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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Why can we assume that samples $X_i$'s are independent if the parameter is fixed (though unknown)?

To put it in context, I was trying to learn Bayesian parameter estimation (by an example of learning the probability of heads of a coin) and was trying to understand the independence of the samples ...
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Bayesian Linear Regression

I have the following question concerning Bayesian linear regression on my machine learning assignment: Consider $f = w^Tx$, where $p(w) ∼ N(w | 0, Σ)$. Show that $p(f | x)$ is Gaussian. I ...
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Prior for the coefficients of a linear regression model

I have a linear regression model $\bf Y=\bf{X}\bf{\beta}+\epsilon$. I want to assign a prior on $\bf\beta$ in order to derive the posterior predictive model $p(y_{predictive}|\bf{y},\bf{X},\beta)$. ...
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Bayesian linear regression question

I am doing a problem on Bayesian regression but I'm having a lot of trouble with it. Here is the question: Consider $f=w^Tx$, $p(w)\sim N(w|0,\Sigma)$. Show that $p(f|x)$ is Gaussian. Find the mean ...
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The speed of convergence to the true probability depending on the prior

In a Bayesian Updating world, suppose there is a true probability of a binary outcome happening, $P_{true}$. My questions is does the speed of convergence depend on the prior.For instance, if my prior ...
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Kalman filter with input control noise?

assume we have a standard Kalman filter with input controls, following wikipedia notation (http://en.wikipedia.org/wiki/Kalman_filter) where the latent state is $x_{t}$ and the observation is $z_{t}$, ...
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Interpretation of absurdly large (but probably correct) Bayes Factors?

I estimated a Bayes factor to compare a hypothetical model against a null-model (which obviously by visual comparison of the posterior predictive with the data) fails to capture a certain aspect in ...
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19 views

Observed versus hidden variables for Bayesian network in this particular context

I am a novice in Bayesian networks. I have a problem which is best described (at least I think so) in the following story. One wants to predict earthquakes. Let's say it has 5 variables, the last one ...
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Is p-value essentially useless and dangerous to use?

First are some background information. This article "The Odds, Continually Updated" from NY Times happened to catch my attention. To be short, it states that [Bayesian statistics] is proving ...
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can someone fix this cognitive dissonance I have about marginals?

Consider two Bayesian updates, where there are two observations. One updates with respect to $x_1$, and then uses the posterior of that as a prior to update with respect to $x_2$. In both cases, $x_1$ ...
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How to compute the expectation of a normally distributed random variable given an imprecise signal?

Given $r\sim\mathcal{N}\left(\bar{r},\frac{1}{\alpha}\right)$ where $0<\bar{r}<1$ and an imprecise signal about $r$, $x_i=r+\epsilon_i$ where ...
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32 views

How to derive the conjugate prior for univariate Gaussian distribution(assume both mean and std unknown)?

From google search, it seems Normal-Gamma is the conjugate prior for univariate gaussian. I am wondering if there is a systematic way to derive this ? (or to derive conjugate prior for exponential ...
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How does the number of components in a GMM relate to the information content?

Say you fit a Gaussian Mixture Model (GMM) to your data using a Bayesian technique, which should tell you the number of components needed to fit your data. Does this also give insight into the ...
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20 views

Developing probability distributions

An event occurs, and after this event has occurred there is a set of conclusions which can be drawn. All of these conclusions have results which are distinct. I am trying to keep this as general as ...
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21 views

Iteratively solving for prior probabilites.

I'm using Bayes theorem to classify data into two groups, where the conditional probability is known but the prior is not. So I assume that the ratio of prior probabilities is 1 and calculate the ...
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14 views

Bayesian inference of marginal likelihood using ABC

I have the following situation: suppose data $D = \{x_i\}$ iid are generated through some process with density function $f(x_i | \alpha, \beta)$ (which I think will be negative binomial) and we'd like ...
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29 views

How can I use the posterior distribution of parameters from one model in another model?

