Combining probabilities with Bayes' Theorem, especially as used for conditional inference.

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“Multiple definitions of node p[1]” Error using WinBUGS [on hold]

I have written the following code in WinBUGS and every time I try to compile the data after loading in the data I get the same error which is "Multiple definitions of node ...
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88 views

What would be an example of when L2 is a good loss function for computing a posterior loss?

L2 loss, together with L0 and L1 loss, are three a very common "default" loss functions used when summarising a posterior by the minimum posterior expected loss. One reason for this is perhaps that ...
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27 views

Posterior predictive test quantities

I've been trying to figure out problem 6.2 from Gelman's book, second edition, page 192 on Bayesian data analysis. Can anyone help? a) Set up predictive test quantities to check the following ...
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43 views

How to simplify $P (X|Y, Z)$ when $X$ is independent of $Y$ but not $Z$? [duplicate]

How do I simplify $P (X|Y, Z)$ if I know that $X$ is independent of $Y$ but not $Z$.
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39 views

Bayes Net: how to calculate joint distribution?

I originally posted this question to Computer Science Stack Exchange, but then I was told that CrossValidated site existed. I've been reading many questions, but none of them seem to answer my doubts. ...
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Am I fine in implementing Naive Bayes Classifier?

I have implemented one Sentiment Analysis using Naive Bayes Classifier. To do this I have taken the following steps. First I checked the problem, the nature of data (continuous or discrete) and ...
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172 views

Why is $p(A) \times p(B|A) = p(B) \times p(A|B)$?

Is there an easy way or simple example to get why it must be that $$ p(A) \times p(B|A) = p(B) \times p(A|B) $$? I get that $p(A) \times p(B|A)$ can be seen as probability of $B$ occurring, weighted ...
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37 views

Why is it called “mode” in MAP estimation?

When estimating parameters with MAP, why is it written that we are estimating the "mode"? I thought it would be the mean of the posterior distribution?
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42 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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How to compute the probability of $P(a|b,e)$ given $P(a|b), P(a|e)$ and $A$ independent of $B$?

The following is given: $$ \begin{align} P(b) &= P(e) = 0.1\\ P(a|e) &= 0.2\\ P(a|b) &= 0.95\\ P(a|\neg b, \neg e) &= 0\\ B &{\perp\!\!\!\perp} E \end{align} $$ Is there any way ...
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Describing a distribution of probabilities/percentages

Is there a better way to describe this probability? \begin{equation} Pr(Y_{sij}|R_n) = \frac{Y_{sij|R_n} \sum_{ij}Y_{sij}}{\sum_{ij|R_n} Y_{ij}\sum_{sij}Y_{sij}} \end{equation} The $R_n$ describes ...
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Conditions under which “explaining away” does vs. does not occur

My understanding of "explaining away" is as follows. If there is an effect, C, that can result from two independent causes, A and B, then observing only C makes A more likely than observing both C and ...
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30 views

Marginal distribution MLE or MCMC

I'm a bit confused about how to maximise the following likelihood: $\mathcal{L}(k, \lambda, p) \sim \mathrm{Binomial}(n, k, p)\mathrm{Poisson}(\lambda, n)$ i.e. my probability is relatd to a number ...
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Resources for the “ah ha” moment when learning Bayes' theorem

I've been studying statistics for a little while and I keep coming back to Bayes' theorem trying to relearn it and have that "ah ha" moment with it. I keep coming back to it because I understand just ...
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86 views

Would a one-tailed t-test technically be considered Bayesian?

I'm just curious whether the expectation implied from a one-tailed test would somehow be considered a prior, and whether this is enough for it to be in the purview of Bayesian statistics.
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48 views

Deriving the Bayes Filter Correction Equation

The correction rule for Bayes filters is: $$p\left(x_{k}|D_{k}\right)=\dfrac{p\left(y_{k}|x_{k}\right)\cdot p\left(x_{k}|D_{k-1}\right)}{p\left(y_{k}|D_{k-1}\right)} $$ For: State at time $k$ is ...
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How to combine probabilities, weighting by strength of evidence?

