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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An example shows the difference between inference in Bayesian network and Junction Tree

Why inference in Junction tree is more efficient? There are directed graph BN and the corresponded undirected graph transformed by Junction tree algorithm. The literature describes that inference in ...
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33 views

Extended Bayes' theorem: p(A | B, C, D)

I'm having some difficulty understanding Bayes' theorem with multiple events. I'm trying to put together a Bayesian network. I have four independent probabilities but I have found that A, B and C can ...
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Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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22 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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18 views

Expectation of higher order moment (Bayesian Stats)

I am taking a course in Bayesian statistics, which is off my field. In the lecture notes the instructor showed E[X^2n] = (2n-1) σ^2n and E[X^3 . Y] = E[Y^3 . X] = 3.ρ.σ^4 where σ = variance E = ...
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how can I Find a 95% credible interval for p using the Bayesian method with the uniform distribution as a prior for p?

When I have a RV X~Geom(p): $x\ Frequency\\ 1 7459\\2 1930\\ 3\ 463\\ 4\ 117\\ 5\ 22\\ 6\ 6\\ 7\ 2\\ 9\ 1$ This is what I am trying to do: Since p is a probability, I say that $ p\sim U[0,1]$ An ...
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29 views

Markov chain: if $X\rightarrow Y\rightarrow Z$, then why is $Z\rightarrow Y\rightarrow X$ true?

in a Markov chain, given three random variables $X,Y,Z$, we have $X\rightarrow Y\rightarrow Z$, which means $p(x,y,z) = p(x)p(y|x)p(z|y)$. The right arrow symbol $\rightarrow$ is used to denote a ...
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In what sense is the Bayesian posterior mean a “convex combination”?

This is related to a previous question that hasn't gotten an answer: Definition of convex combination with matrix-vector multiplication Suppose I want to estimate $x \in \mathbb{R}^n$ from two ...
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23 views

Likelihood of a function of different types of random variables

Is there a general way of expressing the likelihood of some known, but non-trivial function of several random varaibles. For example, suppose that we need to calculate the parameters of a process ...
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36 views

Bayesian Shrinkage Factor

Vasicek(1973), referenced in this paper(See bottom of page 16) explains a method of shrinking individual betas $\beta^{TS}$ toward a cross-sectional mean $\beta^{XS}$ as follows: for each time $t$, ...
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17 views

From Bayes factor to posterior probability?

Statisticians I and II assign prior probabilities $1/2$ and $1/4$, respectively, to $H_0$. Their analyses both yield $BF_{H_0} = 100$. Whose "posterior degree of belief in the truth of $H_0$" is ...
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27 views

Estimate the Measurement Bias of Coin Flip Classifier

Suppose that instead of observing coin tosses directly, you measure the outcome using an instrument that is not always correct. Specifically, suppose there is a probability X that an actual heads is ...
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24 views

Solution to a modified version The Locomotive Problem [closed]

A railroad numbers its locomotives in order 1..N. One day you see a locomotive with the number 60. Estimate how many locomotives the railroad has. Using the Likelihood Approach: Assume prior is ...
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54 views

What is the likelihood of a fair coin given 10 heads (with added component)?

What is the likelihood of a fair coin given that it has landed heads up 10 times? You have a fair coin or a double-headed coin... $\mathsf P($Fair$\mid 10$ heads$) = ...
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how to calculate unknown probabilities in the bayesian network

I am working on a bayesian network problem. I read from one of the website the following network. My problem : "as soon as the cold water becomes low, you have at least a 94% chance of a high ...
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Solve for mean and std deviation of new normal distribution

There are normal distributions with known means and standard deviations. The first distribution is a Bayesian prior distribution with known mean1 and known SD1. The second distribution is a Bayesian ...
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Bayesian Uni-variable ou multi-variable and formulation

I have a parameter that has a prior distribution with mean equals to 30, a variance of 25 and a number of samples $n=30$. I was able to execute 30 more samples, and I got a mean of 25 and a variance ...
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20 views

Why are these variables not conditionally dependent given 'active triplets' and the 'explaining away' effect?

