The approach and interpretation of probability associated with Bayes theorem; usually used as opposed to the frequentist approach. It can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain. A Bayesian probabilist starts with some prior ...

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Bayesian network and unknown probability

I'm trying to solve questions regarding bayesian network, and now I was wondering if it is possible to know the probability of an unknown variable in the tree. For instance, I have this tree, ...
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8 views

Why are A->C<-B conditionally dependent in a directed graph?

$P(A,B,C) = P(A)P(B)P(C|A,B)$. I understand how $A,B$ are marginally independent on $C$, but I'm confused as to how the $A, B$ are conditionally dependent on $C$. $P(A,B|C) = ...
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35 views

Comparing models to smoothed data

I am attempting to fit a model to a noisy data set. I am performing this modeling in two stages - first, smoothing it out by fitting an analytic mixture model to it, and second, fitting my final model ...
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1answer
37 views

Help applying Bayes' Law

my problem is the following: Lets imagine we have a computer with 3 memories (m1, m2, m3). When data is needed it is searched if m1, if not found in m1, it is searched in m2 and so on. P(finding ...
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37 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
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35 views

Bayesian Chain rule

I am going thorugh a Naive Bayes Classifier, and faced the following: $p(y|a,b,c) = \frac{p(a|y,b)*p(y|c)}{p(a|b,c)}$ When I am trying to derive the above, these are my steps: ...
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44 views

Bayesian Estimate Problem

So... I'll be honest, I don't know anything about anything Bayesian, this problem being no exception (from the Society of Actuaries' Exam C sample questions): You are given: (i) The annual ...
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1answer
51 views

Bayesian Parameter Estimation - Notation in Terms of Probability Spaces

As far as I know, random variables are functions form a probability space $(\Omega,\mathcal{A},\mu)$ to real numbers $\mathbb{R}$, i.e. $X:\Omega\to\mathbb{R}$. Let a probability density function ...
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14 views

What should I be learning to combine Bayesian networks with measurement variables?

I've been reading up on Bayesian networks recently and maybe I'm missing something about the intuition. I don't know if I've picked the correct tags for this question, so I apologize in advance. The ...
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1answer
17 views

Conditional PDF Inference

I am attempting to create an inference model, such that given any $y$, I can output an estimated probability density function of $x$. Given $X,Y$ where $f_X$ and $f_Y$ are probability density ...
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54 views

Conditional probability and bayes theorem problem involving a medical test

I have a test that checks if a patient is sick (E = {patient is sick}) and gives either a positive (A={result is positive}) or a negative result. Given that $P(A|E) = 0.95 = P(A^c | E^c)$ and $P(E) = ...
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27 views

Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
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37 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...
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1answer
69 views

How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
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1answer
25 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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11 views

Integration of Bayes-factors from multiple tests

I have been using an Bayesian-centric R package for some genomics analysis to detect mutations in 3 individuals from the same family. I have to do each analysis for each individual separately due to ...
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57 views

Is my interpretation of Bayesian probability and inference correct?

I have the following interpretation of the Bayesian probability and inference (without referring to Measure Theory, I am still at the very beginning of learning it): Let's say we have five random ...
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69 views

Derive/ prove: p(a,b|c) = p(a|b,c).p(b|c)

How can this expression be derived? p(a,b|c) = p(a|b,c).p(b|c) where a,b,c are random variables. UPDATE: from the following ...
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1answer
33 views

Bayesian Inference Problem

We have a Bayesian Network that A to D is Boolean variable. we want to calculate the probability which C and D be True and A be false. my answer sheet calculate the last result and is 0.0424. any ...
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1answer
44 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
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59 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
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84 views

Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
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2answers
66 views

rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
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2answers
47 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...
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9 views

Asymptotic coincidence of the MAP and MMSE estimator

In many works, simulations show that as number of samples increases, the mean-square-error (MSE) of the MAP estimator attains the minimum MSE. Where can I find a theoretical proof to these empirical ...
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64 views

Strange conditional probability problem

Not sure if this problem even makes sense, but anyway: Lets say you have a button which switches on a light. The light lights green with probability $p$ and red with probability $1-p$. If you push ...
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20 views

