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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Understanding error in bayesian inference
Let us say we have:
Data $X$
Parameter that we are trying to estimate is $\Theta$
The Bayesian estimation method is to
Assume a prior on $\Theta$
Sample $x$ from $X$
Use Bayes theorem. Compute the ...
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Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold?
Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold? (Here $\mathbf{1}(\cdot)$ is the indicator function). To me it seems that it holds. Here is the proof:
$E(X|X\in A)=\int_{-\infty}^{\...
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Conditional Distribution of Normally distributed random variable
Let $x \sim \mathcal{N}\left(0,\sigma^2\right)$ and $y = x+\epsilon$ where $\epsilon\sim \mathcal{N}\left(0,\sigma^2_\epsilon\right)$ and independent of $x$. We know that the conditional distribution ...
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Which metric to use to evaluate Quantile Regression?
I have a prediction problem for which I want to predict the 75% Quantile using Quantile Regression. I am a little bit confused on how to evaluate this model (and also compare different models).
If I ...
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How to apply the the conditional probability and chain rule formulas in a multivariable case?
I'm learning about Bayesian Fusion and have a question regarding an expression that I wasn't able to prove.
It poses the following statement:
Assume $y_1, y_2, y_3$ are three observations which are ...
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Understanding conditional notation
Which is the correct way to describe the conditional probability distribution of X conditioned upon Y where X = a, Y = b
$$P_{X \mid Y}(a \mid b) \tag1$$ or $$P_{X \mid Y}(a,b) \tag2$$
What is the ...
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What is the exact role of model $p_\theta$ in Diffusion models for the reverse process?
I'm reading this interesting blog post explaining Diffusion probabilistic models and trying to understand the following.
In order to compute the reverse process, we need to consider the posterior ...
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How can we expand the following probability?
Suppose we have a predictive regression of the form
\begin{align}
y_t=\beta x_{t-1}+u_t\\
x_t=\rho x_{t-1}+\varepsilon_t,
\end{align}
for $t=1,\cdots,T$, such that $cov(x_{t-1},u_t)=0$ for all $t$, ...
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Given all conditional probabilities, find joint probability
For two random variables $A$ and $B$, if we know $\mathbb{P}(A|B)$ for all possible values of $A$ given all possible values of $B$ respectively, and $\mathbb{P}(B|A)$ for all possible values of $B$ ...
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Intuition about the relation between joint distribution, marginal distribution, and conditional distribution
The wording "intuition" might be a bit imprecise. I want to discuss how we visualize in our head going from one to another among the joint PDF, marginal PDF, and conditional PDF.
To make the ...
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Finding the conditional distribution from given normal distributions using Bayes' theorem
Background
This question is related to my previous question: Describing the measurement of a random variable as another random variable, but I've narrowed and clarified my question. I think I've ...
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calculate combination of matrix of probabilities (win rate ranking in game)
Let imagine we have a game with 4 players. And after playing game, we will get ranking of 4 players based on their score, rank 1 is the best, rank 4 is the worst. I have created a model for predicting ...
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Describing the measurement of a random variable as another random variable
Background
Suppose we have a box of resistors. The manufacturer rates these resistors at 100 ohms, but they have some variability. Let $x$ be the true resistance of a resistor chosen from the box at ...
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is P(notB|A) same as not(P(B|A)) [closed]
is P(notB|A) same as not(P(B|A)) if not then what is the difference between them? this is taken from bayes question.
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Under what condition the left stochastic matrix P(X|Y) is an invertible matrix?
Say we have two random variables $X$, $Y$. They are discrete variables (or discretization of continuous variables), both with $k$ categories. Define the left stochastic matrix as $P(X|Y)_{ij}:=p(x_i|...
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Question about probability equation
While I am studying Kalman filter, I saw the following equation
P(A, B, C) = P(A, B)P(C|A)
Is this right? if right, I am wondering why this equation holds.
Thank you :)
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Expand conditional by marginalisation and drop terms from conditional
I have come across this conditional expansion a few times, and I can't seem to make sense of it.
$$p(z|y) = \int{p(z|f)p(f|y)df}$$
I would go about it like this:
\begin{align}
\require{cancel}
p(z|y) &...
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Margin Distribution N-dimensional random vector
I have this exercise on my textbook and I can't understand how I have to do it.
I tried to compute the posterior but I don't understand if it's what the exercise requires.
Thanks
This is what I did ...
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Precise Definition of $\mathbb{E}[X\mid \sigma(A)]$ Conditional Expectation of Random Variable given Sigma Algebra generated by a set
I want to define precisely, exhaustively and constructively the conditional expectation of a random variable given the sigma algebra generated by a set. This question has some discussion on it but the ...
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Conditional Expectation of Random Variable given an event
Suppose $(\Omega, \mathcal{H}, \mathbb{P})$ is a probability space, $(\mathsf{E}, \mathcal{E})$ a measurable space and $X:\Omega\to \mathsf{E}$ a random variable with well-defined expectation $\mathbb{...
