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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Bayes estimates

How do I attempt at solving this problem? Could I use proportionality? Bayes estimate of parameter of lambda with Poisson likelihood with x = (1,5,4,4) and gamma prior for lambda with mean = 2 and ...
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19 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
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11 views

Question regarding the density function of first n prediction

This is an example from Bertsekas' Introduction to Probability 2nd edition example 8.2 Consider now a variation involving the first $n$ dates. Assume that Juliet is late by random amounts $$X_1, ...
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2answers
28 views

In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
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35 views

Proof using Bayes rule?

In Statistical analysis of randomized experiments with non-ignorable missing binary outcomes:an application to a voting experiment by Kosuke Imai a proof is given referring to Bayes rule. Let: ...
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Working out closed form of shifted poisson distribution

In the article "Bayesian variable selection for Poisson regression with underreported responses" the author defines $t_i^0$ as the number of actual occurences in a study in the $i$th covariate ...
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23 views

Bayesian Belief Network, finding probabilities

I draw the Bayesian Belief Network for the question as: $C<-W->M$ But i have no idea how to find prior probabilities $P(C)$ and $P(M)$. Any suggestions?
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1answer
23 views

conditional probability of joints in bayesian net [duplicate]

I have been staring at a bayesian net for an hour and can't understand why this is correct to write: $$P(W|B,E)\cdot P(E)\cdot P(R|E)= P(W,R,E|B)$$ Note that the joint probability of $P(A,B,E,W,R)$ ...
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11 views

distribution of the length for a random walk on an infinite 2D grid

In connection with the flatland paradox, consider a 2D-random walk $(X_n)$ on $\mathbb{Z}^2$: the four moves of length one to W,E,N, and S are equaly likely at each time. For a fixed number of moves ...
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35 views

conditional probability of joints

I have been staring at a bayesian net for an hour and can't understand why this is correct to write: $$P(A|B,E)\cdot P(W|A) = P(W,A|B,E)$$ Note that the joint probability of $P(A,B,E,W,R)$ can be ...
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1answer
17 views

Independent values in joint probability tables

I am looking at a problem in a text book and it asks "how many independent values in a joint probability distribution for eight boolean nodes, assuming no conditional independence relations are known ...
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23 views

a variant of MLE of a normal distribution

It is well-known that if we have "n" sample observations from normal distribution with unknown mean, then the sample mean would be the MLE for the mean of the normal distribution. However, let's ...
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14 views

Evaluating an expected value in Jeffrey's prior for binomial distribution

The material I'm reading derives Jeffrey's prior (or rather, the Fisher information for the Jeffrey's) for single-parameter binomial distribution in a manner quite similar to this Wikipedia article. ...
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16 views

Product of normal densities in a Bayesian context

Two analysts, analyst A and analyst B, are interested in the probability distribution for a multivariate-normal vector $X$ with five dimensions. A estimates a density function $f_X(X=x)$ for $X$, ...
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86 views

Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
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2answers
11 views

Bayesian Network Probability involving intersection

Imagine a node "I" with two children, "W" and "H". "I" means that roads are icy, and "W" means that Watson crashes. "H" means Holmes crashes. If I wanted to know the probability of the roads being ...
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28 views

Bayesian Estimation of the mean of a multi-variate Gaussian

The posterior mean of a multivariate normal distribution is to be estimated with the Bayes rule for densities (http://www.math.uah.edu/stat/dist/Conditional.html), following the approach as described ...
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65 views

Is conditional probability $P(A\mid B)$ proportional to $P(B\mid A)$?

It feels a bit odd but since $$P(A\mid B) = \frac{P(A,B)}{\sum_A P(A,B)} \propto P(A,B)\text{ and }P(B\mid A) = \frac{P(A,B)}{\sum_B P(A,B)} \propto P(A,B)$$ can we say that $P(A\mid B) \propto ...
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34 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
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20 views

Is there any closed form for the integration of multiplication of two multivariate normal probability distributions?

I already computed the following integration but its a messy thing. I wonder if there is any easy way to compute it? or it has any closed form? V and p are known where V and p (p<1) are positive. ...
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19 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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24 views

How to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
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32 views

simplify the division of popular probability density function

This is my first question in Mathematics on Stack Exchange. Please forgive that this is a none sense question... Question I'd like to know a simple form of the division of popular probability ...
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1answer
29 views

In Bayes' theorem, what is little $p$?

In Wikipedia's conjugate prior article, Bayes theorem is given as: $$p(\theta|x) = \frac{p(x|\theta) \, p(\theta)} {\int p(x|\theta') \, p(\theta') \, d\theta'}.$$ What is $p$ here? Is it the ...
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28 views

Difference between Frequentist and Bayesian approach to Statistics

What is the difference between the Frequentist vs. the Bayesian approach to Statistics? Would someone be so kind to come up with a simple example that shows how the approaches and possibly the ...
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1answer
43 views

Maximum likeliood estimation of variances of transformed variables

I use MATLAB's fminunc function in order to find the minimum of a negative log-likelihood function $f(\overrightarrow{\theta})$, parametrized by 3 parameters lets say ...
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1answer
30 views

How to do continuous-time Bayesian updating?

