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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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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2answers
36 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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23 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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19 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
21 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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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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2answers
59 views

Number of tosses needed to determine if a coin is biased [closed]

I toss a coin 3 times to get heads every time. What is the minimum no. of tosses needed to determine with 99% confidence, that this coin is biased?
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1answer
21 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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9 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
18 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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3answers
130 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
25 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
21 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
43 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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2answers
387 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
36 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
17 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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2answers
78 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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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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34 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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1answer
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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0answers
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
45 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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1answer
28 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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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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Why is $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$?

Could anyone derive or explain why the formula $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$ is true? I understand conditional probability definition, but this formula confuses me and makes my head hurt x) Here's ...
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1answer
39 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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1answer
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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2answers
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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1answer
104 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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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
12 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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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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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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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 ...
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82 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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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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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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18 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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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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157 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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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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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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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
24 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 ...