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Questions tagged [bayesian]

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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Is $P(A,B|X)=P(A|X)P(B|X)$ if $A$, $B$ are conditionally independent?

Suppose I have two events $A$ and $B$ that are conditionally independent from each other. Is the following correct?$$P(A,B|X)=P(A|X)*P(B|X)$$ If so, what is this rule called?
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Bayes network probability question

I'm looking at this problem as I review for an exam and I would appreciate it if anyone can give me work/answers to Pr(c), Pr(b), Pr(b,c) and Pr(c,d) so I can check my solutions, and use the work to ...
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discrete bayesian

I have an exercise where I struggle to understand a below sentence: X (prior) is a discrete random variable that takes the value 1 with probability $ p \in \{0,1\} $ , and the -1 value with ...
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Same max entropy for diffirent priors

For a continuous distribution,$f$, we define the entropy with respect to a reference prior $f_{0}$ to be $$\epsilon(f)=\int \log(\frac{f(\theta)}{f_{0}(\theta)})f_{0} d\theta$$ For Lebesgue measure ...
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Detection Theory Test: ML, MAP, Bayes, Neyman Perason

I am looking at a couple of question based on the image below: Question Scenario The question then requires I select all thresholds (vertical lines) which could be used for Maximum Likelihood ...
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Help with gaussian algebra for bayesian inference

I want to better understand the step for calculating the message from the game factor $h_{g}$ down to the difference variable $d_g$ on the TrueSkill factor. Such message is shown in the Rasmussen's ...
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Probability - Similarity Rate in the random Signal

I have a random signal (vector) that consists of random variables varying between -2 and 2. I want to know how many similar patterns do I have in this random signal. In order to achieve this, I select ...
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Which loss function does the maximum likelihood estimator minimize?

I'm trying to understand Maximum Likelihood estimators in the context of general estimation theory. I know Bayesian estimator minimizes mean squared loss, MAP estimator minimizes all-or-nothing loss (...
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Put an expression under a normal form.

I am struggling with the following problem : I have the expression of $$\mathrm{log(p(\theta|y, \alpha, \beta)) = C - \frac \beta2 \|y-\Phi\theta\|^2} - \frac \alpha2 \|\theta\|^2 $$ with $C$ a ...
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Posterior cdf of bernoulli and uniform prior

Let X1, . . . , Xn i.i.d. Bernoulli(θ) with a uniform prior. Show that the posterior density of ψ = log(θ/(1 − θ)) is $$h(ψ|x) = \frac{Γ(n + 2)}{Γ(s + 1)Γ(n − s + 1)}(\frac{e^ψ}{1 + e^ ψ})...
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Derivative of log marginal likelihood

I am trying to differentiate the logarithm marginal likelihood resulting from Bayesian Linear Regression $\mathcal{L}$, which is a real-valued function, with respect to scalars $\alpha$ and $\beta$. $...
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Question using Bayes' theorem with marbles and rolling a die

A urn contains 10 white marbles and 20 blue marbles. You roll a die and pick the amount of marbles without replacement out depending on the number you roll on the die. Find the probability you roll a ...
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Conditional probability for Bayse rule.

So I have the following problem. There are 300 million people and 2 million of them are green. Say there are 10 people who are terrorists. 9 out of 10 of these terrorists are green. What is the ...
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If $X_i\sim N(0,\frac{1}{\theta})$, find $E\left(\frac{1}{\sum_{i=1}^n X_i^2 +2}\right)$

The initial question states that the $X \sim \mathcal{N}(0,\frac{1}{\theta})$, where $\theta$ follows an exponential distribution with parameter equal to 1. We are asked to derive the Bayesian ...
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Step in the dirichlet estimation using posterior of new x

In some notes I'm reading to show how to estimate a new value given a dirchlet distribution parameters, it says: P(x | D) = Integral of P(x|0, d)P(0|D) d0 then ...
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Bayesian Linear Regression

Currently I am trying to understand Bayesian linear regression and there are several things I dont understand. First of all we have $$ p(\beta,\sigma^2|y,X) = \frac{p(y|\beta, \sigma^2, X)p(\beta,...
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Conditional distribution of mean parameters in a Normal mixture

I'm solving an exercise from Casella's book, Introduction to Monte Carlo with R and he askes us to build a Gibbs sampler for the following mixture of Normals \begin{align} p\mathcal{N}(\mu_{1},\sigma^...
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Least mean Square Estimate issue?

