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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Probability distribution for a digit of a number

If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
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92 views

hint with Bayes rule problem

The pirate Captain Queequeg has a lazy crew and suspects they are planning to stage a mutiny. Captain Queequeg's solution is to have every member of the crew roll Queequeg's lucky die. If the roll is ...
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24 views

Bayesian Parameter Estimation Doubt

I was going through a pattern recognition book and in the chapter of Bayesian Parameter Estimation I came across this formula. I cannot understand how the 2nd line is derived from the first line. ...
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226 views

Bayesian networks and joint probabilities

This is my problem My problem is modeled by a basic Bayesian Network with only two layers. So I have parent and child nodes but the children has no children. Essentially a bipartite graph. The ...
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24 views

Naive Bayesian Classifier for Object with Variable attributes

Let say our objects are connected graphs. They are to be classified into two categories, say A and B. However, for our purpose attributes for each graph is equal to the number of vertex of the graph ...
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68 views

Kolmogorov's paper defining Bayesian sufficiency

I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
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72 views

Bayes theorem for calculation with personal probabilities

I'm completely stuck on some homework I have and can't figure it out. The task is to calculate the probability of a bus being late conditional on the weather being snowy and bus driver being ...
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105 views

Posterior density and posterior moments

I would be very grateful to get some help with the following problem. Let $X_1, ..., X_n$ be independent and uniformly distributed on the interval $(0,\theta)$ with $\theta>0$. Let the prior ...
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154 views

Calculating Probabilities for Substitution Ciphers using Frequency Analysis

I have been trying to put together a tool that can take in cipher text encrypted via a simple substitution cipher and calculate the most likely "key" (that is, how the plain text letters were mapped ...
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51 views

Bayesian learning

Imagine we assume there are two different types of coins: Coin A: a fair coin, p(heads) = 0.5. Coin B: biased to heads at p(heads)=0.7. We then want to learn from samples which coin we are ...
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74 views

Hypothesis Testing Bayesian Way

I'm having trouble with the following problem: Suppose a machine is composed of 2 components (1 and 2, independent from each other). Each component has a exponential failure probability distribution ...
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55 views

Laplace approximation of the likelihood Bayesian

I need help with the following question: Consider m observations $(y_1; n_1); ... ; (y_m; n_m)$, where $y_i \sim Bin(n_i; θ_i)$ are binomial variables. Assume that $θ_i \sim w_1Beta(α_1; β_1) + ...
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217 views

Find the posterior distribution of $\theta$

I'm having trouble solving the following problem: Find the posterior distribution of $\theta | x$. Suppose $x$ is a random variable with distribution $f(x) = \theta x^{\theta - 1}$, you observe a ...
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96 views

Find posterior mean

I have this problem, Let $X\sim U(0,\theta)$ with $\theta>0$. Assume a signal random sample $X$, the squared error loss, and the prior $\pi(\theta) = \exp(1)$ i.e. $\pi(\theta) = \theta ...
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122 views

Find the posterior distribution and posterior risk

I have this problem, Let $X\sim U(0,\theta)$ with $\theta>0$. Assume a signal random sample $X$, the squared error loss, and the prior $\pi(\theta) = \exp(1)$ i.e. $\pi(\theta) = \theta ...
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86 views

Bayesian formula for weather exercise

If it is nice weather on one day, the probability that it is going to be nice again the next is $13/15$. If it is raining on one day, the prob. that it is going to be raining again the next day is ...
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149 views

Bernoulli trials conditional probability

Let $\Omega=\{0,1\}^\infty$ and $S_n=X_1+\cdots+X_n$ the number of “successes” or “arrivals” in $n$ steps. $p\in(0,1)$ and $\mathbb P(S_n=k)=\binom{n}{k}p^k(1-p)^{n-k}$ Let $T$ be the time until the ...
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18 views

Conditional distribution for a label given a scalar feature

I am trying to create a simple simulation setup for classifiers on toy data. Each data point can has a scalar feature $X$, which is uniformly distributed between -1 and 1. Depending on the feature, ...
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1answer
88 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
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58 views

Question on Joint Posterior

Likelihood: $f(x^T, n^T|\theta^T) = \prod_{i=1}^{30} \binom{n^T_i}{x^T_i}{\theta^T}^{x^T_i}{(1-\theta^T)}^{n^T_i-x^T_i}$ Prior: $ log(\frac{\theta^T}{1-\theta^T})\sim N(\mu_T,\sigma_T^2) $ I am ...
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231 views

Using Bayes Theorem intuitively without equation (tree-diagrams)

I am working on the following question and I am having some difficulty. The thing is I understand that I must apply Bayes Theorem but to be honest, I like to do problems using Bayes Theorem ...
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126 views

Difference between true positive and true negative in this case

A patient takes a lab test and the result comes back positive. It is known that the test returns a correct positive result in only 98% of the cases and a correct negative result in only 97% of the ...
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67 views

What is the distribution of parameter using Bayesian inference

I am trying something very simple to try to get more "inside" Bayesian statistics but the results are a bit odd. Lets say I have an infinite bowl, inside of which are black and white balls. We do not ...
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1answer
144 views

What's the difference between Maximum a posteriori and Bayes' rule?

What's the difference between Maximum a posteriori and Bayes' rule? They look similar, except that I do understand Bayes' rule and I don't understand MAP. The people I asked - who work in math and ...
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128 views

question related to Bayes' rule and Bays' risk.

Let $X_1, X_2, X_3, \ldots, X_n$ be a random sample for $N(e,1)$. Let the prior p.d.f. of $e$ be $N(0,\sigma^2)$ under the square error loss function $L(e,d)={(d-e)}^2$. Find the Bayes' decision rule ...
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73 views

Is there a formal explanation of the concept of “improper prior” in Bayesian statistics?

