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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Computing evidence for least-squares fit

I'm at a loss trying to implement Bayesian model selection for standard least-squares polynomials fits. I have three polynomials of order $1$, $2$, and $3$, and a sequence of $(x,y)$ data points. ...
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160 views

Maximum Entropy Distribution When Mean and Variance are Not Fixed with Positive Support

I know when the mean and variance of $\ln x$ are both fixed, then the maximum entropy probability distribution is lognormal. When the mean of a random variable is fixed the MEPD is the exponential ...
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126 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
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28 views

Determine which parameter has correlation with result and which is not

sorry for probably silly question, it's the first time when I need to do such work. I have large data set with regarding clicks on some element on web page. It contains some characteristics of such ...
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2answers
113 views

Comparing uniform priors

The background of the problem is this: Assume that we have a parameter vector $\Theta$ which satisfies $\Theta^\prime\Theta=1$. If we let this vector have the uniform prior, the density of the prior ...
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1answer
72 views

Checking independence of variables in a Bayesian network

I need a little help with Bayesian Networks. Consider given the following network (all variables are binary) and we need to check conditional independence of $A$ and $C$ if $X$ and $Z$ are given. Any ...
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1answer
119 views

How to do Bayesian updating on biased information?

You have a coin that you can flip, but you can't see. It's a weighted $3$-sided coin taken (uniformly) randomly from some small known collection of $100$ weighted coins. However, we don't know how ...
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2k views

Bayes rule with multiple conditions

I am wondering how I would apply Bayes rule to expand an expression with multiple variables on either side of the conditioning bar. In another forum post, for example, I read that you could expand ...
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1answer
58 views

Implied prior with relationship $y=\text{arccot}(x)$

I'm trying to solve an exercise, which I think I have almost managed to solve but not quite. Any help would be appreciated! So, what we have is a vector which we obtain by norming the vector ...
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1answer
253 views

Find the Posterior distribution- prior: $exp(1)$, likelihood: $poisson(\lambda)($

I have a prior $\lambda \sim exp(1)$ and a likelihood $X \sim poisson(\lambda)$, and I observed in a sample of $n=5$ a mean of $3$. What is the posterior distribution of $\lambda$? Here is my ...
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1answer
220 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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20 views

Regular Conditional Bayesian Experiment

In "Elements of Bayesian Statistics" (1990), Florens, Mouchart and Rolin describe two basic forms of reduction of a Bayesian experiment: Marginalization and Conditioning (Ch. 1). I don't understand ...
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62 views

Estimating the radius of a circle

I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
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93 views

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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1answer
94 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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1answer
227 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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1answer
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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106 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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1answer
158 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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1answer
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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58 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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2answers
219 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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100 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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1answer
130 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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1answer
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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1answer
152 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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1answer
234 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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1answer
132 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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1answer
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
146 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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1answer
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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1answer
77 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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1answer
140 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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1answer
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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218 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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118 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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62 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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147 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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2answers
126 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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1answer
47 views

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 ...