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 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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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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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) = ...
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24 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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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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27 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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58 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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28 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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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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47 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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10 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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1answer
33 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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1answer
46 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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17 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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47 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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1answer
48 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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39 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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44 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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2answers
48 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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7 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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8 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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18 views

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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79 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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42 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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1answer
16 views

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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32 views

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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32 views

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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13 views

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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42 views

Explain how the following expression was derived?

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

Show covering number $N(\epsilon,\mathcal{P},h) < \infty$ for all $\epsilon >0$

Let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta\}$ be a dominated model of distributions on $[0,1]$. For the parameter space $\Theta$ we have $$\Theta := \{\theta: [0,1] \rightarrow \mathbb{R} ...
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32 views

Conditional probability connecting three terms (chain rule)

How can I express $\Pr(a \mid c)$ in terms of $\Pr(a \mid b)$, $\Pr(b \mid c)$, and $\Pr(c)$? Is it possible? I'm thinking the chain rule might have something to do with it, but I'm having trouble ...
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1answer
26 views

Bayes Network 2 parents one child

I have the following Bayes network: S R \ / H I know that: $$ P(s) = .7$$$$ P(r) = .01$$$$ P(h|s,r) = 1$$$$ P(h|!s,r) = .9$$$$ P(h|s,!r) = .7$$$$ ...
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18 views

Derivation of backward probabilities $\beta_i(s_i)$ of a Hidden Markov Model (message passing). Any help in completing it?

I am trying to formulate in a recursive manner the backwards probabilities $\beta$ of a Hidden Markov Model where $w_i$ are the observed symbols and $s_i$ are the latent states. Is the following ...
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25 views

A baisian estimation problem - How to formulate a Baisian estimation function for a given problem.

Two envelopes are given. Envelope 1 contains $x$ dollars and envelope 2 contains $2x$ dollars. We opened one of them and found in it $100$$. Now we have the option to change envelopes or not. ...
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1answer
61 views

probability calculation for bayesian network

I am studying Bayesian belief networks and in that I am struggling to understand how probabilities are calculated. I found this article here and the network is this: The associated probabilities ...
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159 views

Prove the estimator $\hat{B}$ of ridge regression = mean of the posterior distribution under a Gaussian prior

I want to prove that the estimator of ridge regression is the mean of the posterior distribution under Gaussian prior. $$y \sim N(X\beta,\sigma^2I),\quad \text{prior }\beta \sim N(0,\gamma^2 I).$$ ...
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19 views

Posterior for Beta Binomial Distribution with Repeated Observations

I'm working on a question with simultaneous learning about an underlying population and individual members of the population. The basic setup is: Let $N_g$ be the size of a population. At any point ...
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36 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
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36 views

A Bayesian exercise

I have encountered the following problem in a book I am reading: Suppose you are offered to participate in the following game: Two fair dies are thrown untill '1' will apear (in one of them at ...
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1answer
29 views

How to find the posterior distribution

So suppose I have a coin that has a probability $\mu$ of landing on heads, and $1-\mu$ of landing on tails. I am giving the prior distribution $\mu$ ~ Uniform[0,1], and my realization $D_1 = ...
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1answer
22 views

Assumptions leading to the mutual independence of random variables

I know that $P(AB) = P(A)P(B) \land P(BC) = P(B)P(C) \land P(AC) = P(A)P(C)$ does not imply $P(ABC) = P(A)P(B)P(C)$. But does $P(ABC) = P(A)P(BC) = P(B)P(AC) = P(C)P(AB)$ imply $P(ABC) = ...
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102 views

Bayesian probability with negative conditions

I'm trying to construct a probability model which analyzes signals if someone is in the neighbourhood. There are let's say 20 machines in the neighbourhood (of the wifi router) producing a wifi ...
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30 views

Perfect Bayesian Equilibria of the following game

Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility ...
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31 views

Find parameters of the posterior Gaussian distribution

The question is to find $p(x|y)$ given that $p(x) \sim \mathcal{N}(\mu, \Sigma)$ and $p(y|x) \sim \mathcal{N}(Ax, \Gamma)$. I do realize that I may just obtain a posterior through application of ...
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19 views

What are the different ways to do a rating in a web application?

I tried to do an example using Bayesian Average in order to find the rating for 3 hotels. Following is my example, Hotel A 3 Votes/ 2 Star/ 1 Star,Rating- 3 Star Hotel B 1 Vote, Rating -5 Star ...