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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5 views

Bridge from Bayesian update to covariance matrix

can somebody please explain me the step from the Bayes' Theorem to the covariance matrix or in a more special case from the Bayesian Update to the Kalman Gain. Best regards.
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26 views

Multi armed bandit problem how to calculate $\mathbb{P}(A > B)$ using Thompson sampling [on hold]

Let say that you created 2 marketing campaigns. You sent 200 impressions on these campaigns as follow: Campaign A : Got 100 impressions and 2 successes with a value of 1.5$ per success Campaign B : ...
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52 views

How to calculate $\mathbb{P}(A>B)$ using the Jeffrey Prior

Let say that you created 2 marketing campaigns. You sent 200 impressions on these campaigns as follow: Campaign A : Got 100 impressions and 2 successes with a value of 1.5$ per success Campaign B : ...
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0answers
15 views

How to get best fitting model decision for data X and Y in e.g. Matlab?

I have two sources of data, X and Y, which are basically counts, from 23 individual origins (3D ROIs in my case). For example: ...
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33 views

Bayesian posterior probability [closed]

Let's say that you are in a casino and you have played on 3 different slot machines following this flow: Slot machine A, play 10 times, win 2 times for a total of 2$ Slot machine B, play 100 times, ...
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1answer
11 views

Incentive compatible revenue maximizing multiunit auction

The Vickrey-Clarke-Groves Auction is an example of incentive compatible (truthful reporting) multiunit auction, but it is only maximizing social utility, not the seller's utility. If my ...
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14 views

Inferring the addends of the sum of two random variables

I have three independent Poisson variables: B, C and D, whose parameters $\lambda_B$, $\lambda_C$ and $\lambda_D$ are unknown. I sample once the variable: $$ A_1 \equiv 0.9\cdot B + 0.1\cdot C $$ and ...
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38 views

Bayesian Estimation: calculating an integral

I am reading a book on Bayesian filtering and I have a question regarding calculating transition density $p(X_t|X_{t-1})$. My question is how the term $p(X_t|X_{t-1}, V_{t}=v)$ is converted to the ...
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17 views

Updating Bernoulli probability

I would like to show that the expression $ \frac{E\,\left[\, p^{t+1}\left(1-p\right)^{\left(n-t\right)}\right]}{E\left[\, p^{t}\left(1-p\right)^{\left(n-t\right)}\right]} $ , where $p$ is random on $[...
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22 views

implementing particle filters without a priory distribution

i am implrmrnting the particle filter, and i have some problem understanding the algorithm. given the state equations: $$ x_k = f(x_{k-1},v_k) $$ $$ z_k=h(x_k,u_k) $$ where $v_k, u_k$ are process ...
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1answer
48 views

Assumptions with Bayes's Theorem

After reading extensively on the subject I would like to clarify this apparent problem with "Bayes Rule". Namely the notation often used P(A and B) = P (B and A) has a big assumption that I will try ...
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1answer
29 views

Bayesian Nets and weird probability

I have to solve the following problem: Suppose we have a bayesian net in which we have the following variables: R, PA and PR Let: P(R) = 0.1, P(PA) = 0.5, P(PR|R, PA) = 0.6, P(PR|¬R, PA) = 0.4, P(...
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2answers
35 views

Calculating total probability given some conditions

One machine element is being produced in $3$ series, each series consisting of $20$ elements. In the first series, there are $15$ elements that work correctly, in the second series there're $18$ and ...
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0answers
10 views

Help with finding a particular joint distribution of a Bayesian Network

Consider a Bayesian Network defined by the following matrix: $$\left[\begin{array}{ccccccc} 0&1&1&0&0&0&0 \\ 0&0&0&1&1&0&...
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34 views

Is Bayesian Association mathematically rigorous?