Let's say that I'm trying to classify an item as a member of either class c1, class c2, class c3, etc. out of a large number of classes. In my training set, feature A appears only once; it happens to ...
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21 views

Terminology Numerator Baye's Rule?

I am considering this formulation of Baye's Rule $\mathrm{Pr}(\theta | D) = \frac {\mathrm{Pr}(\theta)\mathrm{Pr}(D|\theta)}{\int \mathrm{Pr}(D|\theta)\mathrm{Pr}(\theta)\mathrm{d}\theta}$ Is there ...
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514 views

What math/stats knowledge does learning Bayesian probability require?

I study undergraduate "pure" math and philosophy. I know that a number of philosophers use Bayesian probability to augment their epistemic logic. My school teaches Bayesian probability as a brief part ...
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3answers
233 views

Why Normalizing Factor is Required in Bayes Theorem?

Bayes theorem goes $$ P(\textrm{model}|\textrm{data}) = \frac{P(\textrm{model}) \times P(\textrm{data}|\textrm{model})}{P(\textrm{data})} $$ This is all fine. But, I've read somewhere: ...
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43 views

Updating belief

Lets say I think that event 'E' has a probability 'p' of happening. Then I go around and ask a number of friends if they think event 'E' will happen, they can either answer Yes or No. Now I know that ...
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Normal learning - multiple signals

I'm having trouble with the exercise below. I know that $E(η_t|z_t)= E(η_t) + [Cov(η_t,z_t)/Var(z_t)](z_t - E(z_t)) $ but still can't show 'b'. I imagine I'm missing something very simple... Can ...
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71 views

Guessing the length of a fish

I would like to solve the following excercise. Any help is appreciated. 90% of the fish in our pond are males, the rest are females. The length of the males are: $X+5$ inches, where $X\sim ...
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80 views

Problem expressing full conditionals

I have this problem, $Y_{i}$~Gamma($\alpha$,$\beta$) 1...N $\alpha$~Exp($\lambda$) $\beta$~Exp($\lambda$) $\lambda$=0.001,Find the full conditionals. I have done the following: ...
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Naive Bayes classifier to identify “mouse-like” human tumor from mouse gene signature

I am a PhD student in biochemistry, and my lab has a mouse model for breast cancer. We have a gene signature that I would like to use to identify "mouse-like" human tumors for further analysis. I have ...
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Complement naive bayes

I would like some help in understanding how does Complement Naive Bayes work. I have googled the paper Complement Naive Bayes I understand that naive bayes works by computing the probability of a ...
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80 views

Implementation of Bayes posterior predictive check

I have a question concerning the implementation of a bayes posterior predictive check. Let us assume i have this model (implementation is in R and jags): ...
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77 views

What is meant by this formulation of Bayes' Rule?

From the Wikipedia article on Bayesian inference, we get the following formulation of Bayes' Rule: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) p(\theta \mid ...
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20 views

Meta analysis, joint posterior distribution of study effect

Meta analysis (with common study variation $\sigma$) often assumes that: $$ X_{i,j} | \theta_i \overset{ind}{\sim} N(\theta_i,\sigma)\\ \theta_i \overset{i.i.d.}{\sim} N(\mu,\tau) $$ where ...
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51 views

Probability of being each demographic

Relatively simple problem, but can't see how to solve it. Rephrasing in terms of everyday events. Assume there is a fair in town for three days. Each day the total visitors are known, and so are ...
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43 views

Bayes' rule and multiple conditioning

This is going to be a stupid question, but I know I am missing something, so here it goes: My textbook (Probabilistic Robotics, p.17) trivially states that Bayes' rule gives: $ p(x |y,z) = \frac{p(y ...
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Am I thinking correctly about this Bayes Problem

