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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23 views

What is this distribution formulated with w, m and sum sign?

I have a binary classification problem, part of which is defined as follows : p(x|y=1) $\sim w (m_1 , \sum_1$) and p(x|y=0) $\sim w (m_0 , \sum_0$) Where $\sum_1$ is a covariance matrix : $$ ...
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105 views

Why a beta distribution with the parameters $\alpha=0$ and $\beta=0$ as a prior is bad

what happened if I define a beta distribution with $\alpha=0$ and $\beta=0$ as a prior? in other words if $p(\theta) \varpropto \frac{1}{\theta(1-\theta)}$. Thanks
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2answers
58 views

Bayesian learning for input “If A, then B.”

Can anyone point me to literature on Bayesian learning when the new information has the form “If A, then B”? I’m familiar with the rule that after one learns X, posterior probability P(Y) equals prior ...
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41 views

multiplication of 2 PDFs

If I multiply the two PDFs, does the variance of the result PDF becomes narrower than the two PDFs always? In other words, if I multiply likelihood and prior to get the posterior, is the variance of ...
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67 views

How do I prove and expand Bayesian Networks?

Attempting to understand Exercise 20 (pdf page 44) in the paper: (Warning: large paper; small exercise) Bayesian Reasoning and Machine Learning The party animal problem corresponds to the ...
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47 views

Does this question work with Bayes formula?

Looking at slide 11, Example 1.10 from: http://www-users.aston.ac.uk/~cornford/probmod/ProbMod310810_Ch1.pdf Luke has been told he’s lucky and has won a prize in the lottery. There are 5 prizes ...
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20 views

Probability of one node given all the others in a bayes network

For a bayes network which has $n$ nodes, $X_1, X_2, ... , X_n$. Is there any efficient way to calculate $P(X_i|X_1,X_2,...,X_{i-1},X_{i+1},...X_n)$, without constructing the full joint distribution?
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52 views

conjugate prior

A class of sampling distribution is a conjugate family of a prior distribution, if the posterior distribution belongs to the same family for all priors and all samples. Why is this phrase incorrect?
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161 views

Estimating conditional probability as a function of time

My question relates to estimating from a time series a time dependent conditional probability without having a prior parametric model of anything. Suppose I have two variables: r and I, and each can ...
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104 views

Bayes factor and Posterior odds

Consider the following posterior odds \begin{equation*} \frac{P(H|D_1,D_2)}{P(\overline{H}|D_1,D_2)}=\frac{P(D_2|H,D_1)\times P(D_1|H)P(H)}{P(D_2|\overline{H},D_1)\times ...
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88 views

determining maximum a posteriori (MAP) hypothesis

I have this problem: You are given a coin that may or may not be biased. Specifically, you have three hypotheses about the coin: ...
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41 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. •Data∼Gamma(α,β) •Parameters α∼Gamma(kα,θα) β∼Gamma(kβ,θβ) I used Winbugs (code ...
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139 views

Bayes' Theorem Question, with a twist

I have a very old past high school exam question I am trying to solve (for interest only). It's a straightforward application of Bayes' Theorem, with the last part of the question containing a slight ...
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27 views

Growing of a score function

The argument that I'm dealing is very specific, I hope to make you understand the problem without going into detail. I have this score function: \begin{align} score = MargL^q + MargL^{\theta} ...
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1answer
51 views

When using Bayes Rule, what are the rules for flipping the conditions and the event of interest?

Here is Bayes Rule: $$P(A\mid B) = \frac{P(B\mid A) P(A)}{P(B)}$$ This paper (http://www.cogsci.northwestern.edu/Bayes/Sivia_1996.pdf) uses Bayes rule on page 21 in the context of model selection ...
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157 views

What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
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91 views

Bayesian Probability Question - Parameter Estimation

I would like help on the following question and I will show my work. Here is the question in my notes and I will follow up with my work: Q: Suppose a forest is segmented into strips, referred to as ...
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31 views

Is a prior distribution always a random probability measure?

Let $(\mathcal{X}, \mathcal{B})$ be a measurable space and let its probability measure be $P$. In Bayesian statistics, we may wish to define a prior $\mu$ on the space of all such probability ...
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21 views

Determining the liklihood in Baye's rule for parameter estimation

I have used Bayesian statistics in classes but what I am trying to do now is different than anything I have done in class. Previously, I was given information and certain numbers adn I could ...
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78 views

Monty Hall Problem Extended Using Bayes's Theory

I know there is a question on the website concerning the extension of the monty hall problem. The question is provided with very good answers given by the participants on the website which I would ...
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1answer
107 views

completing the square for matrices

I'd like to calculate the posterior distribution given the prior distribution $w\sim N(0,\Sigma_p)$ and the likelihood $y|X,w\sim N(X^\top w,\sigma_n^2I).$ Ignoring everything that does not contain ...
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64 views

Monty Hall Problem Solve Using Detailed Algebra

I have been searching the monty hall problem for two days now and I generally understand it but I am having a very hard time solving the monty hall problem using Bayes's theory. I do not know what ...
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67 views

Help with Bayes's theory

I know how to use this form of the Bayes's theory : $P(A | B) = P(A ∩ B)/ P(B)$ But how do I use?: $P (A | B,C) = P (B | A,C) P(A | C)/ P(B | C)$ What does the comma mean? I know its a simple ...
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105 views

Finding a posterior distribution of an exponential distribution parameter theta

Suppose that $X_1, ... , X_n$ each have an exponential distribution with parameter $\theta$, and suppose that the prior for $\theta$ is an exponential distribution with parameter $\lambda$. Find the ...
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31 views

Is this problem suited for Bayesian inference?

Suppose that the quality of a widget is distributed according to a score, given by a normal distribution with mean 1 and variance σ^2. A fraction, π of all widgets are defective. The cost of having an ...
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112 views

Probability of independent events $P(ab)=P(a)*P(b)$

I know there are two ways to say event $a$ and $b$ are independent: $P(a)*P(b)=P(ab)$ $P(a\mid b)=P(a)$ and I can derive one from the other with the Bayes Formula $P(a|b)=P(ab)/P(b)$. My question ...
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55 views

Generalized Bayes Estimator

Consider a decision problem in which the model parameter, $\theta$, is any integer, the distribution for the integer observation, y, given $\theta$ is $P(y|\theta) = 1/3$ if $y \in [\theta - 1, \theta ...
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42 views

Why is the marginalized inverse-Wishart distribution not equal to the inverse-gamma distribution?

Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has ...
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55 views

Give the Bayesian Posterior Mode

Suppose that $X_1, X_2, \ldots, X_n$ are IID Bernoulli random variables with success probability equal to an unknown parameter $\theta \in [0,1]$. Let $A$ and $B$ be nonnegative constants. If we ...
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38 views

Conditional probability with bayes rule??

http://cseweb.ucsd.edu/~dasgupta/103/2b.pdf part 2.1.2 implies $P(X|Y \cap Z) = \frac{P(X|Y)}{P(Y|Z)}$ Seems to imply that this is true but if you take bayes, the left hand side is: $P(X|Y \cap Z) = ...
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90 views

Conjugate priors make calculations easier but at what cost to the model?

As I understand, when we have a parametric pdf and need to estimate the parameter based on some observed fact, we tend to choose a conjugate prior of the pdf for the parameter. Because conjugate prior ...
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37 views

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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36 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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3answers
152 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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59 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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1answer
81 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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2answers
117 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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1answer
182 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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37 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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63 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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131 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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1answer
53 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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27 views

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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110 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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77 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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28 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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454 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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110 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 ...