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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29 views

Is this Bayes estimator result correct

I am trying to see where I went wrong in this calculation of the Bayes estimator or if there is a hole in my understanding. We have a common discrete density $$f(x|\theta)=\frac{\theta^{x}e^{-\theta}}{...
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32 views

Posterior for normal likelihood, normal prior

We are given the following information: $\Theta = \mathbb{R}, Y \in \mathbb{R}, p_\theta=N(\theta, 1), \pi = N(0, \tau^2)$. I am asked to compute the posterior. So I know this can be computed with the ...
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Explanation of the Preece - Baines model?

I'm a bit confused on the Preece - Baines model for growth in humans. It states that for data on a humans height, y[i] at time points t[i] where i = 1,2 ... n and hence t[1] < t[2] < ... t[n] ...
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Sum of Square in Matrix Manipulation and Gibbs Sampling

We know that $\sum_{i=1}^n(y_i - \mu)^2 = \sum_{i=1}^n(y_i - \bar{y})^2 + n(\mu -\bar{y})$ where $y_i$ is a scaler. Suppose $\textbf{Y}_i \in \mathbb{R}^r$, $\boldsymbol\Sigma \in \mathbb{R}^{r \...
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Gaussian process for surrogate model

I am studying for surrogate modeling approaches that can be used for sensitivity analysis. It seems that Gaussian process is one of the main approaches for building surrogate model. Why Gaussian ...
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How to estimate the “innate speed” of a leaping frog?

The motion of a leaping frog is set by a "hidden parameter" $V_\infty$ that we want to estimate: it is the "innate average velocity" of such a frog. The frog jumps a distance $J_i \in \mathbb{R}^+$ at ...
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¿How can I get a posterior distribution about the uniform distribution?

I need to get the posterior distribution for theta parameter: The translate of the image: let $p(x|\theta)=unif(x|0,\theta)$ with unknown theta and a priori distribution on theta given by: $p_1(\...
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how to solve the Bayesian Nash equilibrium when there is payoff unknown

in this situation, I don’t know how to solve this. Is it correct that the answer of (a) is ε E [0,1]? I cannot even solve the problem a.
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Limit of Markov chain of beta distributions

Let \begin{align} x_t &\sim \mathrm{Beta}(\alpha_t,\beta_t) \\ \alpha_{t+1} &= \alpha_t + x_t \\ \beta_{t+1} &= \beta_t + 1 - x_t \\ \end{align} My questions are the following: What is ...
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giving meaning to a prior distribution without invoking “physical probability”

When we refer to a concrete probability space the events to which probabilities are assigned should have a specific well defined meaning. Now when we refer to a prior distribution of some parameter it ...
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Unintuitive result while working with a temporal Bayesian Network

I am working on a temporal Bayesian Network toy problem using BayesFusion GeNIe Software. I have a node (Case_24 in the figure) that models the state (0 or 1) of a time-dependent variable. At every ...
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What is the prior and sampling distribution of this situation?

In an assignment, I have to assume that I'm lost in a city with four areas, let's call them Area A, B, C, and D. All I have with me is a spreadsheet of the city's voting results from a random election....
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Find the parameter of the posterior distribution (Inverse-Gamma distribution).

Suppose $y$ is an independent and identically distributed sample of size $n$ from the distribution $Normal(θ,a)$, where $(θ,a)$ have $Normal-Inverse-χ^2(µ,\frac{τ}{\sqrt{k}};v,τ)$ prior distribution, (...
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What are sufficient conditions such that consistency of ML estimate implies consistency of MAP estimate?

I am interested in under what conditions the frequentist consistency of a Maximum-Likelihood estimator is enough to give the consistency of a maximum-a-posteriori point estimate, with the further ...
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Naive Bayesian expansion problem

enter image description here Known: 1. Y ∈ {0, 1} , x1 ∈ R,x2 ∈ R , xj ∈ {0, 1} (j = 3, 4,...). 2. P (X, |, Y) is independent, that is, P (x1, x2, ..., xn | y) = P (x1 | y), P (x2 | y) ...P (xn | y) ...
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How does the posterior density g(μ|x) change if we find out x could only be observed if it were greater than 0.

