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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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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33 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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31 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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20 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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13 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}), p(X=0|w=\...
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11 views

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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19 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
39 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)= P(A|D)\...
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26 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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1answer
71 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 $Y = y$...
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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
35 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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13 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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15 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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18 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
32 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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14 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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0answers
18 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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30 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: $$ ||V|...
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29 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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15 views

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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16 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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4answers
45 views

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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17 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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25 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 e^{-x\...
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3answers
62 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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10 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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25 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
47 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
68 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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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 \mathcal{N}\left(s|x,\sigma^2(s)\...
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1answer
45 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 ...
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1answer
24 views

Deriving the joint posterior pdf as a decomposition in terms of…

Really struggling with how to approach this question. The lecturer, as per usual, has provided us with the bare minimum in terms of hints on how to approach this. I know how to do it when we want in ...
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0answers
30 views

Calculate probability using Naive Bayes Classification

I'm having problem calculating the probability using Naive Bayes approach Problem First I calculate \begin{align} P(\text{No})P(\text{Sunny}|\text{No})P(\text{Cool}|\text{No})P(\text{High}|\text{...
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13 views

Likelihood of an autoregressive model

I have the following autoregressive model: $Y_I=\lambda_t + \alpha_t(Y_t-\lambda_{t-1}) + \epsilon_t$ where $\lambda_t=\beta_1+\beta_2cos(\pi t/6)+\beta_3sin(\pi t/6)$ and $\epsilon$ has a ...
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1answer
27 views

Finding likelihood for an event using Bayesian Inference

A spacecraft carrying two female and three male astronauts makes a trip toMars. The plan calls for a two-person detachable capsule to land at site A on the planet and a second one-person capsule to ...
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1answer
57 views

Finding probability of a sample using Bayesian Inference [closed]

In a particular water sample, ten bacteria are found, of which three are of type A. What is the probability of obtaining six type A bacteria, in a second independent water sample containing 12 ...
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12 views

Convergence t-student distribution a posteriori

Good afternoon. I want to show that when n goes to infinity, the predictive distribution converges to distribution of $x_{n+1}$. To do this, I need to know the density limit of $x_{n+1}$. So I'm ...
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31 views

Bayesian inference for sum of random variables

Assume that we have a random variable $Z = X + Y$ for $X$ and $Y$ independent. Then if w use two independent data-sets $D_1$ and $D_2$ to try and approximate the distribution of $Z$, i.e. $$p(Z|D_1,...
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1answer
18 views

Conjugate prior distribution

Suppose data consists of a single observation $x$ on Poisson random variable $X$,where $X\mid\xi\sim\mathcal{P}(\xi)$.How do I show that the likelihood function for $\xi$ is $f(x\mid\xi)$ proportional ...
2
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1answer
37 views

Bayes' Net Conditional Probability

I have the following Bayes Net. And I need to calculate $P(R\mid W)$ and $P(S\mid W)$. For, $P(S\mid W)$, is it $.1 \cdot .9$ because I multiply the probabilities of those two events that the ...
2
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1answer
55 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
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23 views

Inference on a factor graph (Sum-product Algorithm)

I was going through the sum-product algorithm which can be used to find marginal distribution efficiently(and exactly) when the factor graph is a tree. I found it difficult to understand the way they ...
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17 views

Relation between Bayesian analysis and Bayesian hierarchical analysis?

I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light ...
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13 views

Finding marginal posterior distributions (Gibbs Sampling)?

When using Gibbs sampling I need to find the conditional distributions of the parameters. In all textbooks and examples they seem to unanimously suggest that "it's obvious". Take for example page 56 ...
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38 views

What do $d\phi$ and $\in$ mean in terms of probability

I'm reading Peter Orbanz's notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf. In it, he uses the following notation, which isn't defined. We have some ...