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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Inferring poisson rate from interval determined by data

I have a dataset of arrivals, which are from a Poisson process. For the purposes of this question, let's say they're arrivals of cars on a particular road. My goal is to infer the gamma posterior for ...
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6 views

Discriminant Functions of two classes sharing same covariance matrix

How can i find the discriminant functions of two classes having same diagonal covariance matrix with different means? (their feature vector is two dimensional) Thank you!
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9 views

Log likelihood function for binary classification

I need help with this following task. There is a binary classification problem where each observation xn is belong to one of two classes (t = 0 and t = 1). The training data points are sometimes ...
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22 views

how to calculate crash probability?

A plane crashes with probability 0.95 if both of its engines fail. On each flight each engine has a probability of failure of $10^{-5}$. Both engines fail with probability of $10^{-9}$ a) Are the ...
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19 views

using bayes' rule

Women carrying a certain gene are ten times more likely to develop breast cancer. Only 1 out of 100 women carries this gene. If a woman has breast cancer, what is the probability that she carries this ...
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25 views

Bayesian networks: What's wrong with my answer?

Consider The following four random binary variables: Given the following Bayesian network: With the following conditional probability tables: I want to calculate the probability that ...
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1answer
32 views

Posterior distribution of $\theta$

Let $X_{ij} ~ N(\theta_i,\sigma^2)$ with $\sigma^2$ known, i = 1,... k, and j = 1, ... ,$n_i$. The prior distribution of $ \theta_i$ is $N(\phi,\tau^2)$, independently for i = 1,...,k and ...
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57 views

Uniform lattice sample inside a particular convex polytope

[update]: hardmath suggests using tools from linear programming. This looks like a good idea indeed. I can now tell that my feasible set is described by: $Set = \{d \in \mathbb{N}^c, -B.d\leqslant ...
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8 views

Finding a bayes estimator

Let $X_1,...,X_n|\eta~\exp(1,\eta)$ and $\eta$~$N(\mu,1)$, where $\mu\epsilon\Re$. Find the Bayes estimator $\eta$ under the squared error loss. After finding the joint likelihood of $exp(1,\eta)$ ...
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48 views

Baye's Classifier for recovering a signal from a measurement

Below is the question i am having trouble with: Independent and identically distributed symbols s(n) = ±1 are transmitted over the channel C(z) = 1 + z −1 . Symbols s(n) = +1 occur with probability p ...
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30 views

Bayes factor for fair and biased coin

There is the following task: Suppose we toss a coin $ N = 10$ times and observe $m = 9$ heads. Let the null hypothesis be that the coin is fair,and the alternative be that the coin can have any bias, ...
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Conjugate priors: wht not binomial-binomial?

Citing from Kevin Murphy's machine learning book: When the prior and the posterior have the same form, we say that the prior is a conjugate prior for the corresponding likelihood. Conjugate ...
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30 views

Bayesian hypothesis testing

Let $x_1,\ldots,x_4$ be a sample taken from the uniform dstribution with the density $$ f_{\theta}(x)=\theta^{-1} \cdot 1_{(0,\theta)}(x). $$ Assume that $\theta$ is a random variable with the ...
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9 views

Calculate CPT of bayes net

I have a Bayes net of a pretty simple construction. I need to find the expressions that the CPT's represent and also the number of entries. A--B--C .....| ....D A is the parent node of B. B is ...
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1answer
34 views

How to prove Laplace distribution is scale mixture of Gaussians?? [closed]

How does one prove the Laplace distribution is a scale mixture of Gaussians? I.e, how does one show that $X \sim \text{Laplace}(\lambda)$ is a scale mixture of Normal $Y \sim N(0,\tau)$ and ...
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51 views

How do I put together a set of modified conditional distribution into a single joint distribution?

I am abstracting my original problem to a simple scenario. Consider a bivariate multi-modal mixture of gaussian distribution, $P(x,y)$. When we slice through $x$ or $y$ we get a univariate multimodal ...
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32 views

Evidence propagation in bayesian network

I'm currently trying to wrap my head around evidence propagation in bayesian network (simple tree propagation) but I'm having trouble finding information about the process. As an example, let's take ...
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2answers
30 views

How do I combine assertions of experts based on trustworthiness?

5 friends have come up to me and asserted that "Fred is coming to visit tomorrow". The more people I hear it from, the more I believe it to be true. How do I model this probabilistically? I think I ...
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30 views

complicated posterior distribution

I have a question concerning a rather specific posterior. It should be a simple application of Bayes' Theorem. However, I am always confused here. I try my best to describe the situation. There are ...
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21 views

Independence Assumption of Recursive Bayes' Parametric Estimation

Background: I am using Bayes' decision theory to obtain the MAP decision by this formula ...
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54 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
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1answer
34 views

Terminology: Probability “with respect to a measure”

The following excerpt is taken from Shen and Wasserman (2001). I have difficulty understanding some terminologies. On line 4, [...] each $P_\eta$ is a probability on $(\mathscr Y,\mathscr ...
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21 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
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2answers
26 views

Bayes theorem with multiple variable question

The below formula is from an article that i red for my work. The author said he used Baysian theorem to get this, but I have no idea why this is true! Can someone please clarify how the first ...
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1answer
19 views

Loss functions for regression

[From PRML Bishop, p:46] The average or expected loss function is given by $$E[L] = \int\int (y(x)-t)^2 p(x,t)\ \ dx\ \ dt$$, where, the loss function $L = (y(x)-t)^2$, given x and the ...
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17 views

Bayesian shrinkage doesn't affect eigenspace

The book Machine Learning a Probabilistic Perspective by Kevin Murphy on page 130 states following fact without proof: Consider the MLE estimate of covariance matrix $\Sigma_{\text{MLE}}$. The ...
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9 views

LDA with fixed topics?

