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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 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 validate my statistical model

I established some Bayesian model. Then reviewer said that I should validate this model. So, I replicate data $D_1,D_2,..D_n$ from known distributions whose parameter $\theta ^*$ (truth), and ...
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Estimate the probability distribution of two parameters

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
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9 views

How to show that MSE of ML estimator is greater than that of Bayesian posterior mean?

This question is based on problem 9 from chapter 4 of Gelman et al.'s Bayesian Data Analysis. Suppose we observe $y\sim N(\theta,\sigma^2)$ and wish to estimate $\theta$, with $\sigma^2$ known. We ...
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31 views

Mean estimator of a Gaussian variable with positive mean for quadratic loss

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian, and $X\sim\mathcal{N}(\theta,1)$, in which $\theta>0$ is positive but othrewise unknown. We want to estimate $\theta$ ...
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Bayesian probability: proving $P(x|y,x) = 1$

I have an exercise where I'm supposed to prove that $P(x|y,x) = 1$ I've come up with the following but I'm not sure if it feels right to me: $P(x|y,x) =$ $P(x,y,x) / P(y,x) = $ $P(x|y,x)P(y|x)P(x) ...
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21 views

Searching for proof - bayesian inference for exponential distribution

According to Wikipedia (https://en.wikipedia.org/wiki/Conjugate_prior) the gamma distribution is a conjugate prior for the exponential distribution (with unknown rate-parameter, $\lambda$, and ...
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5 views

Bayesian approach to function fitting

Assume we are given a dataset $(x_i, y_i)_{i=1}^n$ which consists of samples of univariate normally distributed random variables following the law $y_i \sim \mathcal{N}(f(x_i), \sigma^2)$ with unknown ...
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How can I get the probability given the Bayesian table?

Consider that $A \rightarrow B$. And A has two states 0 and 1 respectively with probability of $0.6$ and $0.4$. $$ \mbox{And for}\ B: \left\{\begin{array}{rcl} {\displaystyle Pr\left(B = 1 \mid A = 1\...
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32 views

Reverse probability problem.

Suppose there are two cards A and B. Both of the cards have a yellow side and a green side. When tossed in the air the probability of the yellow side facing up is %31 for Card A and 35% for Card B. A ...
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30 views

Find likelihood for a given posterior

Let $p(z)=\mathcal{N}(z|0,1)$ be the standard normal density. Is there a conditional density $p(x|z)$ that could be expressed analytically and that would make the posterior $p(z|x)=\mathcal{N}(z|\phi(...
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23 views

On a Bayesian hypothesis testing

Let $X_1,...,X_n$ be a random sample and $\lambda >0$ be a parameter, with $X_i |\lambda \sim Poisson (\lambda)$ and $\lambda \sim Gamma(\alpha, \beta) (\lambda)=\dfrac {1}{\Gamma(\alpha) \beta^\...
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Derivaton of Chernofff's bound on Bayes error for Multivariate Gaussian distribution

I was following derivation of Chernoffs bound for Bayes error Given in the book Pattern recognition by Duda Hart and Stork. However there is a minor difference between the results in book and details ...
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BAYESIAN MODELLING: Using R and OpenBUGS fit the (fixed effects) linear model to the data: yi = β0 + β1xi + bzi + i , i ∼ N(0, σres)

The data set ‘Politeness’ contains data from a study on the voice pitch (measured as a frequency) of 84 subjects. Six individuals, three males and three females, were asked to perform some tasks, such ...
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Eigenfunctions heuristics for self-conjugate priors

I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the ...
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How to reduce credible sets in over-specified linear regression while maintaining global coverage probability?

