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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Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
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37 views

Bayesian probability

recently I bumped into following puzzle and I would like to validate(or correct) my results as I asked several people and got several different answers. You are planning a picnic with your friends ...
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12 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
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51 views

Probability of number of people who know a rumor

Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
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Expectation Maximization (EM) for 3-dimensional parameter $(\alpha,\mu_2,{\sigma}^2)$.

Let $x_i$ where $i=1,...,100$ are iid observations from a mix of two normal distributions with means $\mu_1=0$ and $\mu_2$ and the same variance ${\sigma}^2$. If $\alpha$ is the proportion of the ...
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Hypothesis test in Bayesian statistics

Let $X\sim N(\theta,1)$ and 5 independent observations $X=(4.9,5.6,5.1,4.6,3.6)$. The prior probability that $\theta=4.01$ is $0.5$. The remain values of $\theta$ are given the density of ...
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30 views

How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange? I have a very basic question concerning updating from a prior to a posterior in bayesian statistics. Setting: I ...
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2answers
59 views

Bayesian urn questions

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, ...
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40 views

Facebook Data Science Question (Expected Payout and Probability)

I saw this question on Glassdoor and couldn't seem to find a answer to validate mine anywhere: You're at a casino with two dice, if you roll a 5 you win, and get paid $10. What is your expected ...
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20 views

Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but ...
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27 views

Bayes' Rule Question

I am reading about Bayes' rule, I can solve all the exercise but this one. Suppose you had a checkup, and there is a bad news; you tested positive for "the giggles" and that the test is 99% accurate( ...
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26 views

how to calculate conditional independence

This Bayesian net (click) is given with the binary variables B, F, G and D and the following probabilities $p(B=1) = 0.9$ $p(F=1) = 0.9$ $p(G=1\mid B=1,F=1) =0.8$ $p(G=1\mid B=1,F=0) = 0.2$ ...
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10 views

Bayes Rule and Multivariate Normal Estimation

This is an exercise in this pdf file http://statweb.stanford.edu/~ckirby/brad/LSI/chapter1.pdf and how can I show that by using Bayes Rule?
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14 views

Book recommendations for introductory Bayesian statistics?

Anyone here have some recommendations for a good book introducing the reader to Bayesian statistics? Let me mention my background. My undergraduate majors were in Actuarial Science and Statistics, ...
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1answer
30 views

What does the notation $d|x \sim N(0,14^2)$ stand for?

I'm reading a book about Bayesian data analysis (by Gelman et al.) and I bumped into the following text: $x= \text{Football point spread}$ $y=\text{Game outcome}$ $d=y-x$ For the ...
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19 views

Find pmf for binomial distribution with prior

Let $X$~$Bin(n,P)$ where $P$~$Beta(\alpha,\beta)$. How do I find the pmf for $X$? I have a vague idea that I have to condition on $P\leq \tilde{p}$ to find $Pr(X=x|P\leq\tilde{p})$ but I'm not ...
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11 views

Marginal probabilities

I am stuck on the following problem of calculating marginal probabilities, which I have highlighted in yellow: Given the information below, how do we calculate$ p(X=0|w=\frac{1}{4}), ...
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Bayes risk and Bayes decision

We are considering a sample of size $n$ from an exponential distribution, with parameter $w >0$. We wish to produce an estimate for $d$, for $w$ , with loss function: $L(w, d)=w(w-d)^2$ The prior ...
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18 views

Ranking players and puzzles from performance in a single player game format

I have a 1000 crossword puzzles and a 1000 solvers - each individual is assigned a 100 arbitrary puzzles to solve (so each solver gets exactly 100 puzzles but each puzzle could have 1-1000 solvers) - ...
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1answer
37 views

Integrating a Delta Function of a Sum

As part of an inference project, I'm normalising a prior distribution which vanishes unless the set of $M$ data points $f_1,...f_M$ satisfies $$ \sum_{i=1}^M f_i = 1. $$ Accordingly this is encoded ...
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21 views

How does one compute this bayesian probability?

