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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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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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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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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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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26 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 \frac{...
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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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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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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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18 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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51 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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32 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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21 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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43 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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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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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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52 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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274 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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12 views

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
30 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
48 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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14 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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36 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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33 views

$P(A=0, B=0)$ 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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15 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
27 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, ...
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17 views

Variational inference on a Normal distribution: is my choice of priors passable?

I am trying to understand the basics of Variational Inference. In order to do so I designed a very simple problem: using the free-form mean field method to approximate the posteriori distribution of $\...
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30 views

Which is the best indicator of probability here? Bayes

I am part of a group of teachers in DFW area. We are very competitive when it comes to our profession. So we like to have a little fun throughout the year by having “test battles”. We simply ...
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61 views

How to derive the posterior predictive distribution?

I often seen the posterior predictive distribution mentioned in the context of machine learning and bayesian inference. The definition is as follows: $ p(D'|D) = \int_\theta p(D'|\theta)p(\theta|D)$ ...
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1answer
29 views

Bayes' rule where the realization of random vector is a subset of the realization of a different random variable?

I have realizations of two different random vectors, where one is a subset (is that proper terminology here?) of the other $$s^t = (x_1,x_2,x_3,\dots x_\tau, x_{\tau +1},\dots x_t)$$ and $$ s^\tau = ({...
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18 views

Expected utility of action, given probability model

We record measurements of an appartus every day. If apparatus doesn't break (it has probability equal to $1-p_2$), it will measure zero with probability $p_1$. If apparatus breaks (probability $p2$), ...
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31 views

Find a conditional probability of a Bayes' Net knowing only the prior probability of the root.

Given three nodes A,B,C that form a Bayes Network as the following: (A)-->(B)-->(C) If we know the prior probability of A is 0.3, i.e. P(A)=0.3, is this ...
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30 views

Understanding Bayes' theorem through an example

Suppose I have three nodes A,B,C such that A and B are independent and pointed to C as the following: A --> C <-- B Also Suppose that each node takes a peobability between (0,1) so that the ...
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7 views

Interpreting data from a Gaussian Mixture Model using Gibbs sampling

I have data from a population with suspected subtypes within it. I have used a Gibbs sampler with different numbers of potential subtypes to produce Markov chains and posterior distributions. I am ...
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49 views

how to calculate expected utility for probability decision problem?

consider a decision problem: classifying $x$ as belonging to one of two classes $C_1, C_2$. there are prior probabilities for each class, $p(C_1), p(C_2)$ and likelihood probabilities for data given ...
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What assumptions did I make when I strengthened my independence criterion across a new random variable?

I have an algorithm which tries to calculate some $\operatorname{Pr(X | Y_1 Y_2 \dots )}$ (where juxtaposition means event intersection, "given $Y_1$ and $Y_2$ and ... have happened".) We have some ...
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66 views

Bayesian Approach: Is a die from a 3-D printer fair?

In a recent post "Fair die or not from 3-D printer"on this site @Eumel reported making a die on a 3-D printer, providing data on the faces seen in 150 rolls, and wondered about "the chances that the ...
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19 views

Bayes classification

What are the synonyms for 1) Bayes classifier 2) Bayes decision rule 3) Bayes decision function for uniform distribution I found many terms in literature and got confused because they look similar (...
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2answers
55 views

Simple example of “Maximum A Posteriori”

I've been immersing myself into Bayesian statistics in school and I'm having a very difficult time grasping argmax and ...