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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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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14 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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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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16 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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23 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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3answers
58 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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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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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
67 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 ...
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1answer
43 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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28 views

Calculate probability using Naive Bayes Classification

I'm having problem calculating the probability using Naive Bayes approach Problem First I calculate \begin{align} ...
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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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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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26 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. ...
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1answer
16 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 ...
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1answer
36 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 ...
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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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22 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 ...
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18 views

What is a Markovian time evolution model?

Supose I have to constuct a dynamical model for a random variable $X$ . Then I have read that for atmospheric and environmental purposes, a popular and flexible collection of models are (one-step) ...
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1answer
43 views

Why does this conditional probability formula work?

Following paragraph comes from Page 18 of E. T. Jaynes's Probability Theory: The Logic of Science (http://bayes.wustl.edu/etj/prob/book.pdf) -- I really have no idea how (1.36) works given (1.34) ...
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2answers
22 views

Bayesian Statistics: Finding Sufficient Statistic for Uniform Distribution

The example: let $y_1,\dots,y_n \overset{\text{i.i.d.}}\sim U([0,\theta])$, where $\theta >0$ is unknown. Find a sufficient statistic for $\theta$. Solution attempt: $$g(y_1,\dots,y_n) = c\quad ...
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12 views

Problems in notations in a paper on Bayesian space-time models

Suppose I have been given some process $Y$. Let $Y(s,t)$ denote the value of process at location $s$ and time $t$. For my experiment, I consider a model described as - $$Y(s,t) = \mu(s) + ...
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Bayesian inference exercise

I am learning online Bayesian Statistics and I have a test in a couple of days. I have no idea how to solve this exercise, any help will be appreciated. There might be something similar in the quiz... ...
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35 views

How to derive the conditional given the following joint probability

I encountered this question while reading about MCMC methods to solve image reconstruction problems. Consider a black and white image where $-1$ corresponds to white and $+1$ to black. $X_{i,j}$ ...
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12 views

Meaning of “T-vector of time series values”?

I am currently studying a paper on Hierarchical Bayesian space-time models. In that, we have denoted $Y(s,t)$ to be the process of interest ate location $s$ and time $t$ in a gridded space-time. $Y(s, ...
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13 views

Example for unknown parametrs chosen in Bayesian and frequentist inference?

The difference between Bayesian and frequentist inference is that in Bayesian analysis, parameters are random but in frequentist analysis they are fixed but unknown. Can someone explain to me this ...
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1answer
19 views

Integrating over parameter in Bayes

I am going over the paper "Sparse Bayesian Learning and Relevance Vector Match" by Michael Tipping. There is one equality there which I do not fully understand. He states: $$p(t | \alpha, \sigma^2) = ...
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29 views

What does the distribution of Fourier components indicate about the real-space distribution?

I read a paper that assumes a prior distribution on the Fourier components of a 3D model--specifically that the components are independent and normally distributed: $$ p(\Theta) = \prod_{l=1}^L ...
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What is a parameter in Bayesian analysis?

In any case study, when we use Bayesian analysis to solve our problem we consider a model parameter which is sometimes known and sometimes unknown. And using this parameter(and of course prior data) ...
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3answers
30 views

What is the probability a piece of clothing was made by person 1 if it is defective

Person 1,2 and 3 produce the following proportions of clothes: Person 1: 10% Person 2: 30% Person 3: 60% The probability they make clothes that are defective are: Persion 1: 4% Person 2: 3% ...
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24 views

$I(X,Y,Z)$ and $I(X\bigcup Y,Z,W) \ge I(X,Y,W)$

I am trying to prove if $I(X,Y,Z)$ and $I(X\bigcup Y,Z,W)=> I(X,Y,W)$. I know that $I(X,Y|Z)=I(Y,X|Z)$ and $I(X,W|Z\bigcup Y)$ and $I(X,Y|Z) \Rightarrow I(X,Y\bigcup W|Z)$, unable to use the above ...
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How to find convergence with a learning rule that depends on the outcome of a game?

my first post here and really excited about the community. In a game theory set in which agents choose from a finite set of actions with a probability distribution, how can I look for convergence ...
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1answer
28 views

Bayesian Expected loss integral

Thanks. I don't understand how to calculate the integral for a Bayesian Expected Loss. The problem is from Berger 1985 Stat Decision Theory and Bayesian Analysis page 8. Example 1. Assume no data is ...
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9 views

How to incorporate p-values into Bayesian classification?

I'm interested in using data from multiple studies to assess cancer risk in a patient. Each study has different p-values for the confidence of their result. When assessing an individual's cancer ...
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1answer
32 views

Elementary book for Bayesian statistics

I need to study the applications of Bayesian statistics in environmental sciences. For that I need a good book which can explain concepts from basics. I do have sufficient knowledge in probability but ...
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17 views

Why can prior be swamped? (given a data set, any prior will go to the same posterior) [closed]

Is there any mathematical proof for that?
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14 views

How can I properly define $G_0$ in Chinese Restaurant Processes clustering?

I would like to implement a generative model for clusters as defined in section 2.2 of 1. Assuming I already have the procedure for assigning tables, I would like to now assign each "table" with a ...