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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GARCH vs SV for Forecasting

I believe I am aware of how GARCH family and stochastic volatility models differ in their construction and assumptions on the volatility states, (i.e. GARCH family assumes deterministic volatility ...
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129 views

MLE and Bayesian methods

I saw in some lecture the fact that as the number of data points N goes to infinity, the prediction of the Bayesian method goes to the prediction of the MLE. Can someone explain what exactly this ...
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18 views

Effect of “parameters.to.save” in R2jags/ JAGS

I'm using the package R2jags in R, which uses the parameters.to.save argument to specify parameters. I'm interested in the statistical distinction between a ...
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12 views

I have a hierarchical bayesian logit model with large variation in outcome between units. Should I still use a hierarchical model?

So instead of using country and time fixed effects for a time series cross section data on country-years with a binary outcome, I'm using a hierarchical logit model with country and time random ...
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11 views

how to model and implement bayesian change point with longitudinal data

I am working on Bayesian change point of Poisson data with different identities that is longitudinal. I have understood to degree the hierarchical structure of the posterior with hyper priors for data ...
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17 views

How to use lagged outcome variable in CausalImpact? [on hold]

I would like to use the lagged outcome variable as regressor in the CausalImpact/bsts package. Since I cannot use the ...
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1answer
41 views

What is the limiting distribution of the Bayesian Filtering

I've got a question about the iterative Bayesian filtering, the general form of which is shown as follows: $P(x|z_0,...z_{k+1})\propto P(z_{n+1}|x)P(x|z_0,...,z_k),\,k=0,1,\dots$. $P(x|z_0)=P_0(x)$ ...
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1answer
36 views

Closed form of conditional probability for a specific joint

I have a joint probability of a very specific form: $P(x_1,\cdots,x_n)=\phi(x_1)\psi(x_1,x_2)\phi(x_2)\cdots\psi(x_{n-1},x_n)\phi(x_n)=\prod_{i=1}^n \phi(x_i) \prod_{i=1}^{n-1} \psi(x_i,x_{i+1})$ I ...
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Linear regression with prior on $\arctan \beta_1$

Suppose we have $\hat{y} = \beta_1 x + \beta_0$ (I ask only for the univariate case.) A typical Bayesian approach might involve Normal priors on both parameters. I was thinking today about a ...
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24 views

What are the practical problems where the prior and posterior probabilities derivable from data are not reliable?

In Rough Baysian Model (Rough sets and Bayes Factor), authors always say that this model is very applicable to practical problems where the prior and posterior probabilities derivable from data or ...
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1answer
21 views

Question About Metropolis Hastings Algorithm [closed]

I am currently a Math/Economics Student and I am working with a local bank on modeling operational risk, and one of the models that propose we might use is the Metropolis Hastings algorithm. This ...
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1answer
35 views

Relation between changing the prior and the effect of an additional data point

E. T. Janes writes the following in "Probability Theory: The Logic of Science": A useful rule of a thumb is that changing the prior probability $p(\alpha | I)$ for a parameter by one power of ...
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Prior knowledge and significance thresholding

I am curious whether the following is sensible for a translational or follow-up study: Knowing form a previous experiment that brain regions A and B show highly significant activation upon treatment ...
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25 views

Root-Mean Squared Error for Bayesian Regression Models

I'm trying to get a sense of my prediction errors for a Bayesian regression model and I was using the Root-Mean-Squared Error. My question is, since are predictions are stochastic, would it make ...
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9 views

Extensions of bsts and CausalImpact to non-Gaussian exponential family distributions

The bsts and CausalImpact packages implement a state space time series model with an optional regularized regression component. ...
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18 views

How can I obtain the Simulations in CausalImpact package? [closed]

Currently we are using your package CausalImpact to evaluate the effect of different interventions over the accident occurrence in different firms. I address to you in order to ask you if there is any ...
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1answer
36 views

How to approximate Bayes Factor?