The Bayesian concept of "improper prior" seems to be surrounded with magic. Even formal, Bayesian-oriented books, such as Schervish's "Theory of Statistics", treat it with the heuristic hand waving ...
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137 views

Bayes estimator $p_\theta(x) = \theta x^{\theta -1} 1_{[0,1]}(x)$ and $\pi(\theta) = e^{-\theta}$

Let $X_1,\cdots,X_n$ iid with density $p_\theta(x) = \theta x^{\theta -1} 1_{[0,1]}(x)$ and $\pi(\theta) = e^{-\theta}$ as prior on the parameterspace $\Theta$. I have to calculate the posterior ...
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85 views

Mathematical notation for probability trees and their usage

A commonly used tool for visualising and solving Conditional Probability problems is the tree diagram of events and their associated probabilities. (Tree Diagram). How can one represent particular ...
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63 views

Estimating probabilities using Bayes rule?

I am working on a past exam paper. I am given a data set as follows: Hair: {brown, red} = $\{B,R\}$ Height: {tall, short} = $\{T,S\}$ Country: {UK, Italy} = $\{U,I\}$ Our sample is: ...
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217 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
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115 views

Conditional probability given two variables from conditional probabilities of one variable

I have a question which comes from an example of a textbook, but all I am concerned with is how we go from having probabilities $P(X|A)$ and $P(X|B)$ to having $P(X|A,B)$. In the example events $A$ ...
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61 views

Queries on the Bayesian method

Currently I am working on a bayesian model framework and have questions related to the philosophy of using such techniques of modeling. 1. How do I know that the prior which I have captured from the ...
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1answer
38 views

Independent random variables considering expression

Having $x, y, z, c \in \mathbb{R}$, is it valid to say: $c \propto g(x, z) h(y, z)$ The context here is to say whether or not the random variables $X$ and $Y$ are independent given the value of $Z$. ...
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145 views

Easy Probability Problem

I was told the following probability problem: While doing a math problem today at the contest the probability of Annie, Tom and Karen getting the problem correct first is 1/7, 1/2, and 5/14 ...
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125 views

Separate expression $c(x + y)^2 e^{yz}$

Considering the following expression, with $x, y, z, c \in \mathbb{R}$, is such expression separable into $f(x, y, z) = g(x, z)h(y, z)$? $c(x + y)^2 e^{yz}$ If not, why isn't it separable? Note: ...
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63 views

Does $\hat{x}$ always mean normalized version of a vector $x$?

From this article: "...a maximum-a-posteriori $(MAP_{x,k}^{\,\,\,\,\,1})$ estimation, seeking a pair $(\hat{x}, \hat{k})$ maximizing: $$p(x, k\mid y) \propto p(y|x, k)p(x)p(k).$$ Are ...
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210 views

bayesian estimation on the poisson distribution

Suppose X~Poisson($\lambda$) and $\lambda$~Gamma($\alpha,\beta$). Find the posterior distribution and the Bayesian estimator of $\lambda$. Thus the prior distribution is: ...
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Bayesian Problem… I think

Let X be the number of coin tosses until heads is obtained. Without knowing that the coin is fair, I assume that the probability of heads is uniformly distributed. How would I find the distribution ...
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312 views

Statistics: Finding posterior distribution given prior distribution & R.Vs distribution

I'm now learning Bayesian inference.This is one of the questions I'm doing. Suppose we have R.V.s $X_1,X_2,\ldots,X_n$ each have an Exponential distribution with parameter $\theta$. and prior for ...
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How to make this inference: Degree of a node in a graph is significantly diffenrent from poisson distribution

I am working on Gene-Gene interaction graphs. I build a graph by adding edges between genes (nodes) which show statistical interaction in predicting a quantitative parameter value (say, brain volume) ...
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220 views

Empirical Bayes estimator for a Beta-Binomial parameters

Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
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212 views

Bayes' Theorem: Detection of bomb in a box

There is a bomb that is equally likely to be in any one of three different boxes. Let $α_i$ be the probability that that the bomb will be found upon making a quick examination ( detection ) of box i ...
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106 views

Applying the Bayes theorem

a) A research institute associated with the Olympics claims that its drug test will detect steroid use (that is, show a positive result for an athletic who uses steroids) 95% of the time. Your friend ...
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115 views

What is the meaning of “mean-field”?

In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
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109 views

Relationship between Correlation and Bayes Theorem

Is there some relationship between the correlation of two random variables, and Bayes Theorem? A bit of background intuition, if W = random variable denoting number of women in a room, and L = ...
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187 views

How would you approach this problem on the Bayes theorem?

I've been reading a book on Statistics and I could COMPLETELY understand all of its text. It basically explained the bayes theorem and what priors were, what posteriors were etc. But then in the ...
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Why is $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$?

Could anyone derive or explain why the formula $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$ is true? I understand conditional probability definition, but this formula confuses me and makes my head hurt x) Here's ...
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221 views

Bayes Estimator

Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf: $f_{X}(x|\alpha,\beta) = ...
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29 views

Coin tossing - Two tosses, one is a head, probability other is a tail? [duplicate]

A friend of mine tossed a fair coin twice. Suppose instead that I happen to see the result of one of his tosses, and it is a head. What is the probability that the other toss is tail?
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3k views

Coin toss - probability of a tail known that one is heads

A friend of mine tossed a fair coin twice. Suppose I ask him whether he got a head in the two tosses, and he says yes. What is the probability that one toss is tail? Now suppose instead that I happen ...