Tagged Questions

Combining probabilities with Bayes' Theorem, especially as used for conditional inference.

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Implementation of Bayes posterior predictive check

I have a question concerning the implementation of a bayes posterior predictive check. Let us assume i have this model (implementation is in R and jags): ...
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1answer
62 views

What is meant by this formulation of Bayes' Rule?

From the Wikipedia article on Bayesian inference, we get the following formulation of Bayes' Rule: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) p(\theta \mid ...
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multiple ROC curves in one image (Bayes predictor) [on hold]

I have several Bayes models made in KNIME and I need to plot their ROC curves in one image. In KNIME there is ROC curve node, but it can't plot more than one ROC curves at one time (or I don't know ...
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11 views

Meta analysis, joint posterior distribution of study effect

Meta analysis (with common study variation $\sigma$) often assumes that: $$ X_{i,j} | \theta_i \overset{ind}{\sim} N(\theta_i,\sigma)\\ \theta_i \overset{i.i.d.}{\sim} N(\mu,\tau) $$ where ...
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1answer
39 views

Probability of being each demographic

Relatively simple problem, but can't see how to solve it. Rephrasing in terms of everyday events. Assume there is a fair in town for three days. Each day the total visitors are known, and so are ...
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1answer
34 views

Bayes' rule and multiple conditioning

This is going to be a stupid question, but I know I am missing something, so here it goes: My textbook (Probabilistic Robotics, p.17) trivially states that Bayes' rule gives: $ p(x |y,z) = \frac{p(y ...
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20 views

Am I thinking correctly about this Bayes Problem

I am playing with the following problem. Recently a company has had some resignations. About x out of Total people resigned in one year. Some of the resignations were due to natural movement, ...
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1answer
39 views

Bayesian inference

Assume two demographics $[F,M]$ and each person has a choice of attending only one of four different lectures $[A,B,C,D]$ all occurring at the same time so they can only attend one. The following ...
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20 views

Conjugate prior equivalent prior sample size with respect to the mean

In Cowles's book ([Applied Bayesian Statistics - With R and OpenBUGS Examples–(http://www.springer.com/statistics/statistical+theory+and+methods/book/978-1-4614-5695-7)), page 108, there is a ...
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1answer
19 views

winbugs multiple definition of node [duplicate]

I wrote this code in WinBUGS but I cant run it! It says 'multiple definitions of node tau' ...
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0answers
65 views

Optimal Stopping for Bernoulli One-Armed Bandit with a Fixed, Known Payout

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. Suppose I have two arms $k = {1, ...
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0answers
21 views

MLE == MAP under Uniform Prior? [duplicate]

Does Maximum Likelihood Estimation always yield the same result as Maximum a Posteriori with uniform prior?
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1answer
16 views

Variance of precision in conjugate prior

How can I calculate the variance of the precision in a normal distribution, knowing I used a conjugate prior?
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19 views

How important is the quality of solutions to NP-hard problems arising in machine learning problems?

Machine learning and inference problems give rise to intractable problems. For instance the exact inference in Bayes nets is an NP-hard problem. At the same time there are polynomially tractable ...
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31 views

Two Gaussian Likelihoods with Two Decision Boundaries , 0/1 loss function

we assume that Y := {1, 2}. Then our decision can be re-written as y ∗ = {1 if p(x|y = 1) > p(x|y = 2) , 2 otherwise} with a decision boundary at p(x|y = 1) = p(x|y = 2). How can we construct an ...
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1answer
21 views

Bayesian calculation of parameter multiplying normal

I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I ...
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1answer
35 views

Bayesian Network

I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
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59 views

How to implement a simple Bayesian Network for Time Series Data?

I'm a computer science grad student, with not much knowledge in Bayesian statistics, so I'm seeking for guidance for the simplest start. I have 10 variables, like demand, price etc. and I want to ...
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1answer
50 views

Calcuate Bayes Factor for Adjusted Mean Difference

I have an ANCOVA model shown below (fit) where I calculated the mean difference between two groups while controlling for another variable (x). For the adjusted mean difference (288.72), I'd like to ...
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16 views

linear discriminant analysis, Bayes approach authors?