Vectors $A, B$ and covariance matrix $C$ are fixed and known. I have a vector of measurements, $Y\in\mathbb{R}^n$, sampled from $$ M_1: Y \sim N(A\alpha_* + B\beta_*, C) $$ My goal, roughly speaking,...
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take derivative of differential entropy of beta distribution

In order to find a beta prior for Bernoulli (Binomial) distribution, one way is to find the maximum entropy prior distribution. Now let assume if we only have information about the mean value of beta ...
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32 views

Find optimal Bayesian decision based on a criterion for the posterior mean

I recently ran into the following exercise when practicing for an exam I have about Statistical Inference. The question looks very large and complex and I'm wondering if this is actually true or if it ...
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32 views

Compute the posterior density for r

I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $\alpha=1=\beta$. Then the prior would be uniform in the range of 0 to 1 so $p(...
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22 views

Same even but two different probabilites

I'm having problem wrapping my head around the following scenario: Suppose there's a gory murder in the island of Onewaynia. Two persons have been shortlisted (and it's guaranteed by divine forces ...
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24 views

Why is the Beta distribution used as a prior distribution in this problem?

Let $\theta$ be the proportion of people who are ready to quit smoking within 6 months. Let's say we perform a survey in $2017$ with a $n$ volunteers who ask people this question until they obtain yes ...
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Posterior Model Odds using Monte Carlo Integration

I have a model $M_1$ with three parameters and a model $M_2$ with two parameters and data $y_t$. I want to compute the quantity $$\frac{p(M_1|y)}{p(M_2|y)}= \frac{p(y|M_1)}{p(y|M_2)}$$ for equal ...
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53 views

How To Interpret Two Bayesian Credible Intervals

I come from a frequentist mindset by training, unfortunately. As such, I'm conditioned to interpret experimental results as either a) reject some null hypothesis, or b) fail to reject it, all based on ...
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33 views

Delta functions/Probability

Here in an answer they say Now note that its perfectly reasonable to have a prior that's say 2 delta functions at p=0.23 and p=0.88. Combining this prior with a likelihood coming from an ...
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27 views

Inferring observation time from a Brownian motion

This might be a bit lengthy question. So let me proceed in steps. General description: I have some observations, based on which I want to infer their occurring time. Specific setting: Let $W(t)$ be ...
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37 views

Why Bayesian Approach don't use test data for model validation?

Up to I know the usual way of thinking in machine learning approach is to split the data in a train and test subsets. The first one is for fitting the model (with the support of a validation subset) ...
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53 views

When to use Bayes' theorem to calculate conditional probability?

Given events $E_1,E_2,E_3,...,E_n$ and $A$, I know that $\displaystyle P(E_k|A)=\frac{P(E_k∩A)}{P(A)}$. However sometimes the Bayes' theorem is used instead: $\displaystyle P(E_k|A)=\frac{P(A|E_k)P(...
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67 views

What is the probability that the girl who laughed loudly was from room number 2?

There are 4 boys and 2 girls in room number 1 and 5 boys and 3 girls in room number 2. A girl from one of the two rooms laughed loudly. What is the probability that the girl who laughed was from room ...
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Bayesian interpretation of product distribution?

I am reading this article about product distribution and I am reading it because of bayesian part, but unfortunately I can not understand part of this part (https://en.wikipedia.org/wiki/...
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57 views

Bayesian Probability Question from “Everybody Lies”

I'm reading "Everybody Lies" by Stephens-Davidowitz now and there's a short passage I'm struggling with proving. Here is a simple thought experiment. Suppose that here are 1,000,000 people in a ...
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Relationship between conditional expectation and marginal join?

I have seen in book (Statistical Rethinking) this equation: $$ Pr(w) = E(Pr(w|p)) = \int Pr(w|p)Pr(p)dp $$ Where $$ Pr(w, p) $$ is join probably density function. Can somebody explain me the equality ...
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Indefiniteness of WAIC

In Bayesian model, If we use improper priors, then indefiniteness of WAIC occur ? I calculate it using target+= formulation in Stan then some term of WAIC tends to ...
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Bayesian Games: Belief Functions

One way to formulate the Bayesian game in Game theory is to define the set of type combinations as $T:=\times_{j=1}^{n}T_j$ and say that the type combination $t=(t_1,\ldots,t_n)\in T$ is supposed to ...
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33 views

On the average, how many times must a die be thrown until on gets a 6 — where am I doing wrong

I've already known one of the right solutions which is $$\sum_{n=0}^\infty n \cdot p \cdot q ^{n−1}$$ However I don't know why this logic doesn't work: We are calculating the quantity -- $$E(N|Last=6)$...
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Relation between a Gamma prior and posterior in terms of paramters [migrated]