I am reading a game theory lecture notes. Some parts involve a continuous time Bayesian updating computation which I didn't really understand. There are two states $\{Good,Bad\}$. At time t people ...
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25 views

Trouble understanding how Naive Bayes Classifier is derived

I've come across the Naive Bayes Classifier while studying machine learning, but the trouble I'm having is with some of the probability theory used to derive the formula for finding the optimal ...
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2answers
63 views

How can Bayesian and Frequentist approach be different?

Let's say I am trying to add numbers, like say one to ten. I can either add them in order, or I can notice that I can group them into five groups of eleven, so I suppose which method to use depends on ...
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1answer
17 views

What prior to use given a Poisson likelihood?

I am trying to incorporate a prior into a model I am working on. From available data, I have found that the likelihood follows a Poisson distribution with $\lambda = 1.5$. I have then used R to ...
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1answer
46 views

Cluster probabilites: Bayesian network (sprinkler example, Russel/ Norvig) as a clustered network

like others here I am also learning with Russel's and Norvig's book about artificial intelligence. My question is about the conditional probability tables of a clustered multiply connected network ...
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109 views

Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
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30 views

Bayesian Networks - Probability of variables with a common parent

I'm having some trouble figuring out a homework assignment which requires me to find the probabilities of two different variables that have a common parent. In order to better understand how to do ...
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13 views

Inverse-Wishart distribution pdf is different if we derive it directly from Wishart distribution?

According to Wikipedia, there is the following relation between the Wishart and the inverse-Wishart distribution: "If ${\mathbf A}\sim \mathcal{W}({\mathbf\Sigma},\nu)$ and ${\mathbf\Sigma}$ is of ...
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470 views

Why does Bayes' theorem work?

Why does Bayes' theorem work? I'm not looking for a cryptic math demonstration. Rather, what I'm interested in is the intuition behind the theorem that allows to obtain the a posteriori probability ...
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1answer
35 views

Inferring poisson rate from interval determined by data

I have a dataset of arrivals, which are from a Poisson process. For the purposes of this question, let's say they're arrivals of cars on a particular road. My goal is to infer the gamma posterior for ...
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1answer
46 views

Bayes vs frequentism and the fair coin

Suppose I have a coin, which I want to test for bias. My problem is: surely there's a philosophical problem with defining "bias". Let me illustate with an example. Firstly, I use a Bayesian approach, ...
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8 views

Discriminant Functions of two classes sharing same covariance matrix

How can i find the discriminant functions of two classes having same diagonal covariance matrix with different means? (their feature vector is two dimensional) Thank you!
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26 views

Log likelihood function for binary classification

I need help with this following task. There is a binary classification problem where each observation xn is belong to one of two classes (t = 0 and t = 1). The training data points are sometimes ...
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1answer
26 views

how to calculate crash probability?

A plane crashes with probability 0.95 if both of its engines fail. On each flight each engine has a probability of failure of $10^{-5}$. Both engines fail with probability of $10^{-9}$ a) Are the ...
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1answer
23 views

using bayes' rule

Women carrying a certain gene are ten times more likely to develop breast cancer. Only 1 out of 100 women carries this gene. If a woman has breast cancer, what is the probability that she carries this ...
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32 views

Bayesian networks: What's wrong with my answer?

Consider The following four random binary variables: Given the following Bayesian network: With the following conditional probability tables: I want to calculate the probability that ...
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1answer
48 views

Posterior distribution of $\theta$

Let $X_{ij} ~ N(\theta_i,\sigma^2)$ with $\sigma^2$ known, i = 1,... k, and j = 1, ... ,$n_i$. The prior distribution of $ \theta_i$ is $N(\phi,\tau^2)$, independently for i = 1,...,k and ...
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63 views

Uniform lattice sample inside a particular convex polytope

[update]: hardmath suggests using tools from linear programming. This looks like a good idea indeed. I can now tell that my feasible set is described by: $Set = \{d \in \mathbb{N}^c, -B.d\leqslant ...
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10 views

Finding a bayes estimator

Let $X_1,...,X_n|\eta~\exp(1,\eta)$ and $\eta$~$N(\mu,1)$, where $\mu\epsilon\Re$. Find the Bayes estimator $\eta$ under the squared error loss. After finding the joint likelihood of $exp(1,\eta)$ ...
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52 views

Baye's Classifier for recovering a signal from a measurement

Below is the question i am having trouble with: Independent and identically distributed symbols s(n) = ±1 are transmitted over the channel C(z) = 1 + z −1 . Symbols s(n) = +1 occur with probability p ...
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40 views

Bayes factor for fair and biased coin

There is the following task: Suppose we toss a coin $ N = 10$ times and observe $m = 9$ heads. Let the null hypothesis be that the coin is fair,and the alternative be that the coin can have any bias, ...
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Conjugate priors: wht not binomial-binomial?

Citing from Kevin Murphy's machine learning book: When the prior and the posterior have the same form, we say that the prior is a conjugate prior for the corresponding likelihood. Conjugate ...
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1answer
36 views

Bayesian hypothesis testing

Let $x_1,\ldots,x_4$ be a sample taken from the uniform dstribution with the density $$ f_{\theta}(x)=\theta^{-1} \cdot 1_{(0,\theta)}(x). $$ Assume that $\theta$ is a random variable with the ...
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19 views

Calculate CPT of bayes net

I have a Bayes net of a pretty simple construction. I need to find the expressions that the CPT's represent and also the number of entries. A--B--C .....| ....D A is the parent node of B. B is ...