Introduction. This question is based on the Ph.D. thesis of B.T. Vo, which can be found in this website ("Papers" section). More specifically, in the introduction of the Ph.D. thesis, at page 8, there ...
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1answer
14 views

Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
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1answer
41 views

Hunter and Rabbit Bayesian Probability [closed]

I've been asked to do this by process of game theory and probability (namely Bayesian theory). Here is the problem: There is a Hunter (H) and a Rabbit (R). They are playing the following game: - ...
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15 views

Understanding the posterior of the Dirichlet process

Draws from a Dirichlet process (DP) are discrete, and exhibit clustering behaviour. Suppose I draw $G_{1:5}$ distributions from a DP. Then the posterior probability for $G_6$ is given by (Blackwell ...
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1answer
7 views

Missing approximation to get the Maximum A Posteriori (MAP) estimator of event times with a sparse prior

Assume that a signal $ y $ is a noisy perturbation of time-shifted copies of a given waveform $ f(t) $ defined on K time bins $ \{ 0, \cdots, K-1 \} $: \begin{equation} \forall t \in \{1, \cdots, T\}...
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1answer
13 views

Maximum a Posteriori (MAP) Estimator of Time Shifts with Poisson Process Prior

Assume that a signal $ y $ is a noisy superposition of time-shifted copies of a given waveform $ f(t) $ on a finite time interval $ [0, T] $: \begin{equation} y(t) = \sum_{i=1}^{n} f(t - \tau_j) + \...
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1answer
40 views

How can we prove this equation using marginalization and conditioning? [closed]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
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1answer
23 views

Did I draw this tree diagram correctly?

On the way home from work Chris goes through a traffic light then passes over a level crossing, the probability that Chris stops at a traffic light is $\frac{2}{3}$ while the probability that he is ...
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25 views

Sprinkler Example, how to compute probabilities?

Can you explain how do you compute probabilities like, P(S=F|C=T) and P(W=F|S=F,R=T) ?
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34 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
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1answer
53 views

Does different the initial prior result in the same posterior of a Kalman filter as time goes to infinity, why?

Let $p(x_0) \in \{p_i(x_0)\colon i\in {\mathcal I}\}=:{\mathcal P}$ be the prior of a discrete-time Kalman filter, where ${\mathcal P}$ is the family of nondegenerated Gaussian distributions. Then ...
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1answer
32 views

Calculate Conditional Probability for program that does not crash

I got these 2 questions in exam, but unfortunately i failed to solved these. 1) you want to buy a computer. The probability that you can run the probabilistic program $X$ on it is $97$% ...
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10 views

Find the 1 dimensional empirical conditional distribution with data

I need to find the the conditional distribution function empirically of some data $C$, given the value of a particular predictor $y$. I have attempted to use Bayes as follows: $$ P(C \ |\ y)=\frac{P(...
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1answer
19 views

Simple Bayes? Probability of a state at time t in hidden markov model

Suppose we have a HMM with $2$ states -- $A$ and $B$, with $P(A) = 0.4$ and $P(B) = 0.6$. $A$ has a probability of $0.9$ of outputting "hot," and $B$ has a probability of $0.1$ of outputting "hot." ...
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1answer
25 views

Application of Bayes theorem and Partition Law, total probability

Hi guys, preparing for my finals and trying to get this question out for practice. The exam is in a couple of hours so apologies for being brief. I think I have computed parts 1 and 2 fine. $$0.3*...
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12 views

OpenBUGS: get a sample from a random variable

I'm working in OpenBugs, and I've defined the next model: $Y\sim {\rm Exp}(\theta)$ so I'm asked to assign different initial distributions to $\theta$: Normal, Gamma and log-normal, to this point ...
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20 views

Simple example of “Spike-and-Slab Prior” for Bayesian Inference

I would really like to understand how Spike-and-Slab Priors work in relation to Linearized Models. Can somebody provide a toy example of a Spike-and-Slab Prior with a Bernoulli spike and a Gaussian ...
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16 views

How many people should I ask if a statement (A) is true if the same can be inferred by asking two other statements (X and Y implies A)?

