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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41 views

Conditional independence iff joint factorizes

I have proven that: $X⊥Y|Z\ {\rm iff}\ p(x,y|z)=p(x|z)p(y|z)$ for all $x,y,z$ such that $p(z)>0$. The next question is to prove an alternative definition: $X⊥Y|Z$ iff there exist functions $g$ ...
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3answers
52 views

Regression function of “non-regressible” data

I have some background in probability, and now trying to understand statistics, which sometimes leads to the questions of the following kind. Let $X$ and $Y$ be two random variables that represent the ...
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0answers
20 views

Sampling from conditional copula under R

this is a follow-up thread dealing with sampling from conditional copulas: Original question (with nice answer by whuber): Sampling from conditional copula Trying to sample from a conditional copula ...
4
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1answer
63 views

Sampling from conditional copula

I am having trouble finding anything on sampling from conditional copulas. I am only interested in the bivariate case. So, if $C(u,v)$ is my copula, I want to sample from it given a specific ...
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0answers
15 views

Conditional and joint pmf of sequences of random variables

I have random variables $X_1...X_N$ which are Poisson($\theta$) distributed. I also have inclusion indicators $Y_1...Y_N$ which are Bernoulli($\pi$) distributed. These indicate whether $X_i$ is ...
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12 views

game scoring marginal probability estimation

I have a game scoring time series data, where X[i] shows the score of player 1 minus player 2 after the ith turn. The game is played in turns, meaning if player 1 plays at ith turn, player 2 plays at ...
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46 views

Is reverse of p-value also uniformly distributed?

If p-value, $p(F|H_0)$, is uniformly distributed $U[0,1]$, so is true to say, that $p(F|H_1)$ is also uniformly distributed $U[0,1]$ or even more: is it true $p(F|H_0) = 1-p(F|H_1)$? I'm not sure if ...
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1answer
28 views

Basic summation of conditional probabilities question

I read on Bishop Chap8 P 374 that: sum(P(b|c)P(c|a)) = P(b|a) where the sum is over c. Can you prove that?
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14 views

Conditional Probability Or Condition

What is the general way to look at a conditional probability of the form : $P(A=true|B=true\;OR\; C=true)$
2
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0answers
44 views

joint distribution, probability, calculating probabilities under false independent assumption, when the random variables are actually dependent

Suppose I have random variables $X,Y,Z$ and I would like to compute the probability that random variable $X$ is smaller than $Y$ and $Z$: $$ \pi_X \overset{def}{=} Pr(X < Y, X < Z) = \int Pr(x ...
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13 views

Can you use accelerometer data for classification with Conditional Random Fields?

I want to recognize activities, based on accelerometer data from the smartphone. I studied Conditional Random Fields and the CRFSuite. Now I am Confused. In my opinion CRF training uses static single ...
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0answers
16 views

The probability density function using the Laplace transform

I am reading a paper that is trying to derive the following probability $$ \mathbb{P} \biggl( F \geq T(I+W) \biggl)$$ Assumptions: 1- Assume that the $F$ has a finite first moment and admits a ...
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19 views

Interesting question about full rank of random matrix

I have a question about rank of a random binary matrix. Assume that I have to make a random binary matrix with its size are k rows and n colmuns (k<=n). Each columns only has 1 or 0 values. Now I ...
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0answers
11 views

Conditional probabilty - case study

I am investigating the time on book (t) before write off.The sample data consists of times taken until write off, however these times have been calculated from accounts that have been written off ...
1
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1answer
34 views

Help setting up a Bayes rule problem

I am relatively new to using Bayes rules for continuous variables and am having trouble setting up part of the formula and am looking for help. The example I am trying to work through is the ...
2
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2answers
60 views

What is the mean of this exponential random variable?

