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 ...

learn more… | top users | synonyms

2
votes
2answers
71 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 ...
1
vote
1answer
66 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 ...
9
votes
1answer
145 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 ...
0
votes
0answers
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 ...
0
votes
1answer
34 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 ...
0
votes
0answers
23 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 ...
2
votes
0answers
68 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 ...
0
votes
0answers
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 | ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
64 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. ...
0
votes
1answer
30 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 ...
3
votes
0answers
51 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 ...
0
votes
1answer
65 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 ...
1
vote
0answers
24 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.
2
votes
1answer
123 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
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 ...
2
votes
1answer
49 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 ...
2
votes
0answers
39 views

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 ...
0
votes
0answers
38 views

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 ...
0
votes
1answer
253 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 ...
0
votes
1answer
29 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 ...
0
votes
2answers
98 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 ...
0
votes
0answers
21 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 ...
0
votes
1answer
45 views

Markov chain: if $X\rightarrow Y\rightarrow Z$, then why is $Z\rightarrow Y\rightarrow X$ true?

in a Markov chain, given three random variables $X,Y,Z$, we have $X\rightarrow Y\rightarrow Z$, which means $p(x,y,z) = p(x)p(y|x)p(z|y)$. The right arrow symbol $\rightarrow$ is used to denote a ...
2
votes
0answers
53 views

In what sense is the Bayesian posterior mean a “convex combination”?

This is related to a previous question that hasn't gotten an answer: Definition of convex combination with matrix-vector multiplication Suppose I want to estimate $x \in \mathbb{R}^n$ from two ...
0
votes
1answer
45 views

Likelihood of a function of different types of random variables

Is there a general way of expressing the likelihood of some known, but non-trivial function of several random varaibles. For example, suppose that we need to calculate the parameters of a process ...
2
votes
0answers
88 views

Bayesian Shrinkage Factor

Vasicek(1973), referenced in this paper(See bottom of page 16) explains a method of shrinking individual betas $\beta^{TS}$ toward a cross-sectional mean $\beta^{XS}$ as follows: for each time $t$, ...
0
votes
1answer
40 views

Solution to a modified version The Locomotive Problem [closed]

A railroad numbers its locomotives in order 1..N. One day you see a locomotive with the number 60. Estimate how many locomotives the railroad has. Using the Likelihood Approach: Assume prior is ...
0
votes
2answers
64 views

What is the likelihood of a fair coin given 10 heads (with added component)?

What is the likelihood of a fair coin given that it has landed heads up 10 times? You have a fair coin or a double-headed coin... $\mathsf P($Fair$\mid 10$ heads$) = ...
0
votes
1answer
38 views

how to calculate unknown probabilities in the bayesian network

I am working on a bayesian network problem. I read from one of the website the following network. My problem : "as soon as the cold water becomes low, you have at least a 94% chance of a high ...
0
votes
1answer
31 views

Solve for mean and std deviation of new normal distribution

There are normal distributions with known means and standard deviations. The first distribution is a Bayesian prior distribution with known mean1 and known SD1. The second distribution is a Bayesian ...
0
votes
0answers
16 views

Bayesian Uni-variable ou multi-variable and formulation

I have a parameter that has a prior distribution with mean equals to 30, a variance of 25 and a number of samples $n=30$. I was able to execute 30 more samples, and I got a mean of 25 and a variance ...
2
votes
1answer
77 views

Why are these variables not conditionally dependent given 'active triplets' and the 'explaining away' effect?

I'm following the Udacity Intro to AI course. This quiz gives the following Bayes network and asks whether different variables are conditionally independent or not. (The explanation of the nodes, ...
2
votes
0answers
31 views

Improper Lebesgue prior normalization in Bayesian filtering

Suppose we have a conditionally Gaussian Linear State Space Model (CGLSSM) where $Y_t=(X_t,S_t)_{t \in \mathbb{N}}$ is the Markov chain of hidden states, where for each $t \in \mathbb{N}$, $S_t \in ...
0
votes
1answer
47 views

Bayesian Network vs Markov Decision Process

I am wondering if somebody can tell me anything about the practical differences between using Markov Decision Processes and and Bayesian Networks in reasoning about probabilistic processes?
2
votes
0answers
70 views

Optimal Stopping for One-Armed Bandit with a Fixed, Known Payout.

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. I have a bandit problem with one ...
0
votes
1answer
157 views

Bayes spam filtering

To analyze the words that appear in spam emails, you collect a sample of 1000 emails marked as spam and 1000 emails marked as non-spam. Of the 1000 spam emails, 210 contained the phrase This isn't ...
0
votes
1answer
89 views

Verification of a Bayes Theorem related problem

This was an assignment problem given to me by the professor. I have did it (not sure if its correct). My answer is around $2/10^{12}$. I fear this is wrong. Can someone try this and verify if ...
0
votes
0answers
33 views

Posterior probability of a $p$ borderline result

Suppose that we throw 400 coins and found a result 0f 220 tails. Using a simple test of the null hypothesis that $H_0: \pi = 1/2$, we get that the probability of such a result is very close to .05. ...
1
vote
1answer
83 views

Bayesian update from uniform prior to uniform posterior ?!?

I was working through a signaling game problem recently and the proof suggested the following: Actor A has a type: $\ \mathscr{t} \sim Uniform[-1,1]$ Actor A gives signal $\pi^*$ that perfectly ...
4
votes
1answer
62 views

Bayes' Net Conditional Probability

I have a Bayes' Net with 4 boolean nodes connected in a diamond shape. I want to find the probability of one of the middle nodes being true given that the ones above and below are both true. So ...
0
votes
1answer
17 views

partily undirected Bayesian Network

I am designing a Dynamic Bayesian Network, but I am a little confused about some definition of DBN and markov network. In my network ,the edges from the hidden nodes of last frame to the current frame ...
0
votes
1answer
83 views

Between bayesian and measure theoretic approaches

I was wondering how a bayesian statistician would approach the problem of defining a probability density function for a random variable. In a measure theoretic sense, If the distribution of the ...
1
vote
1answer
42 views

Bayesian Network/ Number of parameters

Please consider the following Bayesian Network out of $Graphical Models in Applied Multivariate Statistics" by Joe Whittaker: Now the factorization property says that the joint probability ...
0
votes
1answer
39 views

Conditional Posterior Distribution Based on Two Simultaneous Signals

I am trapped by such a problem. Assume the state variable $\theta$ is (prior) normally distributed $N(\eta, \sigma^{2}_{0})$. Now we have two independent signals about $\theta$. Signal 1 is ...
0
votes
1answer
56 views

Finding the MLE estimates of a beta, binomial hierarchical model

Consider $M$ observations ($x_i$, $n_i$) where $x_i$ is a realisation from $X_i \sim \mbox{Binomial}(n_i,p_i)$ and $p_i$ is a realisation from $P_i \sim Beta(\alpha, \beta)$. I would like to find the ...
0
votes
1answer
42 views

How this integration is solved?

Can anyone explain how this integration has been performed? This is a Bayes estimator for uniform prior assuming quadratic loss function. Thanks in advance
0
votes
2answers
139 views

Probability of picking balls out of bins

Question: You have two bins with four different balls in each bin. Bin A: 2 White Balls and 2 Black Balls Bin B: 3 Black Balls and 1 White ball You cannot tell which bin contains what balls. Given ...