I am asking a number of participants if they believe a given statement is valid. I have a number of such statements, some of which can be inferred. In the made up example below, ...
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1answer
55 views

Approximation of product of Bernoulli with different proportions

I want to update a variable $Y$ with Beta (uniform for simplicity, $Y \sim U(0, 1)$) distribution, with Bernoulli information each period... But each period the proportion parameter of the Bernoulli ...
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6 views

The Intergration of the Product of Two Conditional Probabilitie

I have been stragling with figuring out this equality: $\pi(\psi|data)=\int_{\Theta}\pi(\psi|\theta)\pi(\theta|data)d\theta$ Can anyone help me go through the proof? Thanks!
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1answer
15 views

Iterated Bayesian Updates

I get a sequence of data that is generated by a distribution with parameter $a_0$ (e.g. $\mathcal{N}(a_0,1)$). I assume a prior distribution $P(a)$, and Bayesian update for the belief according to the ...
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17 views

Non linear model, logistic regression exercise

Let $y_i$ follows $Bin(n_i,p_i)$ and for $p_i$ we consider the logit quadratic model: $\log\frac{p_i}{1-p_i}=\beta_0+\beta_1A_i+\beta_2(A_i-meanA)^2$ where $A_i$ is AGE_i during ith time. As you can ...
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2answers
15 views

Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
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1answer
45 views

Bayesian probability

recently I bumped into following puzzle and I would like to validate(or correct) my results as I asked several people and got several different answers. You are planning a picnic with your friends ...
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81 views

Probability of number of people who know a rumor

Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
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10 views

Expectation Maximization (EM) for 3-dimensional parameter $(\alpha,\mu_2,{\sigma}^2)$.

Let $x_i$ where $i=1,...,100$ are iid observations from a mix of two normal distributions with means $\mu_1=0$ and $\mu_2$ and the same variance ${\sigma}^2$. If $\alpha$ is the proportion of the ...
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14 views

Hypothesis test in Bayesian statistics

Let $X\sim N(\theta,1)$ and 5 independent observations $X=(4.9,5.6,5.1,4.6,3.6)$. The prior probability that $\theta=4.01$ is $0.5$. The remain values of $\theta$ are given the density of $g(\...
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1answer
44 views

How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange? I have a very basic question concerning updating from a prior to a posterior in bayesian statistics. Setting: I ...
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69 views

Bayesian urn questions

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, ...
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49 views

Facebook Data Science Question (Expected Payout and Probability)

I saw this question on Glassdoor and couldn't seem to find a answer to validate mine anywhere: You're at a casino with two dice, if you roll a 5 you win, and get paid $10. What is your expected ...
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Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but $$f(\...
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1answer
30 views

Bayes' Rule Question

I am reading about Bayes' rule, I can solve all the exercise but this one. Suppose you had a checkup, and there is a bad news; you tested positive for "the giggles" and that the test is 99% accurate( ...
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1answer
26 views

how to calculate conditional independence

This Bayesian net (click) is given with the binary variables B, F, G and D and the following probabilities $p(B=1) = 0.9$ $p(F=1) = 0.9$ $p(G=1\mid B=1,F=1) =0.8$ $p(G=1\mid B=1,F=0) = 0.2$ $p(...
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13 views

Bayes Rule and Multivariate Normal Estimation

This is an exercise in this pdf file http://statweb.stanford.edu/~ckirby/brad/LSI/chapter1.pdf and how can I show that by using Bayes Rule?
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1answer
33 views

Book recommendations for introductory Bayesian statistics?

Anyone here have some recommendations for a good book introducing the reader to Bayesian statistics? Let me mention my background. My undergraduate majors were in Actuarial Science and Statistics, ...
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1answer
31 views

What does the notation $d|x \sim N(0,14^2)$ stand for?

I'm reading a book about Bayesian data analysis (by Gelman et al.) and I bumped into the following text: $x= \text{Football point spread}$ $y=\text{Game outcome}$ $d=y-x$ For the ...