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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24 views

Probability of winning one contract

I have been trying to figure out the following question: A private company has submitted bids on two separate federal government contracts. The company president believes that there is a 45% ...
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20 views

Incorporating background knowledge into a model?

I have built a model that predicts the class of an observation, based on explanatory variables A; i.e. predicting P(y|A). From domain knowledge and academic literature, we know that other variables, ...
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21 views

Conditional independence conjecture

Suppose there are four random variables (events), $A$, $B$, $C$ and $X$. If we have $X\perp A|B$, saying $X$ and $A$ are conditional independent given $B$, then I want to ask that whether we have ...
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46 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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2answers
39 views

How do you calculate a Poisson distribution for matched data?

I am trying to find the rate of infections with 2 different treatments and to see if there is any difference between them. I want to work out whether the rate of 'no infections' is different between ...
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25 views

How to find conditional distributions from joint

I want to learn about how to do Gibbs sampling, starting with finding conditional distributions given a joint distribution. While looking for examples, I found this blog post that I wanted to ...
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1answer
71 views

Semicolon in probability expression

I run in to this formula when reading a tutorial: $$ \begin{align} P(\pi|\mathbf L;\gamma_{\pi1}, \gamma_{\pi0}) & =P(\mathbf L|\pi)P(\pi|\gamma_{\pi1},\gamma_{\pi0})\tag{28} \\ &\propto ...
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37 views

Why is Bayes theorem more popular than the normal definition of P(A|B)? [duplicate]

As everyone knows, the conditional probability of A given B is $P(A|B) = \frac{P(A\cap B)}{P(B)}$, and Bayes' theorem is derived from that equation to $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. I'm pretty ...
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13 views

Conditional probability with joint distribution gaussian [duplicate]

Given that $(x_1, x_2)$ are jointly normally distributed with $\mu = 􏰃\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}$ 􏰄and $\Sigma = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{21} & ...
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1answer
49 views

Probability of P(A,B|C)

I just want to double check that this is true. Google doesn't cooperate well with equations. P(A,B|C) = P(A|B,C)P(B|C)
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20 views

Attraction/ likelihood of going somewhere given similar people have already gone

Given a venue which has previously been attended by a demographic of people (e.g. old, young, middle age) I want to compute the affinity those demographics have for a venue and therefore compute the ...
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1answer
60 views

Question about Conditional Probability

I have: P(c) = 0.98 and P(v) = 0.96. Also I have this conditional probability table: ...
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1answer
25 views

What is the predictive distribution of Bayesian supervised Learning? (rigorous argument)

I was trying to understand the posterior predictive distribution for any supervised predictor (by that I mean any classifier or regression predictor $f$). The exact equation I am unsure of is: $$ ...
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1answer
41 views

Is this an instance of the base-rate fallacy?

The following line of probability reasoning is supposedly fallacious, and is an instance of the base-rate fallacy. The argument is that $(1)-(3)$ don't give us enough reason to conclude that $(C)$. ...
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40 views

How to exploit relationships between independent variables?

Data: Each instance (representing a document) is a bag-of-entities (like BOW, except they're Wikipedia entities instead of words), so each feature is a binary or tfidf-like score based upon the ...
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2answers
44 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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2answers
108 views

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 ...
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28 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
12 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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0answers
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
32 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
26 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
19 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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39 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 ...
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29 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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60 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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0answers
14 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. ...
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0answers
24 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
204 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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10 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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24 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
14 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 ...
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1answer
34 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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98 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 ...
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1answer
36 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
62 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
21 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
15 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
127 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
104 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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0answers
47 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
29 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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18 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
119 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
88 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
10 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 ...