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 probability, and evaluates the evidence in favour of a hypothesis by combining the prior with the likelihood function of the observed data.

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How to use Bayesian Inference in this problem? [closed]

I've just learned Bayesian Inference and encountered this math problem: "A box contains 20 of both red anh black balls. Hypothetically, 40% <= the ratio of red balls <= 75%. Pick 5 balls ...
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Bayesian Approach to Hypothesis Testing: Posterior Probability that Two Distributions Differ

I'm trying to take a Baysian approach to Hypotheisis testing but I need a bit of help formalizing what claims I can actually make. Let's assume I have two datasets $X$ and $Y$ that each consist of $N$ ...
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Notation for probability density function in Bayesian context

The Bayes theorem is often quoted as, $$P(\theta|X) = \frac{P(X|\theta)P(\theta)}{P(X)}.$$ In my use case, I'm dealing with Gaussian continuous variables. So, by $P(X|\theta)$ I'm referring to the sum ...
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How do we derive the conditional distribution for a Poisson whose rate is the product of two Gamma distributed rv?

This question is motivated by Gopalan et al. "Content-based recommendations with poisson factorization." Advances in neural information processing systems 27 (2014). https://proceedings....
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Mathematical Definition of "Improvement"?

In the context of Bayesian Optimization, we model the Objective Function we are trying to optimize using a Gaussian Process. The location at which we evaluate the Objective Function at next is decided ...
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a simple question about conditiona probability

known conditions: $$ P(b|a) = 0.6 \\ P(c|b) = 0.8 \\ P(c|\neg b) = 0.7 $$ What I want to solve: $$P(b\cup c|a )$$ (in the condition of "a" happen, at least one of b and c happen. ) I tried ...
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bayesian gauss prior prove

given zero mean Gaussian prior as $\beta~ \sim(0,\Sigma p)$ inference is given by $\log p(\beta \mid y, X)=\log p(y \mid X, \beta)+\log p(\beta)-\log (y\mid X)$ I can't understand how to get the ...
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Necessary conditions for $P(A|B)=P(A|C)P(C|B)$?

Suppose we know $P(A|C)$ and $P(C|B)$ and we want to find $P(A|B)$. What are the necessary conditions under which the $C$ "cancels out" and we have the equality $P(A|B)=P(A|C)P(C|B)$? I have ...
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Naive Monte Carlo Sampling vs. Importance Sampling

Can someone help me understand this paragraph: The naive Monte Carlo estimator introduced in the last section performs well if the prior and posterior distribution have a similar shape and strong ...
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What is a latent spatial process?

What is a latent spatial process in the context of Bayesian hierarchical models? For intensity, we model daily precipitation above a high threshold at 56 weather stations with the generalized Pareto ...
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Bayesian Network calculations given dependence of variables

I've been given this bayesian network 1 where $P(A) = P(A = t) = 0.2, P(B) = 0.5, P(C) = 0.8.$ $$\require{enclose}\begin{array}{c}\enclose{circle}{~~A~~}&&&&\enclose{circle}{~~B~~}&...
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Does something "magical" happen at 20 Dimensions?

In the context of Bayesian Optimization, I have often heard that Bayesian Optimization tends to perform poorly on functions having more than 20 dimensions. However, I have never been able to ...
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From a conditional probability to an indefinite integral? Bayesian Information Criterion

I am trying to understand the Bayesian Information Criterion (BIC) using this article. At page 2 the following equality is given: $$P(y|M_1) = \int f(y|\theta_i)g_i(\theta_i)d\theta_i$$ with $y$ : ...
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Cannot figure out the Integral result

This is a snippet from the book Introduction to Mathematical Statistics. I am kinda of lost how in this book they calculated the integral $g_{1}(x)$, especially the part where they get $\gamma$($\sum_{...
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Weight prediction of edge in direct graph

I have the following problem. Suppose we have a weighted directed graph $G$ with $J$ edges $(E_0...E_J)$ and $K$ nodes $(N_0...N_K)$. The structure of the graph is fully known (we can simplify and say ...
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Coefficients of posterior on a prior which is gaussians mixture

Having known about sample $x \sim \mathcal{N}(\theta, 1)$ and $\bar{x} = -0.5$, size $n = 50$, a prior $p(\theta) = 0.95\mathcal{N}(1, 0.5^2) + 0.05 \mathcal{N}(-1, 0.5^2)$, I need to show that $p(\...
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Conjugate prior using the exponential family method, whith Normal distribution Likelyhood with 2 uknown parameters

Well I have struggled with this for some days now. Let be $n$ observations $y=(y_1,y_2,...,y_n)$, where $y_i|\mu,\sigma^2 \sim \mathcal{N}(\mu,\sigma^2),1\leq i\leq n$ assumed to be conditionally ...
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Unable to prove(may be under some condition) the identity: for random variables $X,Y,W$, we have $p(x|y) = \int p(x|w)p(w|y)dw$.