I would like to model different effects of siRNA treatment on measurements. Cells are grown in 384-well plates, subjected to different siRNA treatment and then imaged to determine parameters. Around ...
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Learn the bayes net structure with latent variables while testing (but observed while training)

I want to use Bayesian network for data which has 5 types of variables which are inter-dependent on each other. Out of that, 1 variable is observed only while training but it is unavailable during ...
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36 views

Bayesian inference when the data are distorted in an unknown manner

Say I make observations of a spatial distribution on a 3D grid. Due to unknown combination of errors, the data on the grid is non-uniformly blurred, and so we can't consider each grid point to be ...
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Bayesian credible intervals: “superiority” even if 1 is included?

In a recent medical publication comparing a cardiac device to anticoagulation ("blood thinners") using a Bayesian statistical model to evaluate the efficacy of preventing strokes and cardiovascular ...
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50 views

Simple Multivariate Bayesian Method in Python

I am trying to follow the Bayesian method described in this text. The python notebook goes through the example of creating two Poisson functions describing a change in SMS frequency at some point tau. ...
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17 views

How to evaluate a Bayesian forecast?

Suppose that I have a predictive posterior, which is an attempt to predict some one-step ahead forecasted value $\hat{y}_{T+1}$. How do I assess if my posterior has done a good job or not? If we had ...
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37 views

Why $p(y_n|y_{0:n-1})$ is a constant in Kalman filter derivation?

In a derivation of Kalman filter, It says that in Equation: $p(x_n|y_{0:n})=\frac{p(y_n|x_n)p(x_b|y_{0:n-1})}{p(y_n|y_{0:n-1})}$ the denominator $p(y_n|y_{0:n-1})$ is a constant. (See the ...
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33 views

Is summing posterior probabilities valid for classification problems?

A classification for two mutually exclusive problem can be formulated by having a decision hinge on whether $P_0(x) > P_1(x)$ or $P_0(x) < P_1(x)$ where $P_0(x)$ and $P_1(x)$ are posterior ...
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What have I done wrong implementing this Bayesian method for fitting a circle to noisy data?

I have noisy measurements of movement along a circle. I want to fit a circle to these measurements. I tried two methods, a straight forward moment fit, and then an ODR fit (described here. However ...
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Question on how to use EM to estimate parameters of this model

I am trying to understand EM and trying to infer parameters of this model using this technique but am having trouble understanding how to begin: So, I have a weighted linear regression model as ...
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14 views

Normal-inverse-Wishart distribution

The Normal-inverse-Wishart distribution is a conjugate prior for the multivariate normal distribution when the mean and covariance are unknown. I understand that conjugate priors are mathematically ...
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12 views

How can i get prior information using my few data set from the whole data? [duplicate]

I have a data set (x1...x500, y1....y500 ) I want to know about bayesian regression I want to know the prior information , few data set(400) from the whole data (500) using MCMCregress( packages in ...
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Is this notation for the improper uniform prior correct?

Can I write: $\mu \sim U(0,\infty)$ ? Or do I have to use the notation $p(\mu) \propto 1$? Thank you.
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Proof that the Chinese restaurant process corresponds to Dirichlet process?

Let $(S, \mathcal{S})$ be a Polish space. Is there a nice proof of the fact that if the people are seated in a restaurant according to Chinese restaurant process, and then for each table, we sample a ...
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26 views

I want to know about bayesian regression using r [closed]

I want to know about bayesian regression I have a data set (x1...x500, y1....y500 ) I want to set the prior information , few data set(400) from the whole data (500) using MCMCregress( packages ...
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10 views

Bayes with non-parametric data

There is some (recent) evidence that neurological activity is log-normally distributed. Does this invalidate the use of Bayes Theorem with these data? I ask because a major branch of computational ...
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Questions about the schools problem

The package R2WinBUGS includes a dataset called "schools": ...
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Can improper priors be implemented in some way?

