The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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20 views

Conditional probability of first ball being blue given that the second ball is red

Start with an urn with 5 red and 3 blue balls in it. Draw one ball. Put that ball back in the urn along with another ball of the same color. Now draw another ball from the urn.Suppose the second ball ...
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1answer
22 views

Clarification on Notation

I'm using Andrew Gelman's 3rd edition of Bayesian Data Analysis and am going through the exercises. For one of the exercises, he supposes that if $\theta = 1$, then $y$ has a normal distribution with ...
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39 views
+50

Distribution and knowledge about a state of the environment

Let's assume I have a frequency distribution for train travel times, estimated based on one month of empirical data: [1:00h, 1:20[ : 20% [1:20h, 1:40h[ : 50% [1:40h, 2:00] : 30% Edit: The data ...
2
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0answers
19 views

full conditional posteriors for bayesian lasso

I am reading the original Bayesian Lasso paper, and its follow up; They look straightforward to implement, mainly because of the conditional posterior probability for the gibbs sampler; however, I ...
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1answer
11 views

Probability of binary outcome based on observed values of correlated variable

How should one approach the following problem? Suppose an object has an unknown binary attribute X in {0, 1} (for example it is only possible to be either ...
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46 views

Evaluating integral to obtain marginal PDF related to Tikhonov Regularization

I am attempting to derive the marginal PDF for an application of the Gibbs Sampler. My joint PDF contains: $P(b,x) = \frac{1}{\sigma^{n}}\exp \left( -\frac{1}{2\sigma^2}\left\lVert ...
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2answers
30 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
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1answer
21 views

r markov chain markov property on binary variable, discrete time

i have a sequence of 1/0's indicating if patient is in remission or not, assume the records of remission or not were taken at discrete times, how can i check the markov property for each patient, ...
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0answers
16 views

Impact on expected frequency of yellow & red balls of drawing blue ball after yellow ball

I have a opaque urn holding a finite number of balls. I am asked: "Are there more balls marked A or B"? The first ball I pull out is marked A- so, at that point, it seems more likely that there ...
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2answers
37 views

Expected number of trials

Consider independent trials, each of which is a success with probability p and derive the expected number of trials needed to obtain k consecutive successes by (a)conditioning on the time of the ...
2
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0answers
23 views

Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline. With that assumption, the mean and trend ...
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0answers
47 views

pyMC: implementing a joint distribution model

I'm attempting to model a multi-modal distribution that's affected by two separate distributions in pyMC and am having trouble implementing a joint or conditional distribution. Suppose I have N data ...
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13 views

Can Platt Scaling to calibrate probabilities be used for classifiers other than SVM?

I am using Gaussian Mixture Models as classifiers and I compute posterior probabilities from them for a 2 class problem. However, the probabilities are pushed towards 0 and 1 due to very skewed ...
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1answer
38 views

Probability of a medical symptom based on different causations

I want to ask something about a common medical situation: Suppose the probability of a medical symptom X (let's say blue tongue:-) ), to occur in a person at a specific time moment is 1/1000. ...
2
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0answers
21 views

Bayes Rule with 1 Signal but 2 Unknowns

This is a question I originally posted in the math.stackexchange site, but didn't get much of an answer. Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 ...
2
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1answer
200 views

PDF of dependent variables

In my recent question an answer was given, and I am able to compute it myself. Still, I'd like to understand where does that answer come from. Hence, what's the approach to handle dependent variables ...
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0answers
9 views

Discretizing Conditional Probability Density Function of varying dimensions

I hope I can explain this well without too big a wall of text. I have a conditional probability density (CPD) function of arbitrary shape that varies in the number of dimensions depending on the ...
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0answers
22 views

Estimating latent probability variables from binary response data (logit GLMM)

Question: How can I calculate the standard error of an estimate derived from two coefficients of a logit GLMM? We're studying the effect of a categorical condition ('volume', 3 levels) on two latent ...
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0answers
11 views

Conditional expectation in mixture distributions

I have a mixture distribution for observed lifetime data $(\delta_i,t_i,L_i)$, where $\delta_i$ is a censoring variable (1 indicating death, and 0 indicating censoring), $t_i$ is the observed lifetime ...
4
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1answer
32 views

Gibbs Sampler output: how many Markov chains?