I am playing with the following problem. Recently a company has had some resignations. About x out of Total people resigned in one year. Some of the resignations were due to natural movement, ...
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47 views

Bayesian inference

Assume two demographics $[F,M]$ and each person has a choice of attending only one of four different lectures $[A,B,C,D]$ all occurring at the same time so they can only attend one. The following ...
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42 views

Conjugate prior equivalent prior sample size with respect to the mean

In Cowles's book ([Applied Bayesian Statistics - With R and OpenBUGS Examples–(http://www.springer.com/statistics/statistical+theory+and+methods/book/978-1-4614-5695-7)), page 108, there is a ...
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37 views

winbugs multiple definition of node [duplicate]

I wrote this code in WinBUGS but I cant run it! It says 'multiple definitions of node tau' ...
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95 views

Optimal Stopping for Bernoulli One-Armed Bandit with a Fixed, Known Payout

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. Suppose I have two arms $k = {1, ...
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MLE == MAP under Uniform Prior? [duplicate]

Does Maximum Likelihood Estimation always yield the same result as Maximum a Posteriori with uniform prior?
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21 views

Variance of precision in conjugate prior

How can I calculate the variance of the precision in a normal distribution, knowing I used a conjugate prior?
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How important is the quality of solutions to NP-hard problems arising in machine learning problems?

Machine learning and inference problems give rise to intractable problems. For instance the exact inference in Bayes nets is an NP-hard problem. At the same time there are polynomially tractable ...
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Two Gaussian Likelihoods with Two Decision Boundaries , 0/1 loss function

we assume that Y := {1, 2}. Then our decision can be re-written as y ∗ = {1 if p(x|y = 1) > p(x|y = 2) , 2 otherwise} with a decision boundary at p(x|y = 1) = p(x|y = 2). How can we construct an ...
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Bayesian calculation of parameter multiplying normal

I am interested in a model like: $y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is. I ...
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45 views

Bayesian Network

I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
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95 views

How to implement a simple Bayesian Network for Time Series Data?

I'm a computer science grad student, with not much knowledge in Bayesian statistics, so I'm seeking for guidance for the simplest start. I have 10 variables, like demand, price etc. and I want to ...
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65 views

Calcuate Bayes Factor for Adjusted Mean Difference

I have an ANCOVA model shown below (fit) where I calculated the mean difference between two groups while controlling for another variable (x). For the adjusted mean difference (288.72), I'd like to ...
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linear discriminant analysis, Bayes approach authors?

I know that in 1936 Fisher proposed the LDA that minimizes the variance within and maximizes between. My question is, the Bayes approach of LDA is attributed to a particular(s) author(s)? and what ...
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How to apply Bayes' theorem to the search for a fisherman lost at sea

The article The Odds, Continually Updated mentions the story of a Long Island fisherman who literally owes his life to Bayesian Statistics. Here's the short version: There are two fishermen on a ...
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Proof of alternating conditional expectation base equations

How do we prove the base equations for Alternating Conditional Expectation algorithm. The statement is thus: We define arbitrary mean-zero transformation $\theta(Y),\phi(X_i)$,$1<i<p$ for ...
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157 views

What is Bayes decision rule?

Assume binary classification i.e. $y \in \{-1,1\}$ and that the underlying joint probability distribution generating the data is known i.e. $P_{x,y}(x,y)$ is known I was told that Bayes decision ...
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55 views

prior predictive distribution negative binomial in R

I have a prior distribution Gamma(1.71,1.05) from a Poisson(2.2), and I know that the prior predictive distribution will be a Negative-Binomial of Gamma parameters i.e. Neg-bin(1.71,1.05). I would ...
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Picking noninformative priors using pivotal quantities

In 'Bayesian Data Analysis' (Gelman, Carlin, Stern and Rubin) on page 64 it reads: "If the density of $y$ is such that $p(y-\theta|\theta)$ is a function that is free of $\theta$ and $y$, say $f(u)$ ...