I have the following question and I am struggling to find the right answer. The random variables $X, \Theta$ are described by a joint PDF which is uniform on the triangular set defined by the ...
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Derivation of prior for non-exchangeable Indian Buffet Process

Can anyone tell me how one derives to this probability mass function for any Z generated by the non-exchangeable IBP (*): $$ \mathbb{P}(Z) = \frac{\alpha^{K_+}}{\prod_{i=1}^N K_1^{(i)}!} \exp(-\alpha ...
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P-value vs. Bayesian statistics

Are there any theoretical considerations between $p$-value and the Bayesian statistics? I mean say, any theorem regarding both of these two concepts at a time.
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Probability given two symptoms calculate probability of disease

I'm a little bit lost with probability and using Bayes theorem in practise. Here's and exercise, I'm trying to figure out if I'm going in the right direction and if I read the exercise correctly (if i'...
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Range of LMS and LLMS estimator

I need help on how to prove or disprove the following: Suppose we know that a variable Y has a range [0,1]. Is it true that the LMS estimator of Y will also always have a range [0,1]? How about the ...
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Solving an integral equation (possibly Fredholm, 1st kind) containing quartic exponentials with Fourier Transforms

I've been reading an economics paper regarding rational inattention by Sims (link: https://www.sciencedirect.com/science/article/abs/pii/S0304393203000291) and have been trying to follow his steps in ...
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PMF with unknown n?

During an experiment trial of $n$ tossed coins I have $X_1 ∼ B(n, 0.5)$. It means I have a binomial r.v. that expresses the chances of success - landing heads $(x = 1)$ as $0.5$. When tossing again ...
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ordered data - probability distribution

I have n items ranked by k experts. I would like to aggregate the ranks using a Bayesian model. However, I do not know if there is a way to model the ranks. P.S.: You can assume that we have a ...
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Bayesian Model for measuring the agreement

Assume that there are k expert measuring some quality x with a number between zero and one, $x_i \in [0,1], i=1,2,...,k$. I would like to first know the aggregated quality of x with respect to the ...
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Difficulties in implementation of Bayesian information criterion (BIC) model selection criterion

I was reading some papers related to Bayesian model selection. The Bayesian information criterion (BIC) model selection criterion is given by $$\text{BIC}=-2\log f({\bf y}|\hat{\theta})+p\log n$$ ...
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Symmetry in Bayesian Hypothesis testing

Let two hypothesis be: $H_0: \rm PMF(\mu)$ with prior $(1-p)$ and $H_1: \rm PMF(\sigma)$ with prior $p$. Is it true that the probability of total error $ \Big( P_e= \rm type \:I \: error + type \: ...
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Poisson probability with Bayes?

The number of goals scored every month is a Poisson with lambda 5 : $$ P(X=x) = \frac{e^{-5}5^x}{x!} (x=0,1,2,3,4....) $$ What is the probability of at least 4 goals scored next month given two ...
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I'm having difficulty understanding this problem to do with statistical inference (specifically, Bayesian) in scientific investigation.

I'm over here from the philosophy page since a very similar question that I asked there a couple times wasn't ever properly answered and I think statisticians here might be able to provide a helpful ...
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How can I find the P(x |w1) and P(x |w0) from a graph

How can I find the P(x |w1) and P(x |w0) from a graph, with only the x axis labelled, I would like to find the values of the y axis, I was told to look at the probability density Function but that ...
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Bayes' rule revisited

I am currently attending a machine learning course and we are reviewing some probability theory which is of course fundamental for machine learning. Suppose you are working at a library. Some ...
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Finding posterior mean, median and mode

Consider the probability, $\theta$, that a randomly selected person in NYC is from Manhattan. We collect data as we walk down Times Square. We ask three people whether they are from Manhattan, and all ...
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Variable Elimination in Dynamic Bayesian Networks

Suppose I have a Dynamic Bayesian Network and I want to observe the evolution of one variable, $v$ in the DBN. This variable has parents, that has parents and so on. Each of their probability ...
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What is the Bayesian prior predictive distribution?

