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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22 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
16 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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9 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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47 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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1answer
31 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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10 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
37 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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43 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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26 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
62 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
43 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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6 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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1answer
48 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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16 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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38 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
49 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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0answers
25 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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0answers
33 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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31 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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1answer
27 views

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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28 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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23 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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7 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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16 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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20 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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1answer
55 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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50 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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23 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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7 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
43 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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33 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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4answers
66 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
36 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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40 views

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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1answer
29 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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1answer
53 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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0answers
26 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 ...
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2answers
80 views

A house is guarded by two alarms

I am trying to wrap my head around the following problem A house is guarded by two alarms. If Alarm 1 fires, p(theft) = 80% If Alarm 2 fires, p(theft) = 70% If both alarms fire at the same time, ...
2
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1answer
35 views

Bayes - dual estimation of parameter value and parameter growth

I am trying to find an bayesian approach to the following problem: Image a bucket with 100 white balls and an unknown number of red balls During each year, one can take a sample with replacement of ...
2
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1answer
20 views

Bayesian statistics, bivariate prior distribution

I've got a simple question buy I'm not sure how to solve it. It's a bit long. Suppose you've got $n$ iid random variables $X_1$, $\dots$, $X_n$ from the normal distribution with unknown mean $M$ and ...
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1answer
21 views

Bayes with conditional independence

I have a problem that I can't work out I've two conditional independent A,B such as $P(A,B|C) = P(A|C)P(B|C)$ Now I've to find posterior formula for: $P(C | A,B)$, now what I got was pretty ...
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1answer
22 views

Improper Prior Distribution

What is the clear mathematics definition about improper prior distributions? Can you give me some book or article links about it?
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1answer
23 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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1answer
21 views

What is this distribution formulated with w, m and sum sign?

I have a binary classification problem, part of which is defined as follows : p(x|y=1) $\sim w (m_1 , \sum_1$) and p(x|y=0) $\sim w (m_0 , \sum_0$) Where $\sum_1$ is a covariance matrix : $$ ...
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0answers
55 views

Bayesian sequential updates of normally distributed variables

Suppose that you can observe data that are independently and identically distributed as $N(\mu, 1)$. Your prior distribution for $\mu$ is $N(m, v)$. After observing $n_1$ data with sample mean ...
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22 views

Bayesian estimate for unfair die

Suppose you have a six-sided die that you suspect is not fair and toss it N times. What would be a Bayesian approach to estimating the probability of the six outcomes given that you suspect the die ...
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1answer
42 views

Why a beta distribution with the parameters $\alpha=0$ and $\beta=0$ as a prior is bad

what happened if I define a beta distribution with $\alpha=0$ and $\beta=0$ as a prior? in other words if $p(\theta) \varpropto \frac{1}{\theta(1-\theta)}$. Thanks
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51 views

Marginal and conditional probability table without joint probability table

I've a Bayesian network, with discrete node values: for every node I've the conditional probability table $p(A|B)$, where $A$ is the node itself and $B$ is the set of the parents nodes. Now I would ...
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2answers
49 views

Bayesian learning for input “If A, then B.”

Can anyone point me to literature on Bayesian learning when the new information has the form “If A, then B”? I’m familiar with the rule that after one learns X, posterior probability P(Y) equals prior ...
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7 views

$X_1,\ldots,X_8$ come from a Pareto with $\beta=1$. $\alpha$ has a prior Gamma($A$,$B$). Find posterior distribution

Can anyone confirm that it would be a Gamma($A+8,B$). I got; $$\left[\alpha^8 x^{-8(\alpha+1)}\right] \left[\alpha^{A-1}e^{-\alpha \beta}\right]$$ Which is proportional to, ...