I'm new to bayesian inference. I've just discovered that improper priors can't be specified in WinBUGS/OpenBUGS. I was wondering if this is common or not in bayesian inference. Are there same cases in ...
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1answer
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Definition of weakly informative prior [duplicate]

According to Gelman, a weakly informative prior is defined in the following way: We characterize a prior distribution as weakly informative if it is proper but is set up so that the information ...
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21 views

How can you judge the statistical confidence and validity of output from a multi arm bandit algorithm like UCB1

To say something about the validity of outcomes in frequentist statistics we have concepts like significance levels and statistical power and in Bayesian analytics we have credible intervals. In a ...
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1answer
56 views

How can a Bayesian analysis say A < B, when both have only 0s?

I've used python to analyse data from AB tests using Bayesian analysis, and for all tests I assume no prior knowledge and so set alpha = beta = 1. However I'm ...
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1answer
38 views

Bayesian Weighted Linear regression

I am currently reading the following paper which formulates the weighted linear regression in a Bayesian setting. In classic weighted LS, we minimise the following: $$ \sum_{i=1}^{N} w_i (\beta^Tx_i ...
3
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1answer
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Handling outliers in Bayesian linear regression

I am reading this post which talks about Robust Linear regression in a Bayesian setting. The particular blog post can be found here: http://twiecki.github.io/blog/2013/08/27/bayesian-glms-2/ There ...
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R, JAGS and missing data

I am currently using JAGS via the program R to run N-mixture models using about 5 years of count data. In 3 of these years, there were 3 counts per site, while in the remaining 2 there were 2 counts ...
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Define precision matrix from covariance matrix WinBugs

I have a 2X2 covariance matrix C with known means c0xx,c0xy,c0yx,c0yy and variances ...
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24 views

Defining prior on variance and not precision

I know that WinBugs uses precision as a parameter in dnorm instead of variance ...
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32 views

Double and single-bisection methods

What is meant by double and single-bisection methods for subjective probability assessment? See, for instance, Garthwaite, P. H. and Dickey, J. M. (1985). Double and single-bisection methods for ...
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27 views

Prior on sum of bernoulli variables

I wish to model my data as follows: $$ y\sim\mathcal{N}(X\beta,\sigma^2)\\ \beta_i\sim\mathcal{N}(5,1)^{z_i}\mathcal{N}(0,1)^{1-z_i}\\ z_i\sim logit(\gamma_i)\\ ...
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1answer
27 views

Question about posterior mean calibration

I'm reading the article "Prior distributions for variance parameters in hierarchical models" by Andrew Gelman(link). This is an extract that I don't understand very well: Posterior inferences can ...
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26 views

Doubt about conditional conjugate priors

I've just read the definition of conditional conjugate prior in this discussion but I have still some doubts. According to the definition given, it seems that the prior distribution of $\theta$, ...
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1answer
23 views

Multivariate t-distribution definition

I have the following marginal posterior of a vector $\phi$ ($p$ by $1$): $$p(\phi | Y) \propto \left[1+\frac{1}{h}\left(\phi - \tilde{\phi} \right)' \Gamma \left(\phi - \tilde{\phi} \right) ...
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63 views

What is the simple standard error for MCMC?

Simply put: suppose that we have observed $X=\left\{ X_{1},\ldots,X_{n}\right\}$. We then need to calculate some statistic $T$ using MCMC, using $M$ loops (By "loops" I mean the number of times the ...
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What does posterior “over” parameters $\alpha$ exactly mean? [closed]

From my understanding the posterior "over" parameters $\alpha$ is $$p(D|\alpha)$$ and not $$p(\alpha|D),$$ is it correct?
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How to do panel data analysis in Bayesian model with pymc [migrated]

everyone. I have a question on how to do panel data analysis in Bayesian model with pymc. The data is like: ...