I have read a paper that says that the following is exponentially distributed $$ Y= \bigl| \sum_{i=1}^n \gamma_i^{-\frac{1}{2}} h_i \bigl|^2$$ where $\gamma_i$ are non-negative constants and ...
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0answers
12 views

Probability generating function of poisson point process

Assume you have a 1D non homogenous PPP $\Xi$ with intensity $$\lambda(x)=\lambda x^{\frac{2}{\alpha}-1} \ x \in \mathbb{R}^+$$ where $\alpha$ is positive integer. Now define $$\gamma_k = ||x_k||$$ ...
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0answers
6 views

a joint probability distribution that passes through all markov network graph without being filtered

from filter view of the Markov Network where only those distributions can pass that satisfy all conditional independence statements given by the graph. • Can we think of distribution that can pass ...
0
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1answer
41 views

How to draw a Markov network graph for two or pair of variables

For $i \in \{1, 2, 3\}$, let $X_i$ be a random variable for the event that a coin toss comes up heads (which occurs with probability $q$). Supposing that the $X_i$ are independent, define $X_4 = X_1 ⊕ ...
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0answers
5 views

Multiplying distributions with different conditioning

I saw this expression in a UBC machine learning class lecture, and I'd like to understand how the math works. Suppose we're trying to predict a class label $y$ given some data $x$. There are prior ...
2
votes
1answer
55 views

Sum of RV : probability for the value of operands when the result is known

A random variable $Z$ is the sum of two independent random variables $X$ and $Y$, with known probability densities $f_X$ and $f_Y$, respectively. Now suppose you sample $Z_1=X_1+Y_1$ but you don't ...
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3answers
507 views

conditional probability trick questions - drawing cards from a deck and the meaning of 'at least'

Suppose that a box contains one blue card and four red cards, which are labeled A, B, C, and D. Suppose also that two of these five cards are selected at random, without replacement. a. If ...
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0answers
22 views

How to calculate distribution of Y|X from distributions of X|Y and Y [duplicate]

I'm trying to solve the following homework problem: Let $X$ given $Y=y$ have a normal distribution with mean $y$ and variance one, and let the marginal distribution of $Y$ be normal with mean ...
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1answer
26 views

PDF of sum of independent Gaussian variables

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
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0answers
18 views

Regarding Naive Bayes and conditional independence

We all have been talking about how Naive Bayes may, in some cases, not perform well due to the fact that this assumes conditional independence of features and MOSTLY, this is not true for real world ...
0
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1answer
46 views

Covariance of a compound distribution

I am trying to find the covariance of a compound distribution. Given $X=x$, where $X \sim \mathrm{Uniform}(0,1)$, $Y$ is (conditionally) normally distributed with mean $x$ and variance $x^2$. I ...
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1answer
35 views

Bayesian Network

I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
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1answer
42 views

Probability distribution of Y in regression?

I'm trying to predict the probability distribution of $Y$ given $X_0, X_1, ...$ with a nonlinear regression. The probability distribution of $Y$ is likely not normal. So far, I've set up and trained ...
2
votes
1answer
65 views

Convolution of random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
0
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2answers
75 views

basic probability problem

Problem 2 (page 33 of text) Assume that every patient with a particular type of disease has the probability 0.1 of being cured within a week, if the patient is given no treatment for the disease. Ten ...
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25 views

Proof of alternating conditional expectation base equations

How do we prove the base equations for Alternating Conditional Expectation algorithm. The statement is thus: We define arbitrary mean-zero transformation $\theta(Y),\phi(X_i)$,$1<i<p$ for ...
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0answers
33 views

Conditional independence: conditioning on an empty set of random variables

Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is ...
2
votes
1answer
37 views

Marginal Distribution from Conditional Distribution

I came across a problem where the marginal distribution of a random variable $Y$, $f(y) = c/y^2$ and $f(x|y) = 1/y$. Can I simply multiply these two to get $f(x,y)$ the joint distribution of $X$ and ...
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27 views

Hypothesis testing and order statistics

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
2
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1answer
64 views

Combination of Random Variables Conditional Probability

If A and B are independent discrete random variables and C = A+B, then how should one compute the pmf of P(A|C)? For example, let X be the result from a coin toss(1 or 0 for H and T) and Y be the ...
2
votes
1answer
40 views

An example of r.v.s such that their distribution has more (conditional) independencies than their directed graphical model

I was trying to form an example where I had 3 r.v.s such that the distribution describing them had more conditional independencies or independencies than the directed graphical model corresponding to ...
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1answer
47 views

In Probabilistic Graphical Models, what does it mean that r.v. X influences r.v Y?