Assessing goodness of fit in Bayesian framework

I am following a Bayesian approach (specifying an underlying class of models and a prior) in order to produce a predictive distribution of some quantity. The question I am troubled with is: how can I ...
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45 views

Bayesian probability re: people vs. coins

Imagine you're a court clerk recording information about court appearances in a munincipal court. Past records show that on a typical day in this court, in 50% of criminal cases heard, the accused ...
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3answers
58 views

Can this conditional probability be answered using Bayesian Theorem (or at all) with the information given

I have a conditional probability problem I'm unsure can be answered given the information I have - as such I'm unsure if Bayesian Theorem is the way to answer it, or if the answer is staring at me in ...
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29 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
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35 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
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32 views

Change of Variables for two level Guassian model

I have a multivariate Gaussian distribution from which two variables, u and v, are drawn. The next variables, U and V, are U = 1/(u^2+5) + N(0,sig_U) and V = v^3 + N(0,sig_V). U and V are known, ...
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Need help with P(D) in a Bayesian model

So I've been reading about Bayesian models so I tried I'd have a toy example I could play with. Consider the following: You are at a bus stop and you observe the bus arriving at various times $t_1, ...
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33 views

Countable Baye's theorem?

Disclaimer: If this is a foolish question, I'm sorry.. this is the first time I've looked at probability theory in very many years, and have begun to re-read everything from scratch... Question: If ...
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29 views

Given the data set is the Bayesian estimation the best solution for solving the expected value?

I am very new to this. I have several measurements that from which I need to estimate a truth value. Each of them comes with an estimated error. I know that the observation error are biased (I don't ...
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12 views

What is a gibbs predictor

I was reading an article and I don't know what a gibbs predictor is. What is it guys? Any suggestions where I should look guys. I would really appreciate some help
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19 views

Simultaneous multiple perturbations in Markov chain Monte Carlo

I'm coding a McMC algorithm for geophysical applications. Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be ...
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30 views

Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
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77 views

Jaynes' taxicab problem

I am currently reading Jaynes' Probability Theory, The Logic of Science and am still trying to absorb everything. On page 190, he poses the following intriguing question, paraphrased here. Suppose ...
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69 views

Game Theory - Bayes Rule, Sequential Game

I am trying to solve the following model, but I get a few weird results. Sorry if it is too long... Nature moves first and with probability $p$ assigns player's 1 type to be High ($1-p$ for Low) ...
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28 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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12 views

Gaussian Mixture Model as Graphical Model

Could anyone show me a simple gaussian mixture model as graphical model (bayesian network) and explain to me the joint distributions? This is a question of an exam I am learning for. I basically know ...
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2answers
61 views

Betting: Gambler's Fallacy vs. Law of Large Numbers

I know this has been asked before, but I think not in this exact way, so here goes: Suppose you're going to bet on the flip of a coin. Your bet is always "HEADS", but the amount of your bet may vary, ...
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39 views

How would I analyze the accuracy of a model that predicts World Cup matches?

Say, someone made a bunch of predictions for each game between Team A and Team B, such that there's a predicted probability for each of the three possible outcomes adding up to $1.0$ : Team A winning, ...
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80 views

Bayes, two tests in a row

I came up with a standard Bayesian example as to point out my confusion. There is an epidemic. A person has a probability $\frac{1}{100}$ to have the disease. The authorities decide to test the ...
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1answer
41 views

Confusion in Posterior Probability Calculation

I know posterior probability as, $P(\theta|x)= [(P(x|\theta)*(P(\theta))/(P(x))]$, as given in http://en.wikipedia.org/wiki/Posterior_probability I am slightly confused with the term ...
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So I have the following question, dont have much info on class notes and not sure how to tackle it, any suggestions, any help?

A seller has a single item for sale (which she values at zero). There are two potential buyers. The seller decides to use the following auction format to sell the object: each bidder submits a sealed ...
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42 views

Bayes - bias of a coin

struggling with a basic question on the bias of a coin. Assume that i believe, as prior, that a coin is 40% probable to be fair and 60% probable to be unfair, with the estimated prior bias following a ...
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61 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq ...
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31 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...