Given prior density g(μ) and observation X ∼ Poi(μ) , you compute g(μ|x), the posterior density of μ given x. Later you are told that x could only be observed if it were greater than 0. Does this ...
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Need some explanation for the following bayesian graph

I am trying to understand some lectures. A equation is presented as follows I am confused with the first liner. Should it be P(w,y |X) = P(y|X,w)P(X|w) P(w) ?? If the above were correct, would it ...
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Calculating Bayesian probability when measurement error is present.

Let's assume I work for a police department, and I receive an email from a colleague. The email states that 10 honest people independently witnessed Jones committing a murder. Let's say the chances ...
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Conjugate prior for a Beta RV

I have a random variable with support on [0,1], and I think it makes sense to model it as a Beta Random Variable (RV). Is there a convenient conjugate prior for this Beta RV? Convenient here is ...
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Finding distribution of difference of Bayesian and single point linear regression

In task the theorem needed to be proven is : If $y = X\beta$ with $\beta \sim N(0, t \cdot I_d)$ and $w \sim N(0, I_n)$ then $$E\frac{1}{n}\|X\hat{\beta}_{mean} - X\beta \| = \frac{1}{n}\sum_{i = 1}^...
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Stuck on the proof conditional probability theorem

Given a random vector $\mathbb{X}$ with joint density $f$, and a set $A=\{\mathbb{X}\in B\}$ with $B\in \mathscr{B}(\mathbb{R})$ Prove that:\ $$f_{(x|A)}=\frac{f(x)}{\mathbb{P}(A)}\text{ if } x\in B $$...
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Covariance for Optimal Bayes Estimator using Gaussian distributions

Lemma : If we have a Bayesian linear model: $$y = X\beta + w$$ where $\beta \sim N(0, I_d)$ (prior parameter) and $w \sim N(0,I_n)$, then $\beta$ conditioned on $y$ (posterior parameter) is Gaussian ...
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Unscented Kalman filter and measurement function

Suppose I have a nonlinear system with states $x$ and measurements $z$. I don't have a measurement function for $z=h(x)+n$ where n is my gaussian noise term. However I do have the sufficient ...
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Probability of choosing M out of N Groups given that all A elements are inside M

Assuming I have A number of individuals who are randomly distributed into N number of groups. What is the probability of finding A (or any number a <= A) by picking M number of groups within N? I ...
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If the attrition rate of the customer is 5% every year, what is the probability the customer will not be there in year 3?

I want to find out what is the error in my logic: P(employee not being there in year 3) = P (employee dropping out in year 2 or dropping out in year 3) . Note she cannot drop out in year 1 as per the ...
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Does Bayesian inference imply a contradiction?

Suppose that there have been $n$ days and that the sun has risen on all of them. What’s the chance that the sun will rise tomorrow? Assuming that we start with a uniform prior on the probability that ...
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Updating the Conditional Distribution in Bayesian Statistics

Suppose that I have a prior distribution $\pi_0(\theta)$ on $\mathbb{R}^k$ and a "naively chosen" sample distribution $f_0(x|\theta)$ on $\mathbb{R}^n$. I would like to online-update both of these ...
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5 photons detected from a star in 1 hour. What is probability that 3 photons will be detected in the next hour?

This is a question I was working on from Phil Gregory's Bayesian Logical Data Analysis and I ended up getting stuck here. The question seems like a simple Poisson rate-problem by taking $rate = 5/...
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Confusion evaluating Pareto function

I am going through a textbook example and am having trouble working it out myself and am unsure where I am going wrong. I understand the model is the likelihood function and that a Pareto can be ...
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How might I understand the mathematical representations of model complexity and accuracy in information theory?

I'm trying to work through the free energy principle (FEP) for biological organisms. This involves the claim that organisms minimise surprisal––encountering low-frequency sensory states––by changing ...
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Proving the existence of a Symmetric Bayesian equilibrium

I am currently faced with the following question: Consider the public goods game. Suppose that there are $I > 2$ players and that the public goods is supplied (with benefit of 1 for all players) ...
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What material do I need to cover to understand the gamma function?

