# Tagged Questions

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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### Prior probability distribution question

Let, $\mathbb H = (\theta_{1},\theta_{2},..,\theta_{k})$, where $'\mathbb H'$ denotes the parametric space. Let $X_{1},X_{2},...,X_{n}$ be $'n'$ i.i.d. observations from a common density function, ...
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### What does Bayesian Method estimate?

I am not quite sure what Bayesian method want to infer from the data? Frequentists assume the parameters generated the data are fixed constants so they can actually estimate those constants. ...
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### Non-independence of three events, given their intersection is independent

Suppose you flip a fair coin three times, find three events $A$, $B$, and $C$, such that no two of them are independent, but $P(A \cap B \cap C) = P(A)P(B)P(C)$ I am given the question above. ...
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### Flipping a fair coin three times: How to find independent/dependent events?

Given that I have a fair coin which I toss three times, I have the following sample space: $S=\left\{HHH , HHT , HTH, THH , HTT, TTH, THT , TTT\right\}$ how do I: a) Find three events A , B, and C ...
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### Finding the Complicated Posterior Probability Distribution of $θ$

Suppose, we are given a likelihood function, $f(x|θ)$ which follows a shifted-exponential distribution and the prior distribution of “$θ$” is Standard Cauchy distribution. Now the problem is – I am ...
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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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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} X_{3}...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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