Assume you have a network as follow (where X->Y implies X is the parent of Y) A->D, B->D How does one compute $P(A,B|D)$? A and B are independent so my intuition tells me $P(A,B|D)= ...
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24 views

Probability of receiving a bit correctly.

Bits are sent through an information channel, the probability of incorrectly receiving a $1$ is 0.02, while the probability of incorrectly receiving a 0 is 0.01. What is the probability of receiving a ...
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67 views

Let. $X \sim \mathcal{U}(0,1)$. Given $X = x$, let $Y \sim \mathcal{U}(0,x)$. How can I calculate $\mathbb{E}(X|Y = y)$?

Suppose that $X$ is uniformly distributed over $[0,1]$. Now choose $X = x$ and let $Y$ be uniformly distributed over $[0,x]$. Is it possible for us to calculate the "expected value of $X$ given $Y = ...
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19 views

Gaussian distribution with Gamma variance

I am using a hierarchical Bayesian model. In one part of it, I have a normal distribution with mean zero and a variance sampled from a Gamma distribution for some hyper-parameters $a_0$ and $a_1$: ...
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1answer
32 views

Posterior Predictive Distribution for a coin toss

In this question, i can work out that the posterior is supposed to be a Beta (r+1, n-r+1) distribution. However, what I am struggling with is how to compute f(X_n+1|theta). Is this the binomial ...
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12 views

Horseshoe estimator posterior

Suppose given the Horseshoe estimator: $Y|\beta,\sigma^2 \sim N(X\beta,\sigma^2 I)$ $\beta|\sigma^2,\tau_{1}^2,...,\tau_{p}^2 \sim N(0,\sigma^2 D)$ $\tau_{j} \sim C+(0,1)$ $\sigma^2 \sim \pi ...
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12 views

Dirichlet process mixture model

I'm reading Nonparametric Bayesian Inference by Peter Müller and Abel Rodriguez. In Chapter 3, there is no proof provided for some formulas but I think I need to know exactly how it was derived if I ...
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17 views

Bayes–If the prior is increased by a factor of n, what happens to the posterior? If using a threshold, does higher prior mean more “false positives”?

If we're using Bayesian inference in two situations where everything is the same, except that the prior in one is n times the prior in the other, is there anything we can say about how the posteriors ...
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1answer
20 views

Using head-to-head results and Bayes' Theorem to modify predictions of sport/game contests that are initially derived from Elo-type ratings

I am working on an extension of the Glicko2 rating system to use in predicting the outcome of sport/game contests that uses the actual head-to-head results of previous meetings of competitors to ...
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10 views

How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
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17 views

Drawing uniform samples from the *range* of a non-invertible function

I am looking for a Bayesian technique to draw samples from a uniform distribution over the range of a non-invertible (that is, there isn’t even a formula) function $\mathbf{f}: \mathbb{R}^N ...
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28 views

Vector Euclidean norm upper bound by his coordinates average.

I'm trying to extend the Rademacher complexity and have the following question: For $ (v_1,..,v_m) = V \in {\mathbb{R}}^{m} $ , I will be glad to find an upper abound for the Euclidean norm: $$ ...
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28 views

Bayes Rule for Multiple Dependend Random Variables for parameter estimation

During implementation of Expectation Maximization algorithm I got stock on this one P(X|Y,Z, theta), which I tried to solve as follows however I do not know if it is correct $P(X=x | Y=y,Z=z, theta) ...
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18 views

Proper/Improper Popsteriors and Sample size

Suppose we have a two dimensional parameter $\theta=(\mu,\sigma^2)$, and a prior distribution $p(\theta)$. Let our sample come from a normal distribution with mean $\mu$ and variance $\sigma^2$. The ...
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Getting a feel for the Normal-Inverse-Wishart conjugate prior to multivariate normal distribution

I am trying to get a feel for the Normal-Inverse-Wishart conjugate prior, which I have started to use, sparingly, in my work, where I am trying to cluster multivariate normal data. As Wikipedia ...
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14 views

