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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Finding P(S1 = 1 | D1=1, D2=1) (Bayes Networks)

I'm scratching my head over something that is probably simple Probabilities, but I guess I can't see it. Essentially, I have the following table: ------ ------ ------ ------ | D1 | D2 | S1=1 | ...
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21 views

proof that a density proportional to Gaussian is Gaussian

I try to develop bayesian estimation for one dimensional Gaussian with unknown $\mu$ and known $\sigma$. I got $$p(x\mid D) = \int p(x\mid\mu)p(\mu\mid D) \, d\mu =\int \frac{1}{\sigma ...
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32 views

Applying Bayes Rule to Cards

I was playing poker with a friend last night when a question occured to us. I had a two Jacks and the flop came out: King Queen and 4. So, suddenly my pocket Jacks are not so great, unless another ...
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47 views

Assistance with Bayesian Random Effects and Mixed Effects Models

I am looking to build either a random effects or mixed effects model for a project I am working on, but have had trouble finding good sources and understanding the general structure of the model. ...
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18 views

Bayesian Nets. No active path from X to Y, versus No inactive paths from X to Y

I am learning d-seperation in Bayes nets for my A.I. class. What this involves is considering all paths from some node X to Y (representing random variables) and seeing whether such paths are active ...
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39 views

What is the math behind calling election seats with confidence, before all votes have been counted?

On election night, predictions are made on the winner of each district, after only a fraction of the vote has been counted up. How is this done? Say there is a seat up for election, and 10,000 votes ...
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27 views

Bayesian posterior variance

Let $Var[\omega]$ be the variance of a population parameter $\omega$ prior to the collection of a random sample $\mathcal{X}=\left\lbrace X_1,X_2,\dots,X_n\right\rbrace$ from the population. Prove or ...
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20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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20 views

Bayesian versus Classical (frequentist) Statistics

Very often in text-books the comparison of Bayesian vs. Classical Statistics are presented upfront in a very abstract way. For example, in the current book I'm studying there's the following ...
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21 views

What does likelihood density f(y|theta)=5 imply?

I just dont understand what dose constant likelihood density imply, e.g. f(y|theta)=5? In addition, when I use this likelihood density to derive posterior function, it cancels out so my posterior ...
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37 views

This textbook on time series analysis says something wacky

This is from a discussion of analyzing a time series with a sinusoid + noise model. The troublesome statement is: ...data values near the beginning and end of a record are most important for ...
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1answer
30 views

probability matching strategy for coin flips

imagine a betting game where we observe $N$ independent coin flips $x_1,...,x_n$ (where each $x_i \in {H,T}$) from the same coin, whose true weight is $\theta$. the task is to predict how many Heads ...
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25 views

How can I infer order from partially ordered discrete sequences?

A really interesting problem that I can't stop thinking about! Have run in to this a couple of times but yet to find a smart approach to either solve or frame this problem. This is my try at ...
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18 views

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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1answer
46 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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1answer
15 views

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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8 views

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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37 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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25 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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13 views

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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32 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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29 views

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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1answer
36 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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37 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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22 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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28 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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60 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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1answer
16 views

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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1answer
14 views

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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10 views

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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25 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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1answer
15 views

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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56 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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1answer
80 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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27 views

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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59 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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22 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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40 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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1answer
48 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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45 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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1answer
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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1answer
29 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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1answer
34 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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37 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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49 views

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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36 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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55 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 ...