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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How many numbers for the full joint?

Suppose you have 3 binary nodes A, B, C. A and B are independent given C. How many numbers do we need for the full joint? How many numbers do we need for the Baysesian Net? I know the answers to ...
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34 views

In Bayesian approaches, comparing the approximated distribution and the true distribution

I'm asked to compare the distance of the approximated distribution and the true distribution in a Bayesian approach. While we used Laplace approximation to find the MAP of the target posterior ...
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139 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
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58 views

Questions about Auctions

I am having a hard time figuring out a problem. In a first price auction with a reserve price R and values of the bidders are U[0,1], how do we find expected revenue given the strategy of both of them ...
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78 views

Probability - when there is an argument between experts?

A, B and C are all expert doctors. When each of them (individually) gives a diagnosis (in a yes/no question), the chance of accuracy is 90%, or 9/10. In a case where A and B argue for a certain ...
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105 views

Coin toss with unknown probability – Bayesian interpretation

I have observed a coin being tossed $n$ times. I do not know whether the coin is fair or not, but in every single toss I observed, the coin came up heads. What should my belief about $p$ (the ...
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155 views

Posterior Distribution with prior standard exponential (mean 1) and data distribution of poisson

So I have the likelihood being: $\prod^{n}_{i=1}(\frac{\lambda^{x}e^{-\lambda}}{x!})$ which is proportional to $\lambda^{\sum_{i=1}^{n}x_{i}}e^{-n\lambda}$ The prior is standard exponential ...
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33 views

Estimate distance between approximated posterior and true posterior

I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior ...
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38 views

Does this Gamma posterior make sense?

quick question about the form of a posterior distribution. Suppose that $\theta \sim Gamma(a, b)$ and that, given $\theta$, $Y$ has CDF $$F(Y\mid\theta) = 1 - e^{-\theta(e^y - 1)},\quad ...
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59 views

What am I doing wrong in calculating Fisher Information of Triangular Distribution?

I am trying to find Jeffrey's prior for the Triangular distribution which has the following probability density function: $$f(x\mid \theta) = \begin{cases} \dfrac{2x}{\theta} & : x ...
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124 views

Posterior predictive distribution in a Bernoulli process.

Suppose there are $k$ successes in a Bernoulli population $ X = \{x_1, \ldots, x_n\}$. I would like to calculate the posterior predictive distribution $f(x | X)$ where $x = \{0,1\}$. I assume the ...
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49 views

Bayesian Predictive distribution with two marginal posteriors

If we have a random variable $Y$ with pdf $P(Y|a,b)$, where $a$ and $b$ are parameters (with range $0$ to $\infty$). As well as marginal posterior distributions for $a$ and $b$, these are $P(a\vert ...
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If $P(B\text{ }|\text{ }A)=1-\epsilon$ and $P(C\text{ }|\text{ }B)=1$ then $P(C\text{ }|\text{ }A)\geq 1-\epsilon$ [duplicate]

If $$1=P(C\text{ }|\text{ }B)=\frac{P(C\cap B)}{P(B)}$$ then we know that $P(C\cap B)=P(B)$. If $P(B\text{ }|\text{ }A)=1-\epsilon$ for $\epsilon\geq 0$ then $$P(A)=\frac{P(B\cap ...
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42 views

Wordy Bayesian Question

Five million boys below the age of five live in Erewon. The priests of Erewon are sure that one of them (chosen by fate at random) embodies the spirit of Captain Coin Tosser, who can predict heads or ...
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103 views

Calculate the posterior probability of the disease

Suppose the prior probability of the germ carrier is 10%. When in tests, germ carriers have the probability of $95\%$ to give positive results and $5\%$ to give negative; non-germ carriers have the ...
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76 views

Showing the Posterior distribution is a Gamma

Assume $X\sim \mathrm{iid}\operatorname{Pareto}(a,b)$, and $b \le \min(X)$, then $$f_n({\bf x}; \theta) = a^n b^{-n} \prod^n_{i=1}\left( \frac{b}{x_i} \right) ^{1 + a} $$ We assume b is known and ...
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27 views

Estimating variance from a combination of processes

I am fiddling with some Bayesian probabilities for some astronomical data analysis. I have a ccd image and am testing the null hypothesis (no signal is present - all contributions due to noise) on it. ...
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401 views