Relationship between 0-1 Loss and Type I and II error in Neyman Pearson

In the context of hyphotesis test $$H_0:\theta\in \Theta_0$$ $$H_1:\theta\notin \Theta_0$$. Find the relationship between the 0-1 loss defined by $$L(\theta,\delta)=1-\delta \theta\in\Theta_0$$ ...
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10 views

Evidence Approximation

Derivation for Bayesian linear regression Can someone explain how 3.80 is obtained from 3.79? What does completing the square mean in this case.
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If I flip $1$ of $3$ modified coins $3$ times, what's the probability that I will get tails?

We have $3$ modified coins: $M_1$ which has tails on the both sides, $M_2$ which has heads on the both sides and $M_3$ which is a fair coin. We extract a coin from the urn and we flip it $3$ times. ...
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13 views

MAP estimation/Bayesian inference

Suppose that $X$ is a uniform random variable taking values in the range {1, 2, ..., t}. I have two hypotheses: H1="t is 10" and H2="t is 20" and I know $$P(H1)=P(H2)=1/2$$ If I observed that ...
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14 views

Multilinear loss in Uniform-Exponential model

Let a prior $\pi(\theta)=\frac{1}{3}(\mathbb{I}_{[0,1]}(\theta)+\mathbb{I}_{[2,3]}(\theta)+\mathbb{I}_{[4,5]}(\theta))$ and $f(x\mid\theta)=\theta e^{-\theta x}$. Taking the multilinear loss ...
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21 views

Jeffrey's prior

I am currently working on a question, however I am a bit confused about which one I need to work out. Question: Derive Jeffrey's prior $J(\phi)$ when $\theta = e^\phi$ for $f(x|\theta) = \theta ...
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1answer
62 views

Extended Kalman filter for the model x_dot=f(x,u,w)

There is a lot of info about EKF out there but everything I find explains it for the simplified model of the form x_dot = f(x,u) + w; i.e. the process noise is a ...
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56 views

Bayes' formula with three probability

I am having trouble with this problem. I believe I have to use Bayes' formula in this problem, but I notice there are three variables(I'm new to prob stat so I don't know if it would be considered a ...
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8 views

Bayesian Inferences: Finding Posterior HPD Interval

I am currently working with Beta-Bernoulli and Beta-Binomial models. I have been searching around for the specific steps in obtaining the Posterior HPD intervals for both. Does anyone know how to find ...
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24 views

Distribution of states given observations in HMM

Suppose you have an HMM with two states $(S_1, S_2)$ and two observations $(a, b)$. We know the following: $P(S_1|S_1) = 0.5$ $P(S_1|S_2) = 0.25$ $P(a|S_1) = 0.25$ $P(a|S_2) = 0.5$ Initial state at ...
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1answer
46 views

Bayesian Hypothesis Testing Example Questions

I've been really struggling with these 2 questions and was wondering if anyone could give me any help/ advice? For the first one I've tried some calculations using the law of total probability but ...
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2answers
66 views

What is the physical meaning of 'infinite variance'?

I am currently reading the book: Bayesian Logical Data Analysis. In chapter 5 it is mentioned like below: "What happens to the average of samples drawn from a distribution which has an infinite ...
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28 views

When joining probability factors, what is the pattern of deciding which factors are conditioned and unconditioned in the final factor?

For example P(V|W)P(X|Y)P(Z) = P(V,X,Z|W,Y), without necessarily doing it manually using Bayes' Rule, how can you tell what variables are conditioned and unconditioned in the resulting factor?
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24 views

integral of heteroskedastic Gaussian

For a Bayesian analysis I need to solve several integrals of the following kind. Let's start with the simplest 1-D form: $$ \mathcal{I}_k = \int_{-\infty}^\infty s^k ...
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1answer
42 views

Bayesian average with penalty when R approaches 0?

In a system with chunks of arbitrary number (5-200) of questions and quantifiable answers, I'm calculating multiple bayesian average values. One for each one of ...