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

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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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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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25 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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21 views

classify with 3 class [closed]

How do I calculate average rate error using Bayes and neural network classification, for example, on the three classes in Fisher's iris data?
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46 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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45 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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14 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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127 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
71 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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20 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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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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28 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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37 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
64 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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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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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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28 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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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
40 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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53 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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44 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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55 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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37 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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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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61 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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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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31 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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64 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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98 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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43 views

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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156 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
34 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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60 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 ...
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114 views

Why likelihood is not always a density function? [duplicate]

I try to self-learn Bayesian machine learning (mostly by studying Bishop and Kevin Murphy's books). While working with formulas I was puzzled by the quote that "Note that the likelihood function is ...
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What methodology should I choose? If hiearchical, what model design?

I am working on a problem that has can beyond my level of understanding. I am quite familiar with R, so that would be my preferred choice but I also have access to SAS. Data I have created a fake ...
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116 views

Bayes Rule. Given I flipped 10 heads in a row, what is likelihood I own a double headed coin?

$$\Pr(A | B) = \frac{\Pr(B | A) \Pr(A)}{\Pr(B)}$$ So my $\Pr(A)$ is probability that a coin is double headed. For the sake of argument let's say 1 in 10,000 coins are double headed. My $\Pr(B)$ is ...
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PyMC Robust Linear Regression with Measured Uncertainties

I posted this same question on Stack Overflow. I use least squares regression of data with measured errors in both x and y and use the reduced chi-square (mean square weighted deviation: mswd) as a ...
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131 views

Intuitive explanation of Bayesian logistic regression?

I'm looking for an intuitive explanation of Bayesian Logistic Regression (I'm using it for texts if that's relevant). It seems that this article presents it, but it's, uh, way too mathy. Thanks!
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36 views

Updating probabilities given sample

Simplified version of our problem. Assume each person can be one of three categories. The true distribution of the entire population, as a whole, of being in the i'th category is (10%, 20%, 30%, ...
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54 views

Conditional probability and Bayes theorem

Precision Tool Company owns a five-year-old truck. After careful consideration, management has decided that there is a one in five chance that the truck will have to have major repairs within the next ...
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36 views

Application of Bayes rule - relation between a pair of words

I do not have a statistical background but I'm trying to understand bayes theorem using a practical example. In my example I plan on using bayes theorem to calculate how often two words occur next to ...
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44 views

Bayes factor for selecting between two beta-distributions

I have two beta-distributions: $H_1 = Beta(\alpha_1, \beta_1) $ and $H_2 = Beta(\alpha_2, \beta_2) $ (parameters are known), and I'd like to estimate whether a new sample $D$ rather comes from ...
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38 views

Bayes rule and conditional independence

I have two conditionally independent random variables $A$, $B$ such that $$ P(A,B\mid C) = P(A\mid C)P(B\mid C) . $$ I have to find posterior formula $P(C \mid A,B)$. My result with a ...
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111 views

Confused about why we would use expected value instead of MLE when estimating some parameter

I have a conceptual confusion about the use of the expected value of a distribution. Often, we want to estimate the most likely value of something. For example, I have X= ten observations. I know X ...
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2answers
80 views

Bayes theorem an experiment design

Let's say I design an experiment to test the hypothesis, "Agent Z causes cancer". Of $N$ people in the experiment group, $x$ get cancer. Of $N$ people in the control group, $y$ get cancer. The ...
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97 views

Computing a Gaussian posterior from a Gaussian prior and likelihood function in R

I'm new to both R and Bayesian statistics, and I have a problem where I have a normally distributed prior that elicits a mean and standard deviation. The introduced likelihood function is also ...