I am searching for a computationally simple way to approximate a Bayes Factor. Currently, I'm using an approach which seems pretty logical to me but I would still be interested to know if this is ...
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21 views

Effect size for contrasts in hierarchical Bayesian “ANOVA”

Kruschke (2014) shows in his book how to compute posterior distributions of effect sizes (standardized mean difference) for the Bayesian analogues of frequentist independent-samples t-tests, and ...
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44 views

“Non-naive” bayesian classification algorithms

Based on the problem description in this post: Relating parameters to a measured variable Based on a suggestion, I thought of studying the relationship between the parameters and a measured metric ...
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Why multivariate distribution is used to model weights (polynomial coefficients) for prior distribution?

Lately, I have been reading Chris Bishop's work on Bayesian linear regression. What was difficult to understand is why is he models $w_0$ and $w_1$ using multivariate Gaussian distribution in prior ...
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28 views

Test-retest correlation for panel data

I've run an experiment in which subjects rate how much they like six different objects on a 1-5 scale on two occasions. I'd like to obtain a summary measure of how consistent are the subjects in their ...
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2answers
53 views

Metropolis Hastings Algorithm - Prior vs Proposal vs Numerator of Bayes Theorem

I've been using this technique in 'black-box' form for a little while as a physics student. I have been struggling to understand what's happening under the hood for some time and I think I almost ...
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Bayesian importance sampling as an answer to a “paradox” by Wasserman

Both in his book and on his blog, Larry Wasserman has discussed an example in which naive application of the Bayesian methods gives nonsensical answers. Intro The problem is to estimate the ...
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36 views

Binomial Distribution Where N is Generated by a Poisson Process (pymc)

I'm not sure if this is the best way to go about this, because I'm fairly new to Bayesian methods. I'm trying to model a process where the number of trials $n$ used in a binomial process is generated ...
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127 views
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Bayesian logit model - intuitive explanation?

I must confess that I previously haven't heard of that term in any of my classes, undergrad or grad. What does it mean for a logistic regression to be Bayesian? I'm looking for an explanation with a ...
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Multivariate linear regression with prior information on variances

I have a slight variation to a classic problem, which might have a simple answer - but if it does, I cannot find it. My problem is a multiple linear regression, of the type that is common in ...
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33 views

CausalImpact - Should I use more than one control?

In the intro document (https://google.github.io/CausalImpact/CausalImpact.html) it suggests that using one predictor is not ideal. Am I current in understand that they mean one control? If so, should ...
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70 views

Particle filter for estimation of static parameters

I am considering particle filtering methods for the estimation of static and dynamic parameters. For the static parameters $\theta$, Liu and West (page 7, equation 3.1) describe an "artificial" ...
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33 views

Fisher LDA is a Bayes Classifier?

I've been going over many material in classification algorithms, and it seems that under the constraint that the covariance matrices are the same for a two-class problem then classifying a vector $x$ ...
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22 views

Simple derivation of a Bayes Risk function

I'm trying to derive the Bayes risk shown below in the first picture. From the definition of Bayes risk, in the next picture Here is my derivation of the Bayes Risk: ...
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22 views

Calculating probability of sale from auction data

I have some data representing the last 6 months of closed auction data from a particular website. The data I have includes market value of product, actual sale amount, and date sold. I have about 600 ...
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1answer
31 views

What is the difference bewteen linear regression using OLS and bayesian linear regression?

Assume that we have Data $D = \{(x_1,y_1), (x_2,y_2),\dots,(x_n,y_n)\}$ where $x_i \in \mathbb{R}^n$ and $y_i \in \mathbb{R}^n$. $y_i = w^Tx_i+\epsilon_i$ Using OLS, I can estimate the values of ...
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How well does a single data point fit a distribution?

