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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Estimating quantities of a posterior distribution.

Consider the following model: $$ \alpha \sim N(0,1)$$ $$ \beta \sim N(0,1)$$ $$ d_i \mid \alpha, \beta \sim \mathrm{Bernoulli}(\Phi(\alpha + \beta x_i))$$ $d_i$ is $1$ if person $i$ has some ...
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2answers
19 views

Can't find intersection of two probabilities.

I have the following problem: While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation ...
3
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1answer
103 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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0answers
18 views

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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1answer
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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87 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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0answers
12 views

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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1answer
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 ...
0
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0answers
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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2answers
337 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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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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6answers
5k views

Bayes Theorem Example in Nate Silver's The Signal and the Noise

In his book The Signal and the Noise, Nate Silver presents this example application of Bayes's Theorem on pp. 247-248: Consider a somber example: the September 11 attacks. Most of us would have ...
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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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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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2answers
153 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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0answers
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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0answers
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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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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1answer
23 views

Conditional distribution for a label given a scalar feature

I am trying to create a simple simulation setup for classifiers on toy data. Each data point can has a scalar feature $X$, which is uniformly distributed between -1 and 1. Depending on the feature, ...
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1answer
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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2answers
34 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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0answers
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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0answers
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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0answers
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 ...
0
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1answer
33 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 ...
2
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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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1answer
428 views

Revising probability using Bayes' rule

Let's say $100$ numbers is picked from a set of numbers one by one with replacement, which is between $000$ to $999$. I'm required to give my subjective probability before the experiment and revise ...
0
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1answer
30 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 ...
2
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1answer
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 ...
0
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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 ...
4
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1answer
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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1answer
26 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 ...
3
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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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2answers
858 views

Calculating Laplace's law for bigrams

Reading this PDF, I encountered a very simple simplification that I can't obtain. Basically, it asks what is the probability of the occurrence of a word $w_{n}$ given that we know another word ...
0
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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 ...
0
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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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1answer
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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0answers
22 views

Textbook recommendation for Non-parametric Bayesian?

I am looking for textbooks which include basics as well as advanced models like latent Dirichlet allocation, hierarchical Dirichlet process. The most important thing is that those books should present ...
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0answers
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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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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471 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
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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1answer
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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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!