Questions tagged [conditional-probability]
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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conditional probabilty and the sample space
BACKGROUND
Allow that an experiment 1 and 2 are defined by a probability space triple $(\Omega_1, \mathcal{F}_1, P_1)$, and $(\Omega_2, \mathcal{F}_2, P_2)$, respectively [1].
Allow that an ...
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Conditional probability of tossing coins with uncertain head probability
Suppose there are two coins A and B. When tossing a coin $i$, "head" happens with probability $p_i$.
The problem is that $p_i$ itself is a random variable. Say that the associated probability ...
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Compound random experiment and product probabilities
As I understand, a random experiment is modeled with a probability space. Then, you have compound experiments, which are those that can be decomposed in several experiments (I am unsure whether this ...
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1answer
54 views
Probability of a box containing a combination of color
Let's say, we have a box containing 3 balls in it, they can be either red or blue.
Someone draw a ball 5 times with replacement and get 4 red and 1 blue (not necessarily in order). Do you know how to ...
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What is the probabilty that X > 2 conditioning on Y = 2? (Homework)
another homework question here.
Let 𝑌 be a binomial random variable with 10 number of trials and 0.2 probability of success.
Let X be a uniformly distributed random variable over the interval [0, 3].
...
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18 views
Conditional Probability Given Multiple Priors
Reference: ilanman Oct 3, '16 at 13:27. Even though years old, this discussion is relevant to my current interests. The P(Ai|B) in the last formula above, reverses the position of the posterior and ...
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Conditional Probability given sample size
I am working with a dataset of fraudulent credit card purchases. I want to represent on of the features as the conditional probability $P($fraud$|$feature$)$. The feature is categorical feature with ...
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28 views
exponential family form of discrete conditional distribution
What is the exponential family form of a discrete random variable conditioned on another discrete random variable? For example, p(y|x) where both x and y are discrete. Is there a standard form for ...
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Equivalence of expressions involving conditional probabilities
Take the following random variables $Y,\epsilon_0,\epsilon_1,V_0,V_1$, with finite supports $\mathcal{Y}\equiv \{0,1\},\mathcal{E}_0, \mathcal{E}_1,\mathcal{V}_0,\mathcal{V}_1$, respectively.
Suppose ...
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1answer
31 views
Need help understanding Toru Tsujishita' theorem on Triple Information
I struggling to understand point (ii) of Toru Tsujishita's theorem (here) on Triple- (Interaction- or Co-) Information
What is meant with the maps $\varphi_j$ and $\varphi_k$? A biunivocal ...
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Deriving conditional probability of bivariate bernoulli by using Dirichlet
While I was working on my research project, I found it difficult to derive a conditional probability from Dirichlet dist.
Consider two Bernoulli trials that are possibly correlated with each other. ...
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What is to be done when PDFs are not Gaussian/Normal in Naive Bayes Classifier
While analyzing the data for a given problem set, I came across a few distributions which are not Gaussian in nature. They are not even uniform or Gamma distributions(so that I can write a function, ...
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1answer
23 views
Notation for conditional probability when the conditioned event is observed
I am trying to understand proper notation for functions related to state transition models (e.g., a HMM). There are two indexing variables $t$ (time step) and $i$ (state). Such that it matters, in my ...
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1answer
21 views
Calculating Conditional Probability without knowing individual probabilities
I have a case where I know the probability that three annotators will agree with each other pairwise, and I'm trying to find the probability that all three will agree on a yes or no question given ...
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What does “Normal distribution conditioned on $x \mod 1$” mean?
I understand what $Pr[X = x | Y = y]$ means, however, on this paper (Lemma 10 in Appendix A), we have the following:
Let $D_r$ denote the continuous Gaussian distribution of parameter $r$,i.e., the ...
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42 views
Sampling with fixed probability from two different distributions. How is the sample distributed?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$
$X$ be real-valued random variable on $(\Omega,\mathcal A,\...
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1answer
63 views
Probability of A given B and C
I'm trying to write an algorithm and I'm rusty on my statistics. Basically my question comes down to how do you get the probability of A given B and C.
I'm trying to walk myself through a made up ...