I'm following the Udacity Intro to AI course. This quiz gives the following Bayes network and asks whether different variables are conditionally independent or not. (The explanation of the nodes, ...
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22 views

Improper Lebesgue prior normalization in Bayesian filtering

Suppose we have a conditionally Gaussian Linear State Space Model (CGLSSM) where $Y_t=(X_t,S_t)_{t \in \mathbb{N}}$ is the Markov chain of hidden states, where for each $t \in \mathbb{N}$, $S_t \in ...
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Bayesian Network vs Markov Decision Process

I am wondering if somebody can tell me anything about the practical differences between using Markov Decision Processes and and Bayesian Networks in reasoning about probabilistic processes?
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55 views

Optimal Stopping for 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. I have a bandit problem with one ...
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74 views

Bayes spam filtering

To analyze the words that appear in spam emails, you collect a sample of 1000 emails marked as spam and 1000 emails marked as non-spam. Of the 1000 spam emails, 210 contained the phrase This isn't ...
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About matching.

A population of $n$ people, each having $m$ "features" $f_1,f_2, \dots ,f_m$ (for instance, where they live, brand of milk they consume, annual profit, etc...). Now, let $f_1$ a basic feature. Is it ...
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51 views

Verification of a Bayes Theorem related problem

This was an assignment problem given to me by the professor. I have did it (not sure if its correct). My answer is around $2/10^{12}$. I fear this is wrong. Can someone try this and verify if ...
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20 views

Posterior probability of a $p$ borderline result

Suppose that we throw 400 coins and found a result 0f 220 tails. Using a simple test of the null hypothesis that $H_0: \pi = 1/2$, we get that the probability of such a result is very close to .05. ...
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32 views

Bayesian update from uniform prior to uniform posterior ?!?

I was working through a signaling game problem recently and the proof suggested the following: Actor A has a type: $\ \mathscr{t} \sim Uniform[-1,1]$ Actor A gives signal $\pi^*$ that perfectly ...
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45 views

Bayes' Net Conditional Probability

I have a Bayes' Net with 4 boolean nodes connected in a diamond shape. I want to find the probability of one of the middle nodes being true given that the ones above and below are both true. So ...
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1answer
13 views

partily undirected Bayesian Network

I am designing a Dynamic Bayesian Network, but I am a little confused about some definition of DBN and markov network. In my network ,the edges from the hidden nodes of last frame to the current frame ...
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37 views

Between bayesian and measure theoretic approaches

I was wondering how a bayesian statistician would approach the problem of defining a probability density function for a random variable. In a measure theoretic sense, If the distribution of the ...
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32 views

Bayesian Network/ Number of parameters

Please consider the following Bayesian Network out of $Graphical Models in Applied Multivariate Statistics" by Joe Whittaker: Now the factorization property says that the joint probability ...
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28 views

Conditional Posterior Distribution Based on Two Simultaneous Signals

I am trapped by such a problem. Assume the state variable $\theta$ is (prior) normally distributed $N(\eta, \sigma^{2}_{0})$. Now we have two independent signals about $\theta$. Signal 1 is ...
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21 views

Finding the MLE estimates of a beta, binomial hierarchical model

Consider $M$ observations ($x_i$, $n_i$) where $x_i$ is a realisation from $X_i \sim \mbox{Binomial}(n_i,p_i)$ and $p_i$ is a realisation from $P_i \sim Beta(\alpha, \beta)$. I would like to find the ...
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36 views

How this integration is solved?

Can anyone explain how this integration has been performed? This is a Bayes estimator for uniform prior assuming quadratic loss function. Thanks in advance
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Probability of picking balls out of bins

Question: You have two bins with four different balls in each bin. Bin A: 2 White Balls and 2 Black Balls Bin B: 3 Black Balls and 1 White ball You cannot tell which bin contains what balls. Given ...
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30 views

Confusion about Notation for Bayesian Statistics

I'm currently trying to learn Bayesian Statistics but I keep losing time trying to figure out what exactly is meant by notation. Could someone answer the following for me? Let's say $X \sim ...
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47 views

About Bayesian formula and rating system

I'm building a scoring system with score from 0 to 5) and I would like to sort products according to the number of reviews and their scores. After some research on the Internet I have found two ...
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Recursive Bayesian Estimation, $p(C_k|x)$ as likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|x)$, from for example a logistic regression, to enhance ...
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32 views

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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1answer
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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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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35 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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31 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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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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50 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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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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44 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 ...