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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Help needed to solve probability problem

I am trying to solve the following problem. A fisherman is equally likely to go fishing at one of the three ponds A,B,C. The probability to catch fish if he cast his rod at pond A is 0.4 , at pond B ...
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13 views

Why is the notion of an XRP, as opposed to an IID variable, useful in programming?

The most general notion which shares the main properties of i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti. Exchangeability means that while variables may not be ...
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14 views

Looking for some good introductory level resources for Gibbs Sampling

In context of a course in bayesian modelling Im following, im looking for some good resources (videos, lecture slides, texts) about Gibbs sampling.
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21 views

Using loss function to find Bayes estimate

I have a 2 part question, the first I believe I have figured out. The question is: Let $Y_1, Y_2, ..., Y_n$ be a random sample from a gamma pdf with parameters $r$ and $\theta$, where the prior ...
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25 views

“Bayesian decisioning device”

I've been asked to consider a system using Bayes' Theorem. It has a Boolean input vector x and binary output $y$ = {$0$,$1$}. i.e., $x = {x_1, x_2, ..., x_n}$. Given equations: $p(y=1|x) = ...
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Find the posterior of $\theta$.

$\theta \sim \text{ Uniform}(0,1)$ and $X|\theta \sim \text{ Bernoulli}(\theta)$. How would I find the posterior of $\theta$? The likelihood of a Bernoulli is $p^{\sum{x_i}} (1-p)^{n-\sum x_i}$. ...
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24 views

Factorization and conditional independence

Let $X,Y,Z$ be three disjoint subsets of events such that $\chi = X \cup Y \cup Z$. I'm trying to show that $X$ and $Y$ are independent given $Z$ if and only if there exist functions $\phi_1, \phi_2$ ...
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20 views

Bayesian Optimization and Averaging

I read in a statistics book that optimizing the likelihood function (or more generally Quasi-likelihood function) in a Bayesian framework is the same as averaging the posterior means. Why is this ...
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24 views

How to show $P(b | a,e) = P(b)$ given some conditions?

$A,B,C$ are binary r.v.'s such that $B$ and $C$ are independent and $P(a | b,e) = P(a | \overline{b}, e) = 1$. I'm trying to show that $P(b | a,e) = P(b)$. We have $P(b | a,e) = P(a,b,e) / P(a,e)$. ...
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23 views

Bayesian network query

I am having a bit of trouble with something that I imagine is fairly easy. I am wondering how to get the probability of alarm, JohnCalls, and MaryCalls if they have no prior knowledge of their ...
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30 views

Probability distribution of data and likelihood in Bayesian and Frequentive Statistics

I have recently been studying Bayesian as well as Frequentive Statistics (mostly null hypothesis significance testing) and am confused as to the meaning of the distribution of the likelihood and ...
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29 views

Calculating Conditional Probability (Maybe using Bayes Theorem)

Well, I've a bunch of variables, viz $Age$, $Gender$, etc.. Age can take: values from $1$ to $6$ (which are actually coded) Gender can take: $0$ and $1$ I know the $$P(A=1),...,P(A=6), ... P(G=0), ...
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19 views

Minimax of Negative Binomial

Studying for the quals and I am in a deep confusion. This is an exercise from Mathematical Statistics: A Decision Theoretic Approach by Thomas S. Ferguson. Let $X$ has density $$f(x|\theta)={r+x-1 ...
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34 views

Notation for probability distribution (capital P) and density function (lowercase p)

I'm confused by the differences in the notation used to denote probability distribution and the density function. My understanding is that the probability distribution is usually denoted by the ...
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29 views

Find conditional probability of a mixture model

given is the following: A mixture model comprises a non-observable $\{ 0,1\}$-valued random variable $X$ such that $P(X=1)=1-P(X=0)=\pi$ and an observable variable $Y$ such that $Y\mid X=0$ is ...
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37 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
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33 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
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30 views

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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25 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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58 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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33 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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33 views

Estimating the number of classes in a finite population [closed]

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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31 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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47 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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17 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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77 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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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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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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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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26 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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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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169 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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27 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
27 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
47 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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41 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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23 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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14 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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37 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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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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30 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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8 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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81 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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45 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 ...