I know that in 1936 Fisher proposed the LDA that minimizes the variance within and maximizes between. My question is, the Bayes approach of LDA is attributed to a particular(s) author(s)? and what ...
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1answer
149 views

How to apply Bayes' theorem to the search for a fisherman lost at sea

The article The Odds, Continually Updated mentions the story of a Long Island fisherman who literally owes his life to Bayesian Statistics. Here's the short version: There are two fishermen on a ...
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25 views

Proof of alternating conditional expectation base equations

How do we prove the base equations for Alternating Conditional Expectation algorithm. The statement is thus: We define arbitrary mean-zero transformation $\theta(Y),\phi(X_i)$,$1<i<p$ for ...
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1answer
88 views

What is Bayes decision rule?

Assume binary classification i.e. $y \in \{-1,1\}$ and that the underlying joint probability distribution generating the data is known i.e. $P_{x,y}(x,y)$ is known I was told that Bayes decision ...
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24 views

prior predictive distribution negative binomial in R

I have a prior distribution Gamma(1.71,1.05) from a Poisson(2.2), and I know that the prior predictive distribution will be a Negative-Binomial of Gamma parameters i.e. Neg-bin(1.71,1.05). I would ...
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21 views

Picking noninformative priors using pivotal quantities

In 'Bayesian Data Analysis' (Gelman, Carlin, Stern and Rubin) on page 64 it reads: "If the density of $y$ is such that $p(y-\theta|\theta)$ is a function that is free of $\theta$ and $y$, say $f(u)$ ...
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30 views

What is the relationship between naive Bayes and Mahalanobis distance

Recently, I found a code project which uses the Mahalanobis distance to compute Bayes value, but I don't know why you can do that. Second, as I know naive Bayes is based on the Bayes rule, and how ...
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1answer
42 views

Why do we need undirected (Markov) graphical models?

I understand the modular nature of directed models, and that each node captures a conditional probability. But why do we need undirected models? As far as I can see they lack intuition in that the ...
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1answer
36 views

How can I determine source type of a sequence?

I have sequences of binary events, e.g. $s=0001101$. I know that there are two types of sources $A,B$ possible generating each sequence. There are different conditional probabilities for a hit given a ...
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1answer
68 views

Calculating feature probabilities for Naive Bayes

I'm reading "Building Machine Learning Systems with Python" by Willi Richert and Luis Pedro Coelho and I got into a chapter concerning sentiment analysis. There is a whole example about classifying a ...
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9 views

Combining weighted evidence based probabilities?

I'm trying to identify people by determining if a data sample matches a set of existing samples (assume DNA if it helps). In addition to the samples I have a function which gives a probability that ...
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0answers
28 views

Word probabilities in a Naive Bayes filter

While implementing a Naive Bayes filter, I stumbled across a problem with the calculation of the conditional probabilities $p(w|c)$ of a word $w \in \mathcal{W}$ given a class $c \in \mathcal{C}$. ...
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25 views

Specify a covariance matrix

I'm trying to replicate the analysis presented on the bottenada.se and described in the ADA repository. Although the demo file is quite detailed, I didn't grasp where those values for the "C0" ...
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31 views

Parameter Estimation for Naive Bayes - Maximum a posteriori and Maximum Likelihood

I am wondering if I understand those terms correctly. To summarize my thoughts: In naive Bayes, our decision rule is basically the Maximum a posteriori (MAP) estimate of our hypothesis. We assign an ...
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16 views

package in R for BMA of a logistic model?