Let $X,Y,W$ are random variables. Let $p(x)$ denotes the pdf $X$. We try proving the identity: $p(x|y) = \int p(x|w)p(w|y)dw$. We start with $$p(x|y) = \int p(x|y,w)p(w)\,dw,$$ $$p(x|y) = \int \int p(...
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Modeling dependent events

Say we have event A (saving a life) and event B (person living after initially getting saved). A and B are dependent events as p(A|B) = 1. I want to model the probability of a person living which is p(...
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Dropping Conditioning in the Prior for the Posterior Distribution Linear Regression

In linear regression, we have the following formula $$ p(w | X, Y) \propto P(Y | X, w) P(w)$$ where $X$ is a sample vector input data (and is random). $Y$ is the corresponding vector of output data. ...
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probability of rolling a consecutive 1, 2, 3, 4, 5, 6 on a die

Questions I was wondering what the probability of rolling a consecutive $1 ,2, 3, 4, 5, 6$ on a dice is? For realism, is there any way to calculate an 'extra' factor, such as someone kicking the table ...
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Reference request: multi-armed bandit problems with analytical solutions

Is there a book/survey on (multi-armed) bandit problems that yield analytical solutions? I.e. has an exactly optimal closed-form solution (e.g. derived using dynamic programming).
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Calculating the posterior distribution - missing dependency

I am reading an old paper from https://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf. In that paper, specifically, Equation 10 says $$ p(w|t,\alpha,\sigma^2) = \frac{p(t|w,\sigma^2)p(w|\alpha)}...
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Derivation of the integral form of the numerator in the Bayesian inference equation???? (not on the denominator)

In the reference Gaussian processes: iterative sparse approximations by Csató, Lehel (Csató, Lehel. Gaussian processes: iterative sparse approximations. Diss. Aston University, 2002), on page 20, ...
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How to maximize expectations based on historical trials?

There are 10 multiple-choice questions, each of which has 4 options $A, B, C, D$. There is only one correct answer. 1 point is awarded for the correct answer, and 0 points for the wrong answer. let $\...
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False positives when mass testing a low prevalence disease

Assuming you have a test for disease X. The false positive rate of the test is 2%, the true positive rate is 100%. You know that it's a low prevalence disease, only 1% of the population has it. The ...
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Reference Request for Bandit Problems where myopic strategies are optimal

I recently found a paper (Banks and Sundaram, 1992) on a class of bandit problems where the myopically-optimal strategy (in each period, choose arm that maximizes current period's expected payoff) is ...
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Math paper translation

Does anybody know if Bruno de Finetti's paper, Sul significato soggettivo della probabilità, has been translated into English? I am teaching myself Italian, but for some reason de Finetti doesn't ...
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The utility of kernel methods like RKHS in machine learning

In machine learning framework, kernel methods are widely used to find the close-form solution of a optimization problem, which restricts the solution in an RKHS. However, it really puzzles me that ...
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(Bayesian Statistics) Finding posterior distribution

I would be very grateful to get some help with the following problem. Suppose $X_1, X_2, · · · , X_n ∼ N(0, θ)$ where $θ ∼ Gamma(3, 0.5)$, that $$p(θ) = \frac{θ^2e^{-θ\over 2}}{2^3Γ(3)}\ , θ > 0.$$ ...
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Prove the Bayes Rule for multi-class classification

Consider a classifier $\delta:\mathbb{R}^{d} \to (1, \ldots, N)$. Let the misclassification error by written as: $L(y,\delta(X)) = {\sf 1} (Y \neq \delta(X))$ where $X \in \mathbb{R}^{d}$ Prove that ...
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How does one define a model-specific random variable for parameters in Bayesian inference?