I wanted to pin down what the intuitive phrase: r.v. X influences r.v. Y as precisely and as rigorously as I could, and wanted to check if my interpretation was correct and complete with the ...
0
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1answer
42 views

Why do we need undirected (Markov) graphical models?

I understand the modular nature of directed models, and that each node captures a conditional probability. But why do we need undirected models? As far as I can see they lack intuition in that the ...
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92 views

Bayesian modeling using multivariate normal with covariate

Suppose you have an explanatory variable ${\bf{X}} = \left(X(s_{1}),\ldots,X(s_{n})\right)$ where $s$ represents a given coordinate. You also have a response variable ${\bf{Y}} = ...
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25 views

Bayesian Updating Process, 3 signals and 2 states of the world

Suppose Nature chooses a state $\omega = \{X,Y\}$ at $t=0$. Long-lived agents observe a signal $s_t$ at every period $t$, where $s_t = \{ x,y,z \}$. Agents all hold a common prior $\mu_0 \in (0,1)$ ...
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0answers
20 views

Determine which variable or variables is/are the most efficient to predict the outcome

I have a small dataset (n=74) with a +/- 50 variables, not the best data but I have to work with it. The variables are used to select a product. I want to determine which variable or variables is/are ...
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0answers
21 views

Fusing probability scores from different independent sources

Consider a situation where two independent sources, $s_1$ and $s_2$ are giving probability estimates regarding the occurrence of a event $e$. I tried to model this as a bayesian network but that ...
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0answers
28 views

Word probabilities in a Naive Bayes filter

While implementing a Naive Bayes filter, I stumbled across a problem with the calculation of the conditional probabilities $p(w|c)$ of a word $w \in \mathcal{W}$ given a class $c \in \mathcal{C}$. ...
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1answer
29 views

How to work with conditional expectations and variances?

I have seen this statement in a lecture note : $$E[(Y-f(X))^2\,|\,X]=Var[Y|X]+E^2[(Y-f(X)\,|\,X]]$$ If $Z,X$ are random variables, then shouldn't be $Var[Z|X]=E[Z^2|X]-E^2[Z|X]$?In which case ...
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20 views

Conditional Forest - Calculate Overfitting, OOB Error

My name is Abhi and I am trying to teach myself conditional forests. I have got a basic example working myModel <- cforest(Survived ~ .,data = ...
3
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3answers
120 views

Conditional Probability with Normal Distributions?

Let's say that I have $3$ independent random normal variables, $A$, $B$ and $C$. They all have a standard deviation of $17.526$, while $A$ has a mean of $143$, $B$ of $139$, and $C$ of $129$. My ...
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2answers
274 views

Probability of two aces given at least one ace (out of two cards) vs probability of two aces given one card is the ace of spades (out of two cards)

So, in the video lectures from Harvard's Statistics 110:Probability course that can be found on iTunes and Youtube, I encountered this problem. https://www.youtube.com/watch?v=JzDvVgNDxo8#t=566 I ...
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2answers
123 views

Conditional expectation of $X$ given $Z = X + Y$

Suppose I have two independent normal variables $X$ and $Y$ with known mean and variance. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$? I am ...
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1answer
23 views

Modelling a probability distribution on different feature sets

I have a binary classification problem, and I use method A and method B to extract features, F1 and F2, for this problem from dataset X. Now, I train two models, y1 and y2, separately on the two ...
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1answer
83 views

about the definition of bayesian network

In this PDF http://people.csail.mit.edu/yks/documents/classes/mlbook/pdf/chapter2.pdf page 5 says: Given a set of functions $f(x_i,pa(x_i))$ non-negative and sum to 1, we define a joint probability ...