I have to come up with a way to measure the 'quality' of a distribution for a research project. We collect data over a a period of time $t_0$ through $t_1$ and then estimate the distribution that ...
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How to interpret the following Baysian time series representation (picture attached)

I am trying to understand this paper on Bayesian Hierarchial model (http://www.umac.mo/fba/irer/papers/past/vol13n1_pdf/01.pdf) in which one of the sub-models is a time series with random-walk ...
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Math notation for bayesian hierarchical models with covariance matrix with LKJ priors

I am fitting a simple hierarchical bayesian gaussian model of the form: $$ Y = b_0 + b_1 X_1 + b_2 X_2 + e $$ $$ b_{0:2} \sim N(\theta_{0:2}, \sigma_{0:2}) $$ $$ \sigma_{0:2} \sim Gamma(.001 , .001) ...
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1answer
35 views

Reasoning regarding non-informative priors

I'm not sure whether this counts as a question. However, I'd be happy to receive feedback for the validity of my reasoning. Recently, I read a bit about Jeffreys' prior and the "problem" with using ...
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15 views

Covariance of noise in Posterior PDF for Bayesian General Linear Model

I'm reading about Bayesian estimation in Steven M. Kay's Estimation Theory vol. 1. I understand the basic philosophy behind the Bayesian approach, but I think there's a fundamental insight I haven't ...
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1answer
41 views

How do I compute the posterior predictive distribution of a logit model?

So I used stan to take samples from a logit model. I want to compute the posterior predictive distribution of this model, but I am having trouble figureing out the logit link function and how it ...
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Reference prior for a three-parameter model and likelihood factorization

Let a (regular) statistical model with three parameters $\phi_1$, $\lambda_2$, $\mu$, and three observations $x_1$, $x_2$, $y$. Assume the likelihood has form $$ L(\mu,\phi_1,\lambda_2 \mid y, x_1, ...
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Simple question on graphical representation of minmax decision rule

In the picture below, I cannot understand why the minmax decision rule is on the line $R_1=R_2$. $R_i=R(\theta_i,d)$, where $\theta_i$ is the parameter and $d$ is the decision rule. $S$ is the risk ...
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64 views

Process with parameters that are themselves statistical

I'd like to work with a pair of statistical processes such that the random variable from one process is the parameter of the second process. The simplest case I can imagine (and which is still ...
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1answer
34 views

Bayesian optimization or gradient descent?

When and why use Bayesian optimization, instead of gradient descent? Which one is better for which cases?
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1answer
14 views

Estimating the number of classes from a sample

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
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45 views

Bayesian neural networks: very multimodal posterior?

Question: How do Bayesian treatments of neural networks address the fact that the posterior has an exponentially large number of modes? Background: There seems to be a lot of interest in Bayesian ...
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28 views

Derivation of the BIC

i am trying to self-study / understand the derivation of the BIC. I have studied that: however - it is not quite clear to me how this leads to the formula below. I don't fully understand where the ...
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1answer
39 views

How do I use Stan to fit a covariance matrix? [closed]

I'm new to Stan (and bayesian methods in general), so this is likely very simple. I'm trying to model some multivariate normal data. All I want to know is the covariance matrix generating the data, ...
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Varying transition probabilities by position

I'm still very new to Bayesian Tables, Hidden Markov Models and the likes, but have an otherwise solid computational and linguistics background. I've been diving into NLTK (Natural Language Toolkit) ...
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Bayesian approach for comparing the predictability of different datasets for another

Suppose I have three datasets A, B and C with not necessarily the same amount of data. Now, I want to know whether dataset A or dataset B is better in predicting C. I thought of using a Bayesian ...
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1answer
24 views

Coin flip experiment with biased coins (and analogy to real-life problem)

In order to determine if a coin is fair by an experiment I flipped it 20 times and received 7 heads. Since the cumulative probability to have 7 or less heads is 13% with a binomial distribution I ...
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I need little help about Bayesian hypothesis testing

Let $x_0\sim Bin(n_0,p_0)$ and $x_1\sim Bin(n_{1},p_1)$. And, suppose we are interested in testing $H_0: p_0= p_1$ versus $H_A:p_0 \ne p_1$. I wanted to consider the following priors, $\pi(p_1\mid ...