I'm taking a course on Bayesian statistics, and my furthest understanding of math only extends to Calculus 1, logic, and elementary statistics. I'm learning about the gamma distribution (and the ...
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Make a fair game

Assume you're playing a game with your sibling (S.) and a friend (F.), in which S. will throw a coin and if the coin shows heads F. will give you 1 dollar. If the coin shows tails, you will give F. $z$...
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Does Bayes theorem work only for PD functions or it can be used for any other family of functions?

I have been reading about Bayes theorem and I have seen some examples where PDs were integrated in equation. Then I was thinking if there are any other functions, which could be used as well? Do you ...
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Bayes law and conditional probability - throwing 2 coins

Person A has a fair coin (only: head/tail) and an unfair coin (head/head) He chooses one coin randomly, throws it and it lands on head. He throws the same coin again and it lands on head again! What ...
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How to solve Bayes' probabilistic network problem?

Given the following Bayesian Probabilistic Network, let's say I am trying to find the probability of P(!FO|HB). I understand basic Bayes theorem, but not sure how to use it here.
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Bayes' Rule broken?!?!

This question has been driving me CRAZY for 4 days now. The question comes from the textbook "One Thousand Exercises in Probability", specifically Exercise 3 in section 1.4. The solution does not make ...
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Maximum Likelihood Estimation: difference between probability and likelihood

In Maximum Likelihood Estimation (MLE), we want to maximize the probability $P(x_1, x_2, x_3,.. x_n | \theta)$, where $x_1, x_2, x_3,.. x_n$ are the datapoints and $\theta$ is the parameter vector. ...
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Inference of parameters given their relation to expected value

I have a pdf, $p(x; a,b)$ such that the expected value is dependent and strictly increasing on two parameters, $a, b$. $x$ is a real number, while $a$ and $b$ each belong to $[-1, 1]$. I want to ...
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computing posterior probability $P(t=0|x)$ from binary classification

Considering that input $x$ is a scalar, the data generation process works as follows: First, a target t is sampled from {0, 1} with equal probability. If t = 0, x is sampled from a uniform ...
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$a$ red balls and $b$ blue balls in an urn. Probability when a game lasts forever.

In an urn there are $a$ indistinguishable red balls and $b$ indistinguishable blue balls. Every round, you take one ball in random from the urn. If this ball is blue, game over. If this ball is red, ...
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Bayesian diagram of a disease question

Consider an epidemic where K out of every 100 people is infected.Blood test is necessary to diagnose whether a given person is infected or not, and we know L out of every 100 people in the community ...
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MAP Estimation of Covariance Matrix of a Multivariate Normal Distribution

I have a general prior multivariate normal distribution and I want to update it with new samples which are more local for my case. I want to do it with MAP estimation. With MAP estimation it is ...
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Probability of choosing an unfair coin

Problem statement: There are 1000 coins. 999 coins have heads and tails. 1 coin is unfair and both sides are heads. You choose a coin at random and toss it 10 times. You get 10 heads. What is the ...
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What is the justification for the “K-L model prior” that makes AIC appear as a Bayesian result

When trying to understand AIC, BIC, and model selection in general, I came across a paper that states "AIC can be justified as Bayesian using a 'savvy' prior on models that is a function of samplesize ...
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Bayesian aposterior inference for distributions

I'm new to the Bayesian framework. I have information about users' logins by weekdays. And I want to get a mass probability distribution of users' logins for simulation of users' behavior. I have a ...
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Probability Bayes Theorem

I'm new to probability. I'm trying to learn bayes theorem. I came across this question. ** Two machines M & N are used to produce chocolates. M produces 60% of total chocolates and N ...
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Classical v/s Bayesian Hypothesis Testing

This question has 2 parts: (1) What is the fundamental difference between classical and bayesian hypothesis testing? How do I interpret this difference. (2) Here is a paragraph quoted from Casella ...
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Help me to understand the part of the derivation of posterior distribution hyperparameters

I've looked on how to find hyper parameters of posterior distribution for normal distribution likelihood with unknown mean and precision. Here is a derivation described https://www.cs.ubc.ca/~murphyk/...
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The length of the third piece in the Stick breaking process

I'm reading Nonparametric Bayesian Statistics Part I: some classical results. In page 14 and section 3.2.1 I can't understand why the length of the stick after the second break is: $$(1-Y_1)(1-Y_2)$...

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