I am doing a maths exercise and I have found out that the prior of my parameter is is a inv.gamma (alpha, beta), the likelihood is an exponential distribution. Finally I have discovered that my ...
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31 views

probability of broken toaster with brand

I'm working on the problems related to Bayes' theorem - which did not have answers provided. I don't have confidence with my answer and ask if I can get some help/correction. You run the appliance ...
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15 views

Updated belief with Bayes' rule

Let $\{x_0,x_1\}$ be to states. Suppose the reward is $1$ in state $x_1$, and in state $x_0$ it is $1$ with probability $r$ and $0$ with probability $1-r$, with $r\in(0,1)$. Bob has a belief $p\in [0,...
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53 views

How to quantify the confidence on a Bayesian posterior probability?

Consider a physical system that depends on a parameter $0\leq \phi <\infty$. I want to (i) find the probability that this parameter is smaller than a critical value: $\phi\leq \phi_c$, and (ii) ...
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18 views

Corss Validation regarding Bayesian Bayes Decision Theory

Given a training data with 2 categories and validation data. From the training data, a gaussian distribution is estimated for each category, i.e. the mean and covariance matrix. The validation is ...
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Bayesian parameter estimation with a pre-computed grid of function calls

I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, \ldots x_k$ (describing things like the ...
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Is this Bayesian Network Probability calculation correct?

I think I understand how to calculate BN and why it is so, but complex net still confuses me. Currently how I understand it is that, if there is any 'result' variable in the probability, it can be ...
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18 views

Is this Bayesian Network probability correct?

I just extended a bayesian network that was on a ppt into this form. I'm trying to get P(A,B,C,D,E) and I think it's p(A)P(B)P(C|A,B)P(D|C)P(E|C) but as I'm not sure, just wanted to check if it is ...
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26 views

Accept - Reject of Normal Distribution with prior Cauchy

If $X \sim N(\theta,1)$ with Cauchy as robust prior $$\pi(\theta) = \frac{1}{\pi(1+\theta^2)} \qquad -\infty < \theta < \infty$$ how to do the rejection sampler in R, and use it to generate 10,...
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How to show this distribution is proper?

This question is based on problem 14 from chapter 3 of Gelman et al.'s Bayesian Data Analysis. We have four data points $y_i$ with covariates $x_i$, $i=1,\cdots,4$. We use the model: $$y_i\sim \...
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26 views

Specification for Bayesian updating with flat prior

I am reading a research paper that adopts Bayesian learning process or Bayesian updating, and I found it very confusing for me to understand the notation or specification used. As I copy the paper, ...
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19 views

Posterior of Normal $(\theta,1)$ with prior Cauchy distribution

If $X \sim N(\theta,1)$ with Cauchy as robust prior $$\pi(\theta) = \frac{1}{\pi(1+\theta^2)} \qquad -\infty < \theta < \infty$$ How to find the posterior distribution when Cauchy is $(-\infty ...
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20 views

Deriving conditional distributions of AR(1) process with drift

I have an AR(1) process with drift: $y_t=μ+ρ_{t-1}+ε_t$ with the errors following AR(1) process: $ε_t=φε_{t-1}+u_t$ for $t=1, ..., T; ε_0=0$; and $u_t$ are iid $N(0, σ^2)$. We have these ...
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25 views

Why does the use of Jeffrey distributions does not satisfy the likelihood principle?

It's commonly used as an example, the Bernoulli experiments seen as a binomial and negative binomial random sample, with a posterior distributions $$\Pi _J ^1 (\theta) \propto\theta^ {-\frac{1}{2} }(1-...
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49 views

Confused by Kullback-Leibler on conditional probability distributions

I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the ...
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Is MCMC (or any sampling for that matter) explainable?

Recently, at an interview, I was asked if you use MCMC to build Maximum a posteriori (MAP), and use it for an inference, will the system you create have an explainability? Now, explainability is ...
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How to derive the optimal bayesian solution to a model of two normal distributed populations

In the "Introduction" section of the paper Support-Vector Networks, it mentioned Fisher's solution to a model of two normal distributed populations: My questions are: How to derive equation (1)? I ...