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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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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probabilities associated with each node in a bayesian network

for the following bayesian network I need to list the probabilities( and/or conditional probabilities) that are associated with each node. Node A is pointing to Node B Node B is pointing to Node C ...
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29 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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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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489 views

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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33 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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35 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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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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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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40 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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47 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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48 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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72 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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60 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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33 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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27 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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129 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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218 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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60 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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102 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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80 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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33 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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28 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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37 views

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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How can I conceptualize the prior of a deterministic variable in Bayesian data analysis?

I have a model which includes the following priors: $\text{prec}_C \rightarrow \dfrac{1}{\sigma_C^2}$ and $\sigma \sim \text{uniform}(0,500)$ Now, as far as I understand the first is a ...
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95 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 $\log p(X)$. ...
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148 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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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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128 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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19 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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86 views

Why do Bayesian Networks use acyclicity assumption?

I am trying to gain an intuition about how Bayesian Networks are built for a stochastic process. I see how the conditional independence assumptions in a Bayesian Network makes probability calculations ...
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36 views

Bayesian Hypothesis test

Suppose that a certain industrial process can be either in control or out of control, and that at any specified time the prior probability that it will be in control is 0.9, and the prior probability ...
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59 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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3answers
134 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 ...
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101 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
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32 views

Confirm my working for the conditional posterior of $\beta$

So I have the following question from my textbook, the answer I get is slightly different from the book's answer, which I think may be wrong, could someone please confirm? Question: Suppose $y_{1:T} ...
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38 views

Conditional probability in continuous distribution

I'm struggling understanding how conditional probability for a continuously distributed random variable is to be calculated. The task is as follows: $f(t) = 1/8 * (4-t)$ for $0 < t <= 4 $ and ...
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343 views

Find the posterior distribution of θ

I have this problem Given the prior distribution is \begin{align}Pr(\theta=i)=\pi_i=\begin{cases} 0.5, & \text{for i=4}.\\ 0.3, & \text{for i=5}.\\ 0.2, & \text{for i=6}.\\ ...
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187 views

How to create a probability density function from a set of multivariate data

I am trying to create a simple implementation of the Bayes decision rule with minimum error criterion and I am running into a problem. Specifically, if I have a data set consisting of a number of ...
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34 views

Why is it valid to use the PDF for a naive bayes classifier?

In my understanding of a Naive Bayes Classifier, one takes the argmax of the probabilities that example $x$ belong to class $c_i$, that is $$\text{argmax}_{c_i\in C}P(C=c_i|X=x)$$ I understand that ...
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Independence of data in the prediction interval of a Bayesian regression

The annotated screen shot below is from page 732 of Probability and Statistics, 4th ed. by DeGroot and Schervish. Please see my question in the red box. It makes sense to me that Y-hat is simply the ...
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Do prior hyperparameters update as you take successive measurements in the case of Gaussian unknown mean?

I am trying to use conjugate priors to estimate the mean $\mu$ of a Gaussian with known variance, $\sigma^2$. Derived was that the choice of prior should be: $p(\mu) = N(\mu | \mu_0, \sigma_0^2)$ ...
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30 views

Statistical inference (limit!)

Suppose that a random sample of size n is taken from the Bernoulli distribution with parameter θ, which is unknown, and that the prior distribution of θ is a beta distri bution for which the mean is ...
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64 views

Bayes estimator (Inference)

An urn contain 5 balls, $ \theta $ white and $ 5 - \theta $ green. The experiment consists in grab 2 balls from the urn and register the pair $(x_1, x_2)$, where $x_i = 1$ if we observe a white ball ...
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33 views

Bayesian analysis problem

The problem in hand is that the prior distribution which I have received from experts (loan recovery data) ranges from 0 to 100%. Thus a beta distribution was assumed. Where as the actual data shows ...
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83 views

In a deck of cards, if the second card picked is a heart, what is the probability that the first card picked was a heart?

Assume its a deck of 8 cards with 2 cards of each suit. My analysis is: A = First card is heart B = Second card is heart P(A) = 1/4 P(B) = 1/4 P(B|A) = 1/7 P(A|B) = P(A) * P(B|A) / P(B) = 1/4 ...
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61 views

Certainty that one has found all of the socks in a pile

Suppose that I have a pile of $n$ socks, and, of these, $2k$ are "mine." Each of the socks that is mine has a mate (so that there are $k$ pairs of my socks) I know $n$, but not $k$. Assume that all ...
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122 views

Probability question using Bayes Theorem

Hello Everyone I am having trouble understanding how to use bayes theorem in this problem: Suppose a physician assesses the probability of HIV in a patient who engages in risky behavior (unprotected ...
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30 views

Simple question about beta prior property

Say that person $A$ thinks that a certain proportion is $0.3$, person $B$ thinks that a certrain proportion in a population is $0.7$. We have a $Beta(4,4)$ prior. How can one mathemtically prove that ...