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Product of kernels vs Composition of kernels
According to Wikipedia there are two main operations between two kernels: product and composition. They look almost identical to me and I cannot figure out what's the intuition between these different ...
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What's the conditional variance of the chain X -> Y -> Z?
If I have a cascade of 3 random variables, represented as a Bayesian Graph: $X\rightarrow Y \rightarrow Z$, is there a simple formula for $\sigma_{X|Z}$?
Further, assume all the variables are normal, ...
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How to calculate conditional probability on student multivariate distribution
I have a multivariate student distribution fitted on some data on 4 dimensions (so I know the parameters).
I am trying to calculate the $P(X_4\le x_4| X_1=x_1, X_2=x_2, X_3=x_3)$ but falling short.
I ...
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Estimate distribution of variable from non-perfect predictions
Say I pull $n$ balls from a box while blinfolded. The balls can be either red or blue. I do not know the distribution of the balls.
After that I receive a list predicting the color of every ball I ...
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On the likelihood assessment of two extremely rare events
I've been pondering the concept of subjectivity to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my question over trivial hypothetical examples below. I did ...
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Integration of the sum of gaussians over a line [closed]
suppose we have two independent gaussian distributions $X$ and $Y$ .
let $Z = X+Y$ which will be a gaussian too.
the goal is to compute :
$ \int_{L} P(X+Y=c) dXdY$
Where L is the line : $X+Y=c$.
$P(X+...
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Central Limit Theorem with Exponential and Uniform Distributions
The waiting time of a customer in a customer service telephone line in company number 1 has the exponential distribution with an expected value of 2.2 minutes.
The waiting time of company 2 has the ...
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Can you have a "PMF-PDF" Together?
This is a question I have always struggled to understand:
For a discrete random variable, you can have a "Probability Mass Function" (PMF) : For several discrete random variables, you can ...
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Matrix Factorization for Recommendation System vs just calculating Conditional Probabilities
I have a matrix factorization model (using Lightfm, without user-features) to recommend items to users, based on a user-item matrix.
Context
A particular recommendation stuck out to me as curious, ...
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Computing conditional distribution of hidden state given observed states?
I am interested in the following Gaussian linear system that describes a Hidden Markov Model (HMM):
$$x_{k+1}=Ax_k + u_k + \xi_k, \xi_k \sim N_2((0,0), 0.01I_2)\\
y_{k+1}=C^tx_{k+1}+\eta_k, \eta_k \...
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Probability of seeing sun rise tomorrow using Bayes theorem
When Richard Price's published the Bayes theorem, he gave the example of a man seeing the sun-rise for the first time and wondering if it happened everyday. With each observation thereafter, he ...
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Find asymptotic variance of the moment estimator
I have that
$$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$
I have the conditional distribution:
$$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$
and we ...
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix
I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix.
How to formulate that a line of Covariance ...
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Consecutive coin flips, what is the appropriate statistical test for this word problem? [closed]
I was listening to a podcast by NDGT (Neil deGrasse Tyson, a prominent scientist) and he posed a simple thought experiment to illustrate the susceptibilities to cognitive bias. What I've come here to ...
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Conditioning of join gaussian over a line
I need to compute the conditional probability of bivariate normal distribution over a line.
Let's suppose that X and Y both are normal distributions and that they are independent.
Let's suppose that ...
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Probability Notation Confusion
In the following image, taken from paper on Variational Autoencoders by Dr. Kingma and Dr. Welling, it is mentioned that probability of the data points given the parameters factorizes into product of ...
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Logistics model on variable with values 1, 2, 3?
I have a dataset containing traffic crash information. One variable in the set is the number of fatalities that resulted in the crash, which has the values 0, 1, 2, and 3.
I am working in R and want ...
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Estimating conditional mutual informations from 2D histograms
I have binned marginal and joint distributions of two event features X and Y, i.e. p(X), p(Y) and p(X,Y) where the marginal distributions in X and Y are obtained by summing p(X,Y) over the bins of the ...
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Inference on a Gaussian random field / undirected graph?
Assume I have an undirected graph with $D$ nodes, and a $D$-by-$D$ matrix with edge strengths between $0$ (implying conditional indepedence given all other nodes), and $1$ (implying complete ...
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What's wrong with this conditional probability working?
A past exam paper for my course (BSc Mathematics, second-year module in statistical inference and modelling, unpublished) has a question,
Let $(X,Y)^T$ be a bivariate random variable with joint ...
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Sample uniformly from unit square conditioned on sum and product
Consider the following conditional distributions:
\begin{align}
X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X + Y = a & a \in [0, 2] \\
X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X ...
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Determining if a die is fair or not by rolling six times and observing only the sum
Suppose you have two bags and each has 6 dice. In one bag all the die are fair. In the other each die has a bias - die “1” has a probability of returning 1 of $\frac{1}{6} +\epsilon$ and $\frac{1}{6} -...
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Can autocorrelation impact joint probability?
This is probably a pretty stupid question... but assume I have two random variables, X1 and