When running a Gibbs sampler (for $n=200$ Iterations) with two full conditionals, I get the output $\mathbf{x} = (x_1^{(n)},x_2^{(n)})_{n =1,...,200}$. So $\mathbf{x}$ is the realizations of a Gibbs ...
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0answers
92 views

Bayesian inference with sampling and mixture models

I'm having some trouble doing Bayesian inference on an experience I have in hands. I apologize in advance if it is too complex, but I couldn't find a trivial way to split it in several parts. Let ...
2
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1answer
33 views

Proof of Chapman Kolmogorov equation

In the proof of Chapman Kolmogorov Equation $p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$ Proof: $p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$ By the total probability it says ...
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1answer
52 views

Expected number of collisions for balls and bins problem

$n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i_{th}$ ball ...
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1answer
20 views

Methods to weight probabilities that are being multiplied together?

I am trying to classify data according to a taxonomy: That is, given a feature vector $x$, I would like to compute: $P(A,\alpha)$ $P(A,\beta)$ $P(B,\delta)$ $P(B,\gamma)$ (You can think of the ...
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1answer
13 views

Factoring a probability distribution containing a latent variable

I distribution which involves 3 parameters, which I'll call (for now) $P(z | y, x)$. However, one of the parameters is a function of another. For instance, let the random variable $y$ be a ...
5
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2answers
126 views

What can we infer if P(A|B,C) = P(A|B)?

I want to understand what we can infer when the following equation holds: $$P(A|B,C) = P(A|B)$$ My understanding is that $C$ does not give us any extra information about $A$ that $B$ does not give us ...
3
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1answer
87 views

Conditional Poisson Distribution

The number of claims, $N$, in a year on a portfolio of policies follows a Poisson distribution with parameter $λ$. Large claims have a probability $p$ and small claims $(1-p)$, independently of ...
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46 views

3 Class Classification based on a Bernoulli feature and gaussian feature

Consider a 3-class classification problem with mixed features, where one feature is Gaussian and another is discrete Bernoulli: Prior class probabilities: P(C1) = .5, P(C2) = P(C3) = 0.25 ...
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0answers
12 views

Help interpreting this result based on conditioning on gender -Contigency Tables

The problem i'm having is how to interpret the results I got. For part a) I got that the abortion laws are independent of generation where I used a $\chi^2$ test. For part b) I got that the the age ...
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0answers
13 views

How to test a probabilistic fertility simulation?

Imagine I want to build a simulation of a certain society. One part of it is a fertility model, which describes under which circumstances people are born. Let's assume that the model is very simple: ...
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0answers
24 views

Variance of multinomial distribution that is product of 4 Beta random variables

I have a system of 4 binary random variables, $A$, $B$, $C$ and $D$. $A$, $B$ and $C$ are conditionally independent given $D$, and I'll call one set of samples $ABCD$ an event (e.g. $ABCD$ meaning all ...
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0answers
14 views

scikit learn gaussian mixture conditional distribution

I am using scikit-learn to fit a Gaussian Mixture Model to a dataset. However, I now need to find the distribution conditional on one or more variables and I have not found a way to do that. Can ...
2
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1answer
107 views

An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the ...
2
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1answer
74 views

Marginal, joint, and conditional distributions of a multivariate normal

Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The ...
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0answers
8 views