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1answer
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Conditional Probability Question in a Diagram
I know Pr(X2 = T | X3 = F) = Pr(X2 = T, X3 = F) / Pr(X3 = F) but I don't know how to figure out each probability individually. Anyone have any idea how to do it?
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Why this could not be represented by a Bayesian network
A, B, C, D are four variables. Why $A \perp C | \{ B , D \}$ and $B \perp D | \{ A , C \}$ could not be represented by a Bayesian network?
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Conditional Probability Question from Diagram [closed]
Anyone have any idea how to approach this problem?
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Monty Hall Problem with a Fallible Monty
Monty had perfect knowledge of whether the Door had a goat behind it (or was empty). This fact allows Player to Double his success rate over time by switching “guesses” to the other Door.
What if ...
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Rewrite joint probability as product of marginals when all the probabilities are $1$ or $0$
I have a question about the possibility of rewriting a joint probability as the product of the marginals when all the probabilities can only take value $1$ or $0$. I start with introducing some ...
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2answers
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Marginalize probability with three variables
I am reading a book and saw the following equation:
$$ P(X|\theta) = \sum_{z}P(X|z,\theta)P(z|\theta)$$
I know that that it is:
$$ P(X) = \sum_{z}P(X|z)P(z)$$
But I don't know how the above equation ...
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2answers
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Writing the Conditional probability correctly
I was reading the book statistical rethinking chapter 3. There is the following example:
To repeat the structure of common examples, suppose there is a blood
test that correctly detects ...
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18 views
Questions Regarding Conditional Entropy
I have a question related to a conditional entropy that I have to solve and I struggle a bit on the understanding of my implementation. So I formalized the questions here as follows. Any suggestions ...
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Which distribution should I use for Naive Bayes algorithm(Gaussian or Rayleigh)? What to do with categorical data?
I am predicting whether credit card application of an individual would be approved or not given his/her credentials.
I have the following dataset:
The variable descriptions are as follows:
I need ...
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1answer
22 views
Conditional probability for binary events [closed]
I have 2 columns of data called level 1 event and level 2 event.
Both are columns of 1s and zeros.
I want to find the probability of a level 2 event given that the previous event was a level 1 event....
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2answers
131 views
Gibbs algorithm using negative binomial produces NAs
I have the following full conditionals distributions:
$$
X_2|X_1=x_1\sim Bin(x_1,p)\\
X_1|X_2=x_2\sim NegBin(x_2,p)
$$
So I'm using the following code to generate a sample from each one:
...
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23 views
Conditional distribution of a function of a random vector given conditional distribution of random vector
Let $\mathbf{X}=(X_1,...,X_n)^T$ be a multivariate normal distribution. Now we have $\mathbf{Y}=(Y_1,...,Y_n)^T$ defined by $Y_i = e^{X_i}$. Let $\mathbf{Y^1}, \mathbf{Y^2}$ be partitions of $\mathbf{...
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1answer
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How do I calculate $s$ from $\mathbb P(X+Y>u\mid X<s)=q$
Suppose $X,Y$ are (not necessarily independently) normally distributed, how can one calculate the maximal limit $s$ that $X$ may reach such that the probability of the sum $X+Y$ overshooting a given ...
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1answer
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How to infer a missing observation in a state space model?
I read here that "structural time series models handle missing values naturally, following the rules of conditional probability. Posterior inference can be used to impute missing values, with ...
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1answer
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Conditional survival probability up to time $T$ given $t > s$
This is a really basic question I know but for some reason I'm failing to convince myself of the right answer here.
Given a survival model that has CDF $F(t) = \mathbb{P}(\text{failure before}\ t)$
...
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1answer
22 views
Calculate a Conditional Expectation via Samples
Consider a binary random variable $Y \sim p(Y)$, a random variable $X \sim p(X)$ (can be discrete or continuous) and a conditional distribution $p(X|Y)$. Suppose that I generate $N$ samples from $p(Y)$...
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Expected value with respect to a conditional distribution
I am reading the thesis Variational Inference and Deep Learning. On page 16, while deriving the ELBO, the expectation is w.r.t. a conditional distribution $q_{\phi}(\mathbf{z} | \mathbf{x})$:
$$
\...
