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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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 ...
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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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23 views

Is Bayesian Inference what I need? [closed]

I'm not sure if I need a mathematician or a developer (or both) for this question. 1)There is a framework called Infer.NET that uses Bayesian inference for probabilistic programming. 2) I'm ...
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26 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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32 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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37 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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237 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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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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32 views

Example of Kalman filter with irregular time steps/weights to observations

I am trying to find a Kalman filter implementation which does not make the assumption that each observation is equally spaced and of equal weight. In particular, I have measurements of a process ...
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18 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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16 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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36 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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12 views

calculation probability : bayesian network

I want to ask some questions based on this link : http://artint.info/html/ArtInt_148.html about Bayesian Network , this is the following conditional probability and the bayesian network: Image ...
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12 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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31 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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18 views

Bayesian serial chain

I am trying to understand Bayesian networks and particularly causal (serial) chains. It is defined as P(C|A&B) = P(C|B).This means that the probability of C, ...
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22 views

Bayesian formula with multiple events

I am trying to predict next location of the user knowing his current (t) and previous location (t-1). I am trying to use Bayesian networks for this. Lets assume that we have following path: ...
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32 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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11 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) = ...
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15 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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23 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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14 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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22 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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50 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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24 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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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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29 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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28 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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42 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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8 views

Bayesian Networks: simple example when to use discrete network and when to use linear Gaussian network

So I am not sure when to use which. Is there a simple example that a non maths pro would understand when to use which? I use libpgm and the pgmlearner provides different functions to train on data. I ...
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28 views

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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41 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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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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44 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 ...
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41 views

Conditional Probability calcualtion

In the following BBN network, 1)what is meant by P(Martin Late|train strike,Norman Late)? Does this mean probability of martin Late given that Train Strike And ...
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33 views

Change of Variable technique for two variables?

If, $\theta_1 = ln \frac p{1-p}$ $\theta_2 = ln \frac q{1-q}$ $\theta_2|\theta_1 \sim N(\theta_1, \sigma^2)$ which means $f(\theta_1,\theta_2) \propto ...
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42 views

When do we expand the numerator of the Bayes' Theorem

I am trying to understand why the proposed solution below to the following question is wrong:- A box contains three cards: a card that is black on both sides, one that is white on both sides and a ...
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43 views

Hidden Markov Model and Viterbi algorithm: Understanding the Casino Problem?

I am deeply struggling with understanding how to apply the Viterbi algorithm. From my course notes, I have the following simple(I'm told) example: If the sequence ...
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6 views

Bayesian serial link d-separation?

I don't get how I prove d-separation for a serial link: $$ (A)\rightarrow(B)\rightarrow(C) $$ I am trying to prove that if $B$ is known with certainty (hard evidence), then the probability of $C$ ...
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7 views

Bayes risk with loss function that penalizes all errors equally

Loss function $L(\alpha(x),y = 1$ if $a(x) = y$, else 0. If $y\in \{-1,1\}$, then $\sum_y L(\alpha(x),y)p(y|x) = -p(y \neq \alpha(x) |x)$. (taken from ...
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Determining posterior gaussian distribution having marginalised over hyperparameters.

When applying gaussian process machine learning to regression problems where we want to determine the value a function $f$ takes at a new input point $x_{n+1}$, given observations of function values ...
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53 views

Conditional independence in Bayesian network with qualitative influences

I have some troubles solving an exercise from the book Probabilistic Graphical Models (pgm.stanford.edu). We are given the bayesian network with binary-valued variables. We do not know the CPDs, ...
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40 views

Integration in solving coin toss problem via Bayesian appoach

The following is taken from here: You have a coin that when flipped ends up head with probability $p$ and ends up tail with probability $1−p$. (The value of p is unknown.) Trying to ...
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Combining two Gaussian posterior distributions from different data to refine estimated distribution.

If we apply Bayesian inference to try and determine the distribution of a multivariate Gaussian $\textbf{x}$, and we have two predictions $$ \textbf{x}\sim N(\textbf{a}_1,\Sigma _1)~~ and ~~ ...
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How to work out $P(B\mid\neg A)$ using Bayes' formula

I am trying to work out the probability of something using Bayes' theorem: $$P(A \mid B) =\frac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid \neg A)P(\neg A)}$$ So in the question I know what $P(B\mid ...
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Correcting multivariate distribution by additional info about its marginal

Assume that I have a posterior distribution $p(\theta_1, \theta_2|X)$ and I obtain an additional information in the form of a marginal density $q(\theta_1|Y)$ that is of the same type as ...
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Bayesian model to estimate the parameter of a Bernoulli law

Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ...
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39 views

Explain how the following expression was derived?

Can someone explain how the author gets to the expression after the words "This leads to:"