Suppose I have a collection of "topic" probability distributions $\{\phi^{z}\}$ for LDA (Latent Dirichlet Allocation) that I have found via alternate methods; is there a closed form MLE for the ...
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61 views

Poker odds: Chances of a straight flush, given H4,H5

I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this. My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that ...
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25 views

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network?

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network, where $A$ and $B$ are boolean values?
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38 views

Conditional probabilities given the evidence(Bayesian network)

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!
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6 views

Gaussian Process: Using partitions of a choleky decomposition solution for conditional deduction.

If I define a GP over observed values, $y$, of a sensor reading over time, $t$, as (for simplicity assuming discrete time series e.g lets say readings after every 5 mins) : $y=f(t)+\epsilon$ where ...
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2answers
47 views

Why would a uniform prior distribution give a different result than a purely frequentist approach?

I would expect a uniform prior to be a good example of an uninformed prior and get the same result as the frequentist approach. However, this is not the case. As an example, let's look the classical ...
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38 views

The speed of learning and prior

If I know $$\frac{\alpha}{\alpha+\beta}<\frac{\lambda}{\lambda+\gamma}$$ can I know the sign of $$\frac{\alpha+1}{\alpha+1+\beta}<\frac{\lambda+1}{\lambda+1+\gamma} $$ And the sign of ...
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9 views

Bayes with Log-Normal Data

There is some (recent) evidence that neurological activity is log-normally distributed. Does this invalidate the use of Bayes Theorem with these data? I ask because a major branch of computational ...
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15 views

A result regarding Hierarchical Bayes

I have the following, $$x_i \mid \theta_i \sim \text{Bin}(m, \theta_i), ~i=1,\dots,n,$$ $$\theta_i \sim \text{Beta}(\alpha,1),$$ $$ f(\alpha) \propto 1.$$ I wish to compute ...
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110 views

Bayesian inference and new information

The Bayesian inference [1] tells how we can update the prior probability based on evidence. My question is that, in real world, we also update our prior probability of an hypothesis based on new ...
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1answer
21 views

Deriving the marginal posterior

Context of the question: You can take everything below as given. $E_2$ is a $k$ by $1$ matrix and $V_{22}$ is a $k$ by $k$ matrix. Let $X$ denote the data. I have derived so far the joint posterior ...
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31 views

Posterior distribution of two independent random variables with distinct Beta distributions

$\def\Beta{\operatorname{Beta}}$ If $X,\ Y\sim \Beta(\alpha, \beta)$ and $x$ is a vector, then $ P(X>Y\mid x) = \iint_{X>Y}P(X,Y\mid x) \,dX\,dXY $ I need to compute $P(X>Y\mid x)$ when ...
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78 views

Explanation of the TrueSkill bayesian ranking algorithm in a two-person game, like Tennis?

TrueSkill is mostly used for ranking and matching players on Xbox Online Games, it is a general rating model that could be applied to any game, including Chess, Tennis or Football. It models every ...
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14 views

Textbook recommendation for Non-parametric Bayesian?

I am looking for textbooks which include basics as well as advanced models like latent Dirichlet allocation, hierarchical Dirichlet process. The most important thing is that those books should present ...
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1answer
46 views

Bayesian Network, Sprinkler Example

In reference to the wet grass / sprinkler Bayesian network problem at this site: http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html Pr(S=1 | W=1) has been determined as 0.430. Could someone please ...
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2answers
63 views

Optimal solution to a statistical decision problem

Setup I'm trying to find condition(s) that characterize the solution to a statistical decision problem. The environment is as follows. $\Omega$ is a finite set of states of the world. A decision ...
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56 views

Estimate the number of trials needed to observe all the possible outcomes of an experiment [duplicate]

I am stuck with the following problem: Each package of Pokemon cards contains 1 of N possible legendary Pokemon. How many packs do you expect to buy to get all N? We assume all N legendary cards are ...
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134 views

Is Entropy = Information circular or trivial?

I have seen several "maximum entropy distributions" used in the mathematical and statistical literature, often with the justification that they are "minimally informed" beyond the assumptions and data ...
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33 views

Posterior distribution of bernoulli distribution with multiple observations

I'm just learning Bayes's Rule so this question might be really simple Suppose I have a random (real) variable $X$ over $[0, 1]$. I assume a uniform prior. In successive rounds, I sample a value ...
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27 views

A question about Bayesian Networks from Judea Pearl's book.

"Given a probability distribution $P(x_1, \dots, x_n)$ and any ordering d of the variables, the DAG(directed acyclic graph) created by designating as parents of $X_i$ any minimal set П$_{X_i}$ of ...
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Posterior of mean given an observation from a bivariate normal with unknown but common mean, and known variance

suppose the sample vector $(x,y)$ is generated from a bivariate normal: $$ \left[\begin{array}{c} x\\ y \end{array}\right]\sim N\left(\left[\begin{array}{c} \theta\\ \theta ...
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Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
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28 views

Finding P(S1 = 1 | D1=1, D2=1) (Bayes Networks)

I'm scratching my head over something that is probably simple Probabilities, but I guess I can't see it. Essentially, I have the following table: ------ ------ ------ ------ | D1 | D2 | S1=1 | ...
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25 views

proof that a density proportional to Gaussian is Gaussian

I try to develop bayesian estimation for one dimensional Gaussian with unknown $\mu$ and known $\sigma$. I got $$p(x\mid D) = \int p(x\mid\mu)p(\mu\mid D) \, d\mu =\int \frac{1}{\sigma ...