How to calculate pseudo-determinant for implementing Naive-Bayes

(People who followed Bayesian tag, please read the third paragraph) Problem: I need to calculate pseudo-determinant of a matrix (preferably in MATLAB, but no ...
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105 views

Interpreting Bayes' thereom with density functions

I'm studying about Bayes' theorem and according to the theorem: $$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)}$$ This I can understand where it comes from etc. But if I use Bayes' theorem on density ...
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37 views

Uniform prior distribution multiple results

When I have a simple Bernoulli trial with a certain variable taking, for instance, values 0 and 1, I have a constant prior distribution for the $\theta$ parameter, i.e. pdf $p(\theta) = 1$ between 0 ...
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31 views

Likelihood function for $\theta = 0$ (Calculate $\sum_{i=1}^4x_i ^2$ given $\sum_{i=1}^4 x_i$)

I need to find $L_x(\theta)$ for $X_1, X_2, X_3, X_4$ c.i.i.d random variables such that $X_i | \theta \sim N(\theta,1)$ when: $\theta = 0$ $\bar{x} = -0.7$ ($\bar{x} = \Large\frac{\sum_{i=1}^n ...
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70 views

Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
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Maximum Posterior: $ p(\bf{w}\mid\bf{x},\bf{t},\alpha,\beta) \propto p(\bf{t}\mid\bf{x},\bf{w},\beta)p(\bf{w}\mid\alpha) $ for Gaussian Distribution

At the moment I take a look at the book Pattern Recognition and Machine Learning from Christopher Bishop and as I try to understand the basics of the probability theory I get stuck trying to ...
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Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn't. We want to estimate ...
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38 views

Posterior distribution as a distribution for a new random variable?

So in Bayesian framework one uses observed data $X=\{x_1,\dots,x_n\}$ to update the prior $p(\theta)$. My question is it justified mathematically to say that $p(\theta\mid x_1,\dots,x_n)$ corresponds ...
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44 views

How to prove theorem about consistency of Markov edge process?

How to prove such theorem: Markov edge process $p_E(y_E)$ with respect to DAG $G=(V,E)$ defined as $p_E(y_E) = \prod_{v \in V} p_E\left(y_{E_{\rm out}(v)} \,\big|\, y_{E_{\rm in}(v) } \right) = ...
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35 views

Hypothesis testing with Baysian methods: How many animals must I test to be sure that a disease isn't present?

I colleague has come to me with a question which I have answered for him but the only statistics I have done was what I did at school and a one semester course on Bayesian methods at university, so I ...
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27 views

Measure theoretic basis of joint distrib of parameters and data in Bayesian analysis

In Bayesian statistics you have a prior density for your parameters $\Theta$, $\pi(\theta)$ for $\theta\in\mathcal{T}\subset\mathbb{R}^k$, have the conditional distribution of the data given the ...
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43 views

Reversing a conditional probability

If I'm given a P(X|Y) table and P(Y), how can I find P(Y|X)? I understand that $P(Y|X)=\frac{P(X|Y)P(Y)}{P(X)}$ but how do i find P(X)? Furthermore, If i'm told that random variable X is given a ...
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57 views

Calculating Bayes factor

Example: Integer-valued data $y = (y_1, ...,y_n):$ $M_1 = Geometric(\theta_1)$ likelihood with $Beta(\alpha_1, \beta_1)$ prior on $\theta_1;$ $M_2=Poisson(\theta_2)$ likelihood with $Gamma(\alpha_2, ...
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60 views

Please explain to me why $ \int_b p(a|b) db \neq p(a) $

I have one question that bugs me. How is it that: $ \int_a p(a|b) da = \int_a \frac{p(a,b)}{p(b)} da = 1 $ but $ \int_b p(a|b) db = \int_b \frac{p(a,b)}{p(b)} db \neq p(a) $ I don't understand ...
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88 views

Bayesian Network - unclear homework example

I am not sure if it is me or the example: A doctor gives a patient a drug dependent on their age and gender. The patient has a probability to recover depending on whether s/he receives the drug, ...
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77 views

Conditional Independence - Bayesian Network

May the probability distribution $ P(A,B,C,D) $ given as: $ P(A,B,C,D) = P(A)P(B)P(C|A,B)P(D|C) $ The task is to show that this holds $ A \bot B | \emptyset $ and $A\bot D|C$. First thing I'd like ...
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34 views

