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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Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
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16 views

Sequential information discovery in minimum number of steps when some items have information about other items

There are N items, say three: call them A B and C. For each item, there is an associated bit (0 or 1) and there is a prior probability that the bit is 1, call them p(A), p(B) and p(C). There is some ...
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53 views

Ten marbles put in a box, colour of each set by toss of a fair coin. You draw (with replacement) ten white marbles. Probability all marbles are white?

The following question comes from the probability section of the Titan Test*. * I will avoid the debate around whether this test accurately measures what it aims to, nor whether such aims are ...
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31 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
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17 views

Markov-Chain Monte-Carlo: Are transformations on the inputs valid?

The problem: I am trying to solve a high dimensional (up to ~50) class of data fitting & modelling problems. The user specifies the problem, so I would like to make the configuration as easy as ...
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30 views

Estimating the number of classes in a finite population [on hold]

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
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8 views

Creating pdfs froms Sample Data and Bayes Theorem for Continuous Probability

I am not much of a math guy, but know some basics of pdfs, pmfs, Bayes theorems, probability distribution and stuffs. I am actually trying to build a Bayesian Network that models the personality of ...
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29 views

Bayesian Parameter Estimation - Parameters and Data Jointly Continuous?

This is a follow up to my previous question regarding viewing parameters as random variables in a Bayesian framework. If we apply Bayes' theorem to model parameters $\mathbf{\Theta} \in \mathbb{R}^n$ ...
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41 views

Bartlett's paradox in Bayesian evidence

I've come across Bartlett's "paradox" (not to be confused with Lindley's paradox, also known as the Lindley-Bartlett paradox) in Bayesian statistics. The paradox originates from Bartlett's 1957 paper, ...
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16 views

Bayes' Rule for Parameter Estimation - Parameters are Random Variables?

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathbf{X}: \Omega \to \mathbb{R}^n$, $\mathbf{Y}: \Omega \to \mathbb{R}^m$ be jointly continuous random vectors. That is, there exists ...
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40 views

Understanding Conditional Probability Basics

In many online sources I've read a statement similar to: Probability of B happening given A is equal to the probability of A and B both happening divided by B happening or $p(A | B) = p(A \cap ...
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75 views

Baye's Theorem Conditional Probability with multiple conditions

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer - Gender (M/F) & Smiling (Y/N). Assume this is historical data of purchases: ...
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1answer
33 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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20 views

Bayesian Updating - plug in previous posterior for prior?

Let's say I have two sequences of observations, $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$. For each sequence I'm going to estimate the probabilities of certain events occurring, namely event $A$ in ...
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1answer
23 views

Delaunay Triangulation on Convex Polytopes — Uniform Sampling

My goal is to uniformly sample from a convex polytope. I know that for the simpler case, where I have to uniformly sample from a simplex, I can use Bayesian Bootstrap, discussed in these posts: ...
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21 views

Can anybody help with this state space model for filtering

I need an urgent help in an issue with a state space model for filtering. My state model is like: $\mathbf{d}_k = \mathbf{d}_{k-1} + \boldsymbol{\eta}_k$ with $\boldsymbol{\eta}_k \sim \mathcal{N}(0, ...
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1answer
24 views

Bayes' Rule where the probabilities are taken as conditional

I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below. ...
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10 views

ANSML - Proof of Naive Bayes Derivation

I was working through one proof of the Naive Bayes and got stuck at the last step. The setup is as follows: Given a dataset $\left\{ (x^{(i)},y^{(i)}), \cdots\right\}$ for $i=1,\cdots,m$, $y$ can ...
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1answer
19 views

Is a Bayesian credibility estimate in the presence of conjugate priors always a linear function of the data?

I only know four examples of families of distributions with conjugate priors: Poisson/gamma binomial/beta exponential/inverse gamma normal with known variance/normal The Bayesian credibility ...
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150 views

Facebook Question (Data Science)

Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview) You're about to get on a plane to Seattle. You want to know if you should bring an umbrella. You call 3 ...
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1answer
26 views

Application Problem of Expected Value of Posterior Distribution

I am trying to understand the following: Suppose that the number of people who visit the grocery store on any given day is Poisson($\lambda$) and the parameter of the Poisson distributed has a ...
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1answer
24 views

Derivation of Likelihood Function for Random Effects Parameters

I initially posted this question in CV, but getting no responses or interest, I am deleting it there, and trying my luck in math.stackexchange, hoping that the math details of the following derivation ...
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1answer
46 views

Posterior Distribution and Expected Value of a Coin Toss where Probability of Heads is a Random Variable

I am trying to solve the following: Suppose X is the number of times a coin is tossed until a heads is observed. Let Y denoted the probability of observing heads and assume $f_Y(y)=ky^2$, ie the ...
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1answer
37 views

Combining probabilities from different sources

Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the ...
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1answer
22 views

Non-integer $n$ in sample size problem

Setup Consider a sample size determination problem with the maximization of expected utility approach (as in Lindley 1997). Let $\theta$ be the state, $x=(x_1,\dots,x_n)$ a sequence of $n$ iid ...
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13 views

Adjusting share based on location

Pardon me for the inaccurate title - I just do not know how to phrase it better. Let's assume I have a data table. The table describes results of a business survey in a country. An excerpt from the ...
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35 views

Confusion with Bayesian Linear Regression

In the book Gaussian Processes for Machine Learning in Chapter 2 p. 11 (see http://www.gaussianprocess.org/gpml/chapters/RW2.pdf), eq. 2.9 states: $p(f_* | X, y) = \int p(f_* | x_*,w) p(w|X, y)dw$ ...
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25 views

Player's View: Probability of number of certain die on table given dice in hand

I'm trying to make some AI for a single player version of a dice game named Dudo. The relevant aspects are that there are six players with six dice each (which only they can see until the end reveal), ...
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10 views

Performing inference on a further area of study, Bayesian model.

Consider the following model: $y_i \sim \text{Poisson}(n_i \theta_i)$ $\theta_i \sim \text{Gamma}(\alpha, \beta)$ $\theta_i \sim \text{Gamma}(\gamma, \delta)$ All other variables are constant. $ i ...
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1answer
29 views

How to use Bayes's rule with mixed distributions?

On page 81 of The Likelihood Principle by Berger and Wolpert (1988) I find the following claim (which references example 20 on page 75). We consider a certain statistical problem from a Bayesian ...
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46 views

Translation:Bayes Classificator -> precise math?

I want to understand the most simple form of the Bayes classificator (see here) but I want to understand it in a really precise, clean, mathematical way. Math description of the setting: Let us ...
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6 views

Deriving conditional distributions for a normally distributed change point problem

Considering the change point problem of $y_i \left\{ \begin{array}{ll} y_i \tilde{~} N(u_1, \sigma) & i=1,..,t \\ y_i \tilde{~} N(u_2,\sigma) & i= t+1,...,n \\ \end{array} ...
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79 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: \begin{equation} p(x)=\int ...
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1answer
41 views

Dynamic game of incomplete information

Consider a 2-player game: You and a robber. The robber tells You to give him all your money, otherwise he will kill You. However, the robber could be a 'Good' person (i.e. he would not kill You ...
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26 views

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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22 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 ...
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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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21 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
13 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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45 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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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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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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85 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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36 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
22 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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24 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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19 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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103 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 ...