimating the "mode"? I thought it would be the mean of the posterior distribution?
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1answer
33 views

Slope estimate dependent on covariance?

I am trying to perform a linear regression with equal errors on x and y (ex =1 and ey=1) in a Bayesian framework (using WinBugs). Using Winbugs (solid line in the Figure), I managed to reproduce the ...
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9 views

What models allow the study of the relation between a set of response variables and a set of covariates?

A first technique that comes to mind is Canonical Correlation Analysis. Bayesian Networks and other graphical models, I guess, can also be used to analyse such things. Any else that I should be aware ...
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1answer
13 views

Prior for gamma distribution in “mean form”

I need to specify priors for the parameters of a gamma distribution. Normally the gamma distribution is parametrized in either the "rate-form'': ...
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0answers
17 views

Do mildly informative prior distributions tend to mitigate false positives (i.e. Type I error rates)?

I am curious if others have sources that speak to the matter that providing informative and/or mildly informative prior distributions on a parameter tend to mitigate false alarm rates? I know from the ...
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Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then ...
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21 views

Bayesian fixed effects model and invariant variables

Within a fixed effects approach, the effects of invariant variables cannot be estimated. Their effects are captured by the fixed effects. However, when I estimate following Bayesian fixed effects ...
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1answer
70 views

Gibbs Sampler transition kernel

Let $\pi$ be the target distribution on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R^d}))$ which is absolutely continuously wrt to the $d$-dimensional Lebesgue measure, i.e : $\pi$ admits a density ...
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0answers
18 views

prior for integer-valued random variable taking values 1 or greater

In my model I have an integer-valued random variable which should only take values one or greater. I would like to specify an appropriate prior for this which has most of the mass say around 1 to 5 ...
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44 views

How I can analyze a data survey obtained with non probabilistic sampling?

I got data from a survey. The survey was designed with quotas of the population in that neighborhood (near the 10% of the total population of that place). But, the people was surveyed in the ...
5
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1answer
164 views

Gibbs Sampler contradiction proof

I want to prove that the systematic scan Gibbs sampler yields an aperiodic chain $X$ on a general state space. Let $\pi$ be the stationary distribution for the resulting chain. Suppose to get a ...
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1answer
22 views

What happens if I choose a bad prior for Bayesian AB testing?

I'm trying to go through this article http://developers.lyst.com/data/2014/05/10/bayesian-ab-testing/ and I see that they choose a Beta(3, 50) prior and make an argument for that. However, if you use ...
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1answer
57 views

Simplifying P(x|y,z) to P(x|z)

I am looking for the probability of an outcome (x in Win, y in Opponent Actions and z in Opponent Cards) in a poker scenario. I need a mathematical proof to simplify $\Pr (x \mid y, z)$ to $\Pr (x ...
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10 views

jags posterior distribution mimics prior [migrated]

I have this model to adjust Schechter luminosity function but no matter what prior values I choose, the posterior distribution of each parameter is pretty much the same as the prior. And if I run ...
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27 views

Is this a true inequality over multinomial parameters?

Let $p_1,\ldots,p_k$ be the multinomial parameters sampled from a Dirichlet distribution. Assume $\bar{p_1},\ldots,\bar{p}_k$ be the mean of Dirichlet distribution. Then, $$Pr( ...