The question goes as follows: A shoe factory produces brown shoes and black shoes. They look the same but differ only in their weight characteristics. Brown shoes have their weight distributed as ...
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How to use Bayesian Inference for a large set of data?

I have a set of large data and need to come up with a way to quantify correlation. I am thinking that I should use Bayesian Inference to tackle the problem. The question in mind is to see how the ...
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Conditional Probability problem with Bayesian components

Suppose there are two species of Tigers, $T_1$ and $T_2$ which are indistinguishable and exist in equal proportions, but differ in how they lay children. Species $T_1$ gives birth to twins 10% of the ...
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Using Bayes' with a Multinomial Distribution

This is a problem in a bioinformatics class, and I believe it shouldn't be too difficult probability-wise, but I've a novice in this area. I think I have about what I need, but I'm very unsure.   ...
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Conditional probability from Bayesian network

Based on the Bayesian network given below: Network https://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html How would I calculate p(S = T|C = F, R = T, W = F)?
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Is possible the posterior conditional expected value of a random variable is dependent of the same random varible?

Let a posterior X|Y1,Y2 ~ N(a1*Y1+a2*Y2;v) where a1, a2 and v are constant values. a) Thus E[X|Y1,Y2] = a1*Y1+a2*Y2 this conditional expected value is independent of X, but is a random variable ...
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Bayesian probability estimation

Consider a sequence of independent bernoulli random variables $X_1 X_2 ... X_n$ with parameter $\theta$ and $0<\theta<1$,where $P(X_i=1)=1-P(X_i=0)=\theta$. Assume the prior of $\theta$ follows ...
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I wake up in a random class and hear 6 biology-related words. How certain should I be that I'm in Biology class?

Suppose I'm sleeping in some class. I wake up and I hear 6 topic-specific words that seem related to biology. I'm asked to guess whether I'm in Biology class? How confident should I be? I think this ...
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How to construct a simple Bayes network?

Suppose I'm interested in the probability that event $A$ will occur. I'm uncertain about $P(A)$, but I believe that it has a uniform distribution on $[0.1,.9]$. Moreover, I know that event $B$ and $C$ ...
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Analytical form of Jeffrey's prior

Derive, analytically, the form of Jeffery's prior for $p_J(\lambda)$ for the parameter $\lambda$ of a Poisson likelihood, where the observed data $y = (y_1, y_2,...,y_n)$ is a vector of i.i.d draws ...
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Some questions about Bayesian States

So I wonder if I'm understanding them right: 1.We can use other prior for a Bernoulli data $\theta$. This prior follows the beta distribution if it conjugate with the $\theta$. 2.If we have a Markov ...
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Why do many textbooks on Bayes' Theorem include the frequency of the disease in examples on the reliability of medical tests?

A "standard" example of Bayes Theorem goes something like the following: In any given year, 1% of the population will get disease X. A particular test will detect the disease in 90% of individuals ...
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Uniform distribution Bayes estimator for given parameter distribution

I need to find the Bayes estimator for a $U[0, \theta]$ distribution, with $\theta$ distribution being $q(t) = \frac{1}{t^2}$ for $t \geqslant 1$. This is my first time attempting to do something of ...
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MAP estimation of a diagonal linear state transition matrix with bounded support

I have a stochastic linear dynamical system: $x_{t} = A x_{t-1} + w_t,$ where $x_{t}$ is a latent state vector, $A$ is a linear state transition matrix and $w_t$ is a process noise vector drawn from ...
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1answer
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Is normalizing $P(\Omega)=1$ in probability theory arbitrary?

When I first learned probability theory I was told that under the Kolmogorov axioms, normalizing the probability of the whole sample space to $1$ was arbitrary, and any other positive real number ...
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Does the theory of Gittins Indices solve the Multi-armed Bandit problem?

For example, both Wikipedia and Reinforcement Learning: An Introduction (page 33) seem to claim as much, which would suggest that the problem has been solved for over 40 years. However, doing as ...