Dependence of PDF of LLR of symbols

I have a system model with $y=hs+\sum_i^n gx+n$ where h is rayleigh fading desired channel, g is interfering channel x is interfering symbols. $\hat{s}=w*y$ where w is MMSE filter. On what factor pdf ...
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0answers
34 views

conditional distribution question

I have a joint distribution which factorizes as follows: $$ P(y, w, \beta) = P(y|\beta, w) P(w) P(\beta) $$ Now, I want to write the conditional distribution for $P(w|y, \beta)$, so this should be ...
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0answers
24 views

t-student distribution [duplicate]

I've got this problem: Here, if $Z,W$ are independent random variables, and $Z$ has normal standart distribution and $W$ has $\chi^2$ with $n$ degrees of freedom, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$. I ...
0
votes
1answer
32 views

conditional probability, change of variable and Jacobean

I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique. I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can ...
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0answers
11 views

Estimating conditional probability of bernoulli data

Assume I have $i=1,\dots,N$ fathers, each with $j=1,\dots,n_i>0$ sons. Now there is a binary event $A_{i,j}$ with outcomes 1 and 0 and the respective probabilities $p$ and $1-p$. Now I want to ...
2
votes
1answer
45 views

On an implication of the memoryless property of the exponential random variable

I know that if we take $X \sim Exp(k)$ then we have this property: $$P(X \ge s + t | X \ge s) = P(X \ge t)$$ But why does this imply that $X | X > x$ has the same distribution of $X$ only ...
0
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1answer
17 views

Posterior probability of an image when the posterior of local features are known

Lets assume the local features $x_1, x_2, \dots x_n$ of an image $I$ are independent. I know if $p(x_i|c)$ are given $p(I|c)$ can be defined by $\prod_{i=1}^N x_i$ But I dont know how to calculate ...
0
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0answers
31 views

A question about raising the power of the integrand

Given that we know $P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$, is there any way to express $\int_{\theta}[F(x|\theta)]^{n}dG(\theta)$ also in terms of $\alpha$?
0
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35 views

Is is possible to determine conditional conjugacy in this case?

I'm working on a problem where I have to extract sufficient statistics for parameter estimation in a state-space model. Usually these come from the quantities used for conjugate updates. I'm OK with a ...
5
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2answers
147 views

A Question on Elementary Statistical Inference

A box contains $5$ white and $2$ black balls. A coin with unknown $P(Head)=p$ is tossed once. If it lands HEADS then a white ball is added, else a black ball is added to the box. Then a ball is ...
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1answer
39 views

What does “if” mean in this question?

What is the probability that Sam is guilty if Tom and Devi gave conflicting testimonies? Is it conditional probability? Or intersection simply?
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vote
1answer
97 views

Sufficient statistic for a Gamma distribution

I am confused about the steps I need in order to solve the equation below. I must use conditional distribution (and NOT the factorization theorem). Q: $X_1, . . . , X_n$ is a random sample from a ...
0
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0answers
44 views

Bayes Net: how to calculate joint distribution?

I originally posted this question to Computer Science Stack Exchange, but then I was told that CrossValidated site existed. I've been reading many questions, but none of them seem to answer my doubts. ...
0
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0answers
30 views

conditional density wrt lebesgue measure

$X,Y$ are two r.v. $(\Omega,\mathcal{A},\mathbb{P}) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ and have joint density wrt to $\lambda^2$, the two dimensional lebesgue measure. So $f_X(x) = ...
0
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0answers
52 views

Conditional distribution function of sum of correlated Bernoulli variables

Let $X_i$ be a Bernoulli variable with probability $p$ for $i=1,...,N$. Hence $\sum_i X_i$ is binomial and approximately normal$(p,(1-p)p/n)$ for large $N$ The conditional probability distribution of ...
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0answers
23 views

Conditional distribution of current state of HMM given past observations and state

I want to compute the following conditional probabilities for an HMM, where I shall refer to the state at time $t$ as $X_t$ and the observation at time $t$ is $O_t$: $$\text{Pr}\left(X_t | O_1, ...