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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125 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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62 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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50 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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39 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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37 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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60 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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48 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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37 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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39 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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59 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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19 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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18 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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73 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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30 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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9 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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71 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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40 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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12 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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20 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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136 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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23 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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41 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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150 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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22 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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109 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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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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73 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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149 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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36 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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35 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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24 views

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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80 views

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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29 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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28 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 ...
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44 views

Applying Bayes Rule to Cards

I was playing poker with a friend last night when a question occured to us. I had a two Jacks and the flop came out: King Queen and 4. So, suddenly my pocket Jacks are not so great, unless another ...
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65 views

Assistance with Bayesian Random Effects and Mixed Effects Models

I am looking to build either a random effects or mixed effects model for a project I am working on, but have had trouble finding good sources and understanding the general structure of the model. ...
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33 views

Bayesian Nets. No active path from X to Y, versus No inactive paths from X to Y

I am learning d-seperation in Bayes nets for my A.I. class. What this involves is considering all paths from some node X to Y (representing random variables) and seeing whether such paths are active ...
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53 views

What is the math behind calling election seats with confidence, before all votes have been counted?

On election night, predictions are made on the winner of each district, after only a fraction of the vote has been counted up. How is this done? Say there is a seat up for election, and 10,000 votes ...
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89 views

Bayesian posterior variance

Let $Var[\omega]$ be the variance of a population parameter $\omega$ prior to the collection of a random sample $\mathcal{X}=\left\lbrace X_1,X_2,\dots,X_n\right\rbrace$ from the population. Prove or ...
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25 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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174 views

Bayesian versus Classical (frequentist) Statistics

Very often in text-books the comparison of Bayesian vs. Classical Statistics are presented upfront in a very abstract way. For example, in the current book I'm studying there's the following ...
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25 views

What does likelihood density f(y|theta)=5 imply?

I just dont understand what dose constant likelihood density imply, e.g. f(y|theta)=5? In addition, when I use this likelihood density to derive posterior function, it cancels out so my posterior ...
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41 views

This textbook on time series analysis says something wacky

This is from a discussion of analyzing a time series with a sinusoid + noise model. The troublesome statement is: ...data values near the beginning and end of a record are most important for ...
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56 views

probability matching strategy for coin flips

imagine a betting game where we observe $N$ independent coin flips $x_1,...,x_n$ (where each $x_i \in {H,T}$) from the same coin, whose true weight is $\theta$. the task is to predict how many Heads ...
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How can I infer order from partially ordered discrete sequences?

A really interesting problem that I can't stop thinking about! Have run in to this a couple of times but yet to find a smart approach to either solve or frame this problem. This is my try at ...
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An example shows the difference between inference in Bayesian network and Junction Tree

Why inference in Junction tree is more efficient? There are directed graph BN and the corresponded undirected graph transformed by Junction tree algorithm. The literature describes that inference in ...
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342 views

Extended Bayes' theorem: p(A | B, C, D)

I'm having some difficulty understanding Bayes' theorem with multiple events. I'm trying to put together a Bayesian network. I have four independent probabilities but I have found that A, B and C can ...
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31 views

Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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106 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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22 views

how can I Find a 95% credible interval for p using the Bayesian method with the uniform distribution as a prior for p?

When I have a RV X~Geom(p): $x\ Frequency\\ 1 7459\\2 1930\\ 3\ 463\\ 4\ 117\\ 5\ 22\\ 6\ 6\\ 7\ 2\\ 9\ 1$ This is what I am trying to do: Since p is a probability, I say that $ p\sim U[0,1]$ An ...