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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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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60 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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33 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 ...
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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 ...
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1answer
56 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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26 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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18 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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14 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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20 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
45 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
23 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) + M(t;\beta(s)...
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49 views

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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13 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
29 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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30 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 \frac{...
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14 views

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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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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18 views

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
33 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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19 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 ...
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1answer
53 views

Predictive Distribution with Normal Prior

Given $\Theta = \theta$, let $X_1, X_2, \dots, X_n, X_{n+1} \sim \mathcal{N}(\theta, \sigma^2)$ be independent. $\Theta \sim \mathcal{N}(\theta_0, \tau^2)$. What is the easiest way to find the ...
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1answer
33 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim \mathcal{N}\left(\theta_0,\frac{\sigma^2}{\...
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2answers
22 views

Survival bias and probability

Imagine the following situation: A new virus is discovered that is believed to have infected 20% of the population. Anyone infected with the virus has a chance of 50% of dying in their sleep every ...
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1answer
51 views

An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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1answer
34 views

Determine how likely it is that a set of boolean data is produced by a distribution

Suppose we have a collection of independent Boolean random variables $X_i$ and $Y_i$ (for $1 \le i \le N$), and are told $p_i = P(X_i = 1)$ for all $i$. We are now given a set of values $x_i$ that was ...
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30 views

Nando de Freitas' Machine Learning Homework 2 Questions 1 & 2 Solutions

I've been following Nando de Freitas' Machine Learning course from UBC. While I have been enjoying the course I thought it would be good to see if I could do the homework along with it. So I'm on ...
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56 views

A priori probability in Bayesian inference problem

The problem A psychic uses a five-card deck to demonstrate ESP, claiming to be able to guess a card correctly with $0.5$ probability (of course, ordinary guessing is $0.2$). A single experiment ...
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275 views

Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference

In yesterday's Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in ...
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1answer
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Prior that incentives dissimilarity of 2 parameters

I have some binary data. I have a proposed partition of this data into partitions 1 and 2. I want to test whether the data in models 1 and 2 were generated by two Bernoullis such that their ...
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1answer
32 views

Bernoulli Naive Bayes Classification

I am having trouble understanding the following text regarding Bernoulli Naive Bayes. Specifically, the author mentions that $i$ is a feature. However, what is the difference between $x_i$ and $i$?
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28 views

Bayesian estimation of $x_m$ for Pareto distribution

The Pareto distribution has pdf $$f\left(x\right)=\alpha x_m^\alpha x^{-\alpha-1}$$ for $x\geq x_m$ with $\alpha,\,x_m$ positive parameters. I've been researching maximum-likelihood and Bayesian ...
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1answer
49 views

How to solve conditional probability problem using bayesian algorithm

I am trying to solve An agent learning to categorise news articles in two topics, World (W) and Finance (F). Out of $100$ articles, $40$ were classified as W, and $20$ of the articles were ...
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15 views

How to sort list with Bayesian inference?

I have long list of Instagram accounts with the following data: number of followers of the account (N); number of users, who follows both this and mine accounts (n). I would like to get list of ...
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37 views

Conditional Probability with two subsets

Question: A man plans to ship six boxes. Two of the boxes are insured, while the other four aren't. Each package that is shipped has a 10% chance of being damaged. What is the probability that: ...
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$P(A=0, B=0)$ is what given the following graph?

Graph and Probabilities Given this graph and respective probabilities, what would be the value for $P(A=0, B=0)$? I computed $P(A=0, B=0)=P(A=0)P(B=0)=0.24$ because A & B are independent of D. ...
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17 views

How to find the conjugate prior of a probability distribution?

I am looking for a procedure for finding the conjugate prior, given a probability distribution. I am more interested in the exponential family of distributions of the form $$ F(x|\theta) = A[x]exp(B[...
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1answer
31 views

Is it possible to have multiple Conjugate Priors?

In Bayesian probability theory, can a probability distribution have more than one conjugate prior for the same model parameters? I know that the Normal distribution has another Normal distribution ...
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24 views

Undefined notation in Causality book

I'm reading the book Causality - Models, Reasoning and Inference (Second edition). On page 11 the Decomposition property uses the notation $YW$, which is not defined before. What does $YW$ mean, ...