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

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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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32 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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43 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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41 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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25 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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13 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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11 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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43 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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29 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
55 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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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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80 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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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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49 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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31 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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47 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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109 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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1answer
107 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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121 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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34 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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53 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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34 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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42 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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37 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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88 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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75 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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63 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 ...
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63 views

Problem with estimating probability using the multivariate Gaussian

I'm working on a problem where I need to find the probability of a given data sample belonging to a give class using the Bayes Theorem for classification. From everything I've been able to find so ...
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76 views

Bayes in English

I am not a statistician or mathematician but am trying to learn. My question: In Bayes Theorem, $p(C|X)=p(X|C)p(C)/p(X)$, what are the English terms for $p(X|C)$ and $p(C)/p(X)$? In other words, is ...
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69 views

What is the empirical Bayes estimator for a gamma-Poisson with more than 1 observation for each Poisson parameter?

I am looking at the Wikipedia entry for empirical Bayes, but it's a bit confusing - it seems to me the solution must apply only to the case in which there's only $n=1$ sample $y$ for each $\theta$ and ...
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Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...
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35 views

Performance comparsion study on real data in MATLAB (machine learning)

I need to compare performance using 2-class classifiers–an LDA classifier and Exact Bayes in MATLAB. I have to use this dataset. Can anyone give me any advice how to do that (at least the steps of the ...
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I have a problem with bayes and train lines

I have this problem which I can't fully understand it: Assume you have entered in a foreign city of unknown size. At entrance you see a tramcar with number $100$. Let us assume that the tramcar ...
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Variational inference engines

After doing some research on the topic, I have noticed a surprising deficit of inference packages and libraries that rely on message-passing or optimization methods for Python and R. To the best of ...
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67 views

What is the difference between these two probabilities?

I am working on a word sense disambiguation task and I have a dataset consisting of labelled sentences for two meanings of the word 'line'. I am trying to find 'trigger' words that are good at ...
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Does smoothing of Bayes classifier will increase precision?

I have implemented Bayes multinominal and Bernoulli's model and my question is does the smoothing have any impact of the performance of both models (Laplace’s law of succession or add one smoothing)?
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Calculation of a time until the absorbing state given chemicals are initially separate

Biochemist from Siberia has just patented a new super genome formed by amalgamating genome X and genome Y. These genomes are harmless when separate, but when mixed there is a 45% chance of an ...
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48 views

Solve discriminant functions to get decision boundary in R

I am trying to solve the discriminant functions in R for two, 2-D gaussian distributions and trying to find the decision boundary. The discriminant function is given by, $g(x) = -.5x'\sigma^{-1}x + ...
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55 views

Is it possible for a bayesian model to “forget” about some points?

Say you are doing some online bayesian inference over observations. You want to infer $\mu, \sigma$ for $X$, with a model $X \sim \mathcal{N}(\mu, \sigma)$. Now everytime you observe a $X_i$, you ...
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65 views

Expected value of posterior vs. success probability

Context Suppose I have two models, $H_1$ and $H_2$ for which I know the prior probabilities $p(H_1)$ and $p(H_2)$. Furthermore, I know the class-conditional distributions $p(x|H_1)$ and $p(x|H_2)$ of ...
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78 views

Randomly given a coin from an unfair set of 6 coins, how can I determine which coin I was given?

Suppose I have 6 different coins that have wildly different increasing chances of flipping a heads, and I know the chances of flipping a heads for each. (ie. Coin 1: 1/280 chance of heads ... Coin 6: ...
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39 views

Applications of bayesian inference to external ballistics?

I'm reading "The theory that would not die" by Sharon Bertsch McGrayne (Fine book. Strongly recommended to everyone). The author says that bayesian inference has been used for ballistics applications ...
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170 views

Bayes factor calculation

I am a newbie to Bayesian stats...I came across an article for calculating Bayes factor by rounder etal.. In this article how am I supposed to put t value..n scale r on effect size parameters. If you ...
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429 views

Laplace smoothing and Dirichlet prior

On the wikipedia article of Laplace smoothing (or additive smoothing), it is said that from a Bayesian point of view, this corresponds to the expected value of the posterior distribution, using a ...
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1answer
55 views

Bayesian references [duplicate]

Any good places to start getting into Bayesian statistics? I'm a graduate student in social sciences, with a decent amount of stats classes under my belt, but I'm far from fluent. Any references would ...
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2answers
142 views

Long-run behavior in coin tossing experiment

I have read the following information on this site and I am not sure if it is actually true (or if I misinterpret it): Events that are random are not perfectly predictable, but they have long-term ...
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97 views

The Conjugate Beta Prior proof

Hello. I'm having a problem with trying to figure out this proof that shows the beta distribution is conjugate to the binomial distribution (picture attached). I understand it until the third row, ...