My understanding of using Bayes Theorem for inference is that we treat both the observed data, $\vec{d}$, and the possible values of the model parameters, $\vec{\theta}$, as values taken by (vectors ...
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Bayesian Statistics Example clarification

lets assume we have the following data: $$ \text{Test Result} | \text{Has Disease}\\ 1 | 1\\ 1 | 1\\ 1 | 0\\ 0 | 0\\ 0 | 0\\ 0 | 0\\ 0 | 1\\ 1 | 1\\ 1 | 0\\ 1 | 1 $$ Lets denote P(B) as the ...
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how to find a posterior with min of observation and constant

There is a variable Y with exponential distribution with parameter theta. The prior distribution is gamma distribution with parameters alpha and beta. If we don't have an actual observation of Y, but ...
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Resources with examples for Bayesian Data Analysis in R

I'm taking a course on Bayesian Data Analysis, and currently I need to do a project that requires the use of $R$ as a programming language. The goal of the project is to use basic Bayesian Inference ...
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In what situations is this true: $P(A|B) = 1 - P(A|B^c)$? [duplicate]

I am working on some software that uses Bayes' rule to find $P(B|A)$ It asks the user to define $P(A|B)$ but it says that $P(A|B^c)$ is optional, if the user chooses not to define it then it sets $P(A|...
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bayesian probability problem

Question: You are currently quarantining in a house with 2 other people. All three of you decide to try and experimental vaccine, that is either effective(70% chance of preventing transmission) or ...
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Bayesian analysis for multiple unknowns

I've encountered this problem but my stats is rusty and I've been having trouble formalizing it enough to tackle it. I have something like a family tree where each node in the tree has two parents and ...
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Spike and Slab prior

I am looking for a simple example of posterior density with a spike and slab prior. Suppose we have the prior $$b|\pi_0 \sim (1-\pi_0)\mathcal{N}_K(0,I_K) +\pi_0\delta_0$$ where $\delta_0$ is a dirac ...
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Question regarding continuous-time bayesian updating

How to do continuous-time Bayesian updating? On the above link, the answer says Simplifying it after ignoring $dp_tdt$, we get $dp_t/dt=−λp_t(1−p_t)K_t.$ Why can we ignore $dp_tdt$?
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Introductory Bayesian statistics problems book with solutions

I am studying a statistics course and I will have an exam on it in a few weeks. For bayesian stats, our professor has provided us with one example, a couple of questions which differ from the example ...
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Bayesian Estimation and Multivariate Linear Regression [closed]

Please help me understand a step to calculate the mean $\mu$ of this distribution. Setting: $Y_{1}, \ldots, Y_{n}$ are independent given the pair $\left(\beta_{0}, \beta_{1}\right)$ $Y_{i}=\beta_{0}+\...
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Posterior with a Beta prior and a liner function of Bernoulli data.

Let $(a,b)\in\mathbb{R}^2$ be fixed, known constants that satisfy$|a|+|b|<0.5$.$^1$ Let $\theta$ have a $\text{Beta}(\alpha,\beta)$ prior. Given an observation of $X\sim\text{Bernoulli}(a+b\theta)$,...
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Sequential Bayesian updating with a four-parameter Beta distribution and Bernoulli data

Suppose data $\{z_t\}_{t=1}^T$ are $z_t\substack{{\small i.i.d.} \\{\Large \sim}\\ \color{white}{.}}$ $\text{Bernoulli}(\theta)$ and a four-parameter Beta prior $\theta\sim \text{Beta}_4(\alpha,\beta,...
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Non-independent state variables in a Markov random finite set

In my master thesis I need to implement a non-linear Markov random finite set (RFS) model for multi-object tracking (MUT) model to generate the input data to a deep learning algorithm. My motion model ...
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Bayesian inference of normal distribution

Find your posterior distribution for $\mu$ when observing $n$ data with sample mean $\bar{x}$ from $X \sim^{iid} N(\mu, 1)$, with a prior distribution for $\mu \sim N(m,v)$. What I have tried $$\...
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1 vote
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Updating priors

Let $y_1, y_2$ be two independent random samples from $f(y|\theta)$, where $\theta$ has prior $\pi(\theta)$. Consider two possible situations: 1. we observe $y_1$ first, update the pror to $\pi(\theta|...
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Bayesion estimation with Gamma prior

Suppose $X_1,..,X_n$ are i.i.d from $N(\mu_0, \sigma^2)$, where $\mu_0$ is known. Let $\phi = \frac{1}{\sigma^2}$ denote the precision parameter. Suppose we know $\phi \sim Gamma(\frac{m}{2},\frac{m \...
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What does this symbol mean in bayesian estimator example?

Is this symbol equivalent to 'multiplication', or equivalent to 'equal'?
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Ranking Basketball Players Based on Probabilities and Data

Suppose you have the following problem: There are 100 basketball players : Player 1 is a pro, Player 100 has never played basketball before - and the rest follow some sort of Normal Distribution. ...
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