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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18 views

Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
3
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1answer
29 views

Explanation of Aumann's “agreeing to disagree” in modern notation

I'm attempting to understand Aumann's classic 1976 paper Agreeing to Disagree, which claims, under certain assumptions, that if two Bayesian agents share knowledge of each others' posteriors then they ...
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1answer
20 views

p(a,c) vs p(a∧c)

In this paper: https://www.aclweb.org/anthology/J/J16/J16-2006.pdf, the author breaks the Pointwise Mutual Information of a phrase up into several components: They use the ...
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0answers
19 views

Find conjugate prior of an exponential family distribution

I read on Wikipedia that all exponential family distributions have conjugate priors. I have not, however, been able to find a reference that describes how to find it. So given $$f_X(x\mid\theta) = h(x)...
3
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1answer
36 views

Can every factorization be represented by a Bayesian network?

A Bayesian network is defined as a directed acyclic graph with a set of random variables as its nodes, and it satisfies two axioms, 1) Root nodes (nodes without parents) are independent. 2) Given ...
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1answer
17 views

Bayesian probability calculation [closed]

You have one fair coin, and one biased coin which lands Heads with probability $3/4$. You pick one of the coins at random and flip it three times. It lands Heads all three times. Given this ...
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4 views

How to compute the probability and CI of replicating multiple previously observed statistically significant p-value?

The FDA often requires a sponsor to conduct multiple clinical trials prior to approval. Given the following observations in a ph2 and ph3 trial, how would you go about predicting the probability of ...
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0answers
33 views

Given $n$ heads out of $n$ tosses. What is the posterior probability that coin is fair? [closed]

I am given an $\sigma$-fair coin with the probability of head $(\theta)$ being in the interval $[\frac{1}{2} - \sigma, \frac{1}{2} + \sigma]$. Also I am given: For a Bayesian analysis of the ...
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1answer
45 views

Bayesian Example

Ex. suppose that $x=2$ denotes the number of successes in $n=5$ independent trials with probability $θ$ of success, that is $x$ has a binomial distribution with the parameters $n=5$ and $ θ$. ...
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13 views

Maximum-a-posteriori estimation with Gamma prior and scale-invariant likelihood

Let $\mathbf{X}$ be a vector of parameters with prior distribution $X_i \sim \text{Gamma}(\alpha, \beta)$ i.i.d. for $i = 1, \ldots, n$. Let us denote this prior by $p(\mathbf{X})$. We get to observe ...
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1answer
26 views

Setting up the liklihood distribution for Bayesian Estimation

Here is the exact problem: Suppose that the random variables $Y_1,\ldots,Y_n$ satisfy $$ Y_i=\beta x_i+\varepsilon_i, \quad i=1,\ldots,n. $$ where $x_1,\ldots,x_n$ are fixed constants, and $\...
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45 views

How can I generate random samples from following probability density function?

Let $\mathbf{\alpha}=(\alpha_1, \ldots, \alpha_m)$. The posterior density function of $\mathbf{\alpha}$ is given by $$h_0(\mathbf{\alpha}|\mathbf{x})=‎\frac{\prod_{i=1}^{m}\alpha_i^{a_i}}{\left(1+\...
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22 views

find the maximum where gamma is attained

I have this problem and I am not figuring the beginning
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1answer
48 views

Determine probability based on observation

Suppose there is an urn with 100 balls, of two colors, say white and black. Let $p$ be the probability of drawing a white ball. You draw one ball, replacing after the draw. After 100 draws, each with ...
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0answers
58 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
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2answers
95 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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0answers
21 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
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9 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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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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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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1answer
16 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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0answers
15 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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1answer
42 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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0answers
18 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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23 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
50 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
33 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
37 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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12 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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35 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
17 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 $\alpha^{*}$ ...
2
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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
24 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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36 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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votes
1answer
55 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 ...
3
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1answer
33 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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12 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(...
2
votes
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
26 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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23 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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0answers
17 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
56 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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0answers
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 ...