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

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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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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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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“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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12 views

Why multivariate distribution is used to model weights (polynomial coefficients) for prior distribution?

Lately i have been reading Chris. Bishops work on bayesian linear regression what I am finding difficult to understand is why is he modelling w0 and w1 using multivariate gaussian distribution in ...
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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 the subjects in ...
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1answer
27 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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1answer
28 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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67 views

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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2answers
23 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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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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32 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
28 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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3answers
96 views

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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33 views

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
34 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
40 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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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
31 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
30 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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12 views

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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46 views

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 ...
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37 views

Bayesian Inference Using Functions of Parameters

Assume $\mathbf{X}$ has a pdf that depends on the two parameters $\mu, \sigma$ given by $h(\mathbf{x}|\mu, \sigma)$. Traditionally, Bayes' theorem allows the computation of the posterior pdf $g(\mu, ...
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25 views

Use Linear Regression to Estimate Conditional Probability for Bayes Net?

When reading and watching video regarding building and using Bayes Nets, the examples typically use binary outcomes for the nodes. 'Probability of it raining', 'Probability of x disease', ect... ...
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52 views

Conditions on transformation function in Monte Carlo expectation

If I have an i.i.d. set of samples $\theta_1, \ldots, \theta_n$ from my posterior $p(\theta | y)$ then: $ E(f(\theta | y)) = \int f(\theta) p(\theta | y)\, \mathrm{d}\theta \approx \frac{1}{n} ...
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Weighted log-probabilities in generalised gamma distribution

This question is related to the problems I mentioned in this question. I am not sure if there is a good solution, but am hoping someone more experienced with this type of thing can help out. I am ...
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112 views

What is the difference between logistic regression and bayesian logistic regression?

I'm a bit confused whether these two are the same concept. If they are different what's the difference? Thanks!
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Bayesian model selection in PyMC3

I am using PyMC3 to run Bayesian models on my data. I am new to Bayesian modeling but according to some blogs posts, Wikipedia and QA from this site, it seems to be a valid approach to use Bayes ...
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Hierarchical Bayesian Regression, Can an Inverse-Gamma distributed Variance look Normal or t?

Using Peter Hoff's book, A First Course in Bayesian Statistical Methods, I used some of my own data to fit a Hierarchical Bayesian Regression following his example. In his book, he utilized a Gibbs ...
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Bayesian Inference and Conditional Probabilities

From Wikipedia, under "Formal description of Bayesian inference:" $\theta$, the parameter of the data point's distribution, i.e., $x \sim p(x|\theta)$ $\alpha$, the hyperparameter of the ...
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1answer
26 views

MAP and MLE estimation

i stumbled upon the following formula that describes making predictions based on the MAP estimate. i understand that we "plug in" in the MAP estimate in the predictive posterior, but i do not ...
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1answer
7 views

Strong correlation between prior and posterior for a simple bayesian model?

Consider the following pymc3 model. ...
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Inference from the posterior predictive distribution [duplicate]

I want to use Bayesian model to predict the values of signal in the future. The process is like: a. 1000 observations are given. First 800 consecutive observations are training data, and 200 ...
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32 views

Identifiability of a state space model (Dynamic Linear Model)

Take a general linear Gaussian state space model (SSM)(aka Dynamic Linear Model DLM): $X_{t+1}=FX_t + V_t$ $Y=HX_t+W_t$ $V_t \sim N(0,Q)$ $W_t \sim N(0,R)$ I am interested in the ...
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Fastest way to solve Bayes estimator problem

The below problem is from an old PhD qualifying exam in our department. My own solution below is time-consuming and quite possibly wrong. It also relies on recognizing a less common distribution, so I ...
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Interpreting mean of coefficients by accessing $Beta in BMR package in R

I've been using BMR (Bayesian Macroeconometrics in R) package to carryout BVAR(Bayesian Vector Auto Regression). When defining the Minnesota prior for my monthly dataset and have obtained mean of each ...