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Sample Proportion for Estimating Conditional Probability
Suppose I have $n$ observations of binary data $(X, Y, Z, \ldots)$ and suppose I want to estimate $\theta = P(X = 1 | Y = 1)$. It seems clear to me that
$$\frac{\sum_{i}\mathbb{1}_{X_i = 1, Y_i = 1}}...
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Conditional probability density function (PDF) of bivariate normal distribution
Let $X$ and $Y$ have bivariate normal PDF with correlation coefficient $\rho$, i.e.,:
$f(x,y)=\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp{(-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_X)^2}{\sigma_X^2}+\...
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1answer
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Independent variables in conditional probability
So I am reading this paper about reasoning(Hall, Ali, Chater, & Oaksford, 2016). Consider this conditionals.
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1answer
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Probability of an event at the outcome of a binary source
Let a binary (and independent) source $S$ generating a binary sequence with the following probabilities: $p$ for the symbol '0' and $1-p$ for the symbol '1'.
What is the probability that $S$ gives $n$...
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1answer
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Self study of basic probability: Can't solve exercise
I am currently working through 'Concepts of probability theory by Pfeiffer' and can't solve the following problem:
A carnival man hides a pea under one of three nut shells. By a series of complicated ...
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2answers
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Why not to use Bayes theorem in the form $p(\theta | x) = \frac{L(\theta | x) p(\theta)}{p(x)}$?
There are a lot of questions (like this) about some ambiguity with Bayesian formula in continuous case.
$$p(\theta | x) = \frac{p(x | \theta) \cdot p(\theta)}{p(x)}$$
Oftentimes, confusion arises ...
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1answer
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Rearranging Conditional Probability Equation to Show Dependencies
Given the random variables $X$, $Y$, and $Z$, with joint pdf given by $p(x,y,z)=kf(x,z)g(y,z)h(z)$ for some constant $k$, my task is to show that $p(x|y,z)$ is a function of $x$ and $z$. My work is as ...
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Conditional density under conditional indepencence?
Let $X,Y,Z$ three random variables such that the joint density can be factorized as
$$f(x,y,z) = f(x \mid z) f(y\mid z) f(z).$$
This is, I am assuming conditional independence of $X$ and $Y$ given $Z$....
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Conditional probability formula for continuous random variables [duplicate]
Let $V,T$ be two random variables with supports $\mathcal{V},\mathcal{T}$, respectively. Let
$P_{V|T}$ denote the probability distribution of $V$ coditional on $T$
$P_{V,T}$ denote the probability ...
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1answer
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Application of law of total probability for continuous random variables
Consider 3 random variables $Y,V,T$, with supports $\mathcal{Y},\mathcal{V},\mathcal{T}$, respectively.
Let
$P_{Y,V}$ denote the probability distribution of $(Y,V)$
$P_{V}$ denote the probability ...
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1answer
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Is the following definintion of Discrete Random Variable given in the book of Joe Blitzstein correct?
The following is the definition given in the book:
A random variable $X$ is said to be discrete if there is a finite list
of values $a_1, a_2,\ldots, a_n$ or an infinite list of values $ a_1, a_2,...
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1answer
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Why $E$ and $F$ are conditionally independent given $C$ and $D$?
Below is a Directed Acyclic Graph (Fig.a). From this figure, it is said that:
$E$ and $F$ are conditionally independent given $C$ and $D$.
I am confused about it. Let's assume the causal ...
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How the conditional probability is being calculated in Rejection sampling
In a class lecture, the "Acceptance-rejection algorithm" was presented as follows:
To generate $𝑋 \sim 𝑓(𝑥)$, Find density $g$ satisfying
$\frac{f(t)}{g(t)}<=c$ for some constant $c$ for ...
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1answer
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Computing a marginal distribution of a joint involving a delta function
Suppose that we have four continuous random variables $x,y,z,$ and $v$ and we want to compute the following integral:
$$\int f(x\mid y)f(z\mid x,y)f(v\mid z,x,y)\,dx$$
There are a few conditions:
$...
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Correct terminology to refer to a collection of conditional probability distributions
I would like your help to find the correct terminology to refer to a collection of conditional probability distributions.
Consider two random variables $Y$ and $X$, with supports $\mathcal{Y}$ and $\...
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Changing a conditional probability to a deterministic function
Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