I am trying to perform analysis similar to Gerlach et al. (2002). it involves predicting the posterior probability of a particular binary outcome using the previous 5 observations. I was just ...
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1answer
41 views

Credit Risk and Concentration

I am working with a UK credit-union and we are looking to build a model to assess our credit risk and changes to this over time. We have a number of loans to borrowers who each have a credit rating ...
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0answers
54 views

Problem with Finding Likelihood: Bayesian

I am really unfamiliar with Bayesian methods particularly parameter estimation. Suppose I have a test to find a parameter, theta which is the number of packaged bag for retail sale that could contain ...
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1answer
47 views

MCMC algorithm to generate samples

I read that MCMC algorithm is used to draw samples from a distribution. The example mentioned in the text book is about a 6x6 matrix which after 1000 iterations will converge to a steady state 1x6 ...
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90 views

Estimate SVM a posteriori probabilities with platt's method does not always work

I have a problem.. I'm trying to create a multiclass SVM with probability output. The SVM is working so far, what means, that the accuracy is ok (see the last picture). But the probability estimation ...
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47 views

Predict random effects in a multilevel model with Empirical Bayes

In multilevel models, it is possible to predict (not estimate) the random effects by Empirical Bayes after the model parameters have been estimated. I know how to use the ...
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14 views

Bayes Interval Estimator using Neyman-Pearson

This problem is 9.56 from Casella-Berger. Let $X \sim f(x|\theta)$ and suppose that we want to estimate $\theta$ with an interval estimator $C$ using the loss $$L(\theta, A) = bLength(A(X)) - ...
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2answers
69 views

Negating conditional probability

I'm refreshing on bayes theorem and conditional probability and I ran across these practice problems. I was trucking along until problem 9, which states: ...
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53 views

Normal Bayesian Model: Marginal distribution of $\bar X$ with unknown mean and unknown variance

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{ind}}{\sim} N(\mu_i, \sigma_i^2) \\ \mu_i & ...
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1answer
33 views

Obtaining posteriors for exclusive and exhaustive hypotheses

I'm trying to solve a Bayesian problem where I have two mutually exclusive and exhaustive hypotheses: $H_1$ and $H_2$. Given Baye's formula: $$P(H|D) = \frac{P(H)P(D|H)}{P(D)}$$ (where $D$ is my ...
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1answer
57 views

How to determine whether an indirect effect is statistically significant using Bayesian statistics?

I've used bayesian estimation to test the indirect effects within a model and identified 95% credible intervals. I'm typically used to using the Sobel's z test to identify significant mediation, what ...
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0answers
75 views

Some doubt in reading Machine Learning A Probabilistic Perspective ( chapter 3.2 )

When I am reading Murphy's Machine Learning A Probabilistic Perspective. In chapter 3.2. I have some doubt. I think the author want to express is two things. First, we can use Bayes formula to ...
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2answers
106 views

Bayes decision theory: Classification error probability

In Bayesian decision theory: Given $\omega_1$ and $\omega_2$ as two classes for classification, $P\left( \omega_1 \right)$ and $P\left( \omega_2\right)$ their prior probabilities, $x$ the feature ...
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2answers
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Mixed variable, joint distribution, How do we know which one is continuous distribution, which one is discrete

If we have one continuous r.v. $x$ and a discrete r.v. $y$ which takes one of the two values $y_1$ and $y_2$. Let's say we know the prior probabilities $P(y_1)$ and $P(y_2)$. From Bayes theorem we ...
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1answer
230 views

Three-prisoner problem and Bayes rule

Here is the wiki of Three-prisoner problem, in which only one prisoner is pardoned, and the Bayes solution is given in the wiki. My problem is pretty much the same, except that only one prisoner is ...
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1answer
37 views

Bayesian alternative or complement to the Student t-test

I was just recently using the Student t-Test to check whether values from two samples could have an identical mean or not. I was wondering whether there is a complementary technique in bayesian ...
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1answer
72 views

bayesian logistic regression - gaussian distribution on parameters?

I'm trying to read this article about Bayesian logistic regression. In general, to classify instances, they use: $p(y=+1 |\beta) = \sigma(\beta^TX) $ (where $\sigma$ is obviously the sigmoid ...