Difference of a likelihood function for a vector and a single value

$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector? I'm finding hard to ...
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32 views

Conjugate Bayesian analysis

Suppose that conditional on $\tau$, the random variable $X$ has normal distribution with mean zero and variance $1/ \tau$. The prior distribution for $\tau$ is Gamma with parameter $\alpha$ and ...
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178 views

Gaussian with a linear combination random variable mean

A very simple (looks like...) statistical problem, however I don't even know how to name it in a formal way... Suppose in a Bayesian framework I have random variables $y, x_1,$ and $x_2$, $$f(x) = ...
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413 views

Find Bayes estimator of $\theta$

I've got this exercise, which I'm trying to work off using an example, but the example seems very different so I'm not sure if what I'm really doing. I've got a loss distribution for $\theta$: ...
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79 views

Find marginal distribution for Pareto prior

I have the following problem: The prior distribution for $\theta$ is distributed $\pi(\theta) = \frac{aP^a}{\theta^{a+1}}$, $\theta >P$ The likelihood for X is uniformly distributed, i.e. ...
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118 views

Find joint probability P(X=0, Y=0)

I have this problem where I'm not too sure on how to proceed. I need to calculate $Pr(X=0 $ and $ Y=0)$ using the following information: The conditional distributions $f(x|\theta)$ and ...
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91 views

Finding the marginal using Bayes Theorem

I am trying to find the marginal distribution f(x) when given the prior distribution $\pi(\theta)$ (Gamma $\alpha, \beta$) and conditional distribution $f(x|\theta)$ (Poisson, $\theta$). I know the ...
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41 views

Does a Markov Blanket allow connections between Parents of a Node?

In a Markov Blanket, we can connect the childredn of a node between them, as a child can be parent (or spouse) of another child. Does this rule apply as well for Parents of a node? In advance, Thank ...
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35 views

What is Bayesian Evidence?

Could someone explain this concept or give a link to the explanation of this concept please? I know what "Bayesian" is, but I don't know what "Bayesian evidence" is. A good explanation of "evidence" ...
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Is it possible to get 3 decision criteria using Bayes theorem?

I was wondering if it is possible to get 3 intersection points if use Bayes's theorem $$P(B|x) = \frac{P(x\mid B) \times P(B)}{P(x)}$$ Where $P(x\mid B)$ is a gaussian function.$$ P(x\mid B) = ...
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105 views

How to I find the distribution of $\log p(X)$ given an $X$ drawn from $p$?

I have a feeling there's no general solution to this problem, but I'll ask anyway. I have a multivariate PDF $p$ and, given a random vector $X\sim p$, I'd like to find the the PDF of $\log p(X)$. ...
2
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1answer
418 views

Computing posterior distribution for AR(1) model

Question: For this question, note that the notation $y_{1:T} = (y_1, y_2, \cdots, y_T)$, ie, a vector of random variables. Consider the following AR(1) model: \begin{align*} y_{t+1} = \phi y_t + ...
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22 views

Gibbs sampler for local linear trend model

Question: Consider the local linear trend model given by: \begin{align*} y_t = \mu_t + \tau \varepsilon_t \ \cdots \ \text{Observation equation} \\ \mu_{t+1} = \phi \mu_t + \eta_t \ \cdots \ ...
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229 views

Introductory question to the Bayesian Search Theory

Can someone help me with a basic Bayesian Search Theory question? I'm not sure how to approach these questions and after reading up about the Bayesian search theory I'm still wondering how I should ...
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20 views

Addition of distributions in statistics

Is it possible to add distributions? I've worked out "Say that you are given ten identical coins for which you assume Beta(4,4) prior distribution on the unknown probability θ of any of the coins ...
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73 views

If X and θ are both random variables and θ is the parameter of the distribution of X, are X and θ independent?

The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the ...
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208 views

Probability question: given $P(A|B)$ and $P(B)$ how do I find $P(A)$?

I have a probability distribution for some quantity $A$ given a fixed $B$, i.e. $P(A|B)$. I also have a prior distribution $P(B)$ for $B$. I'm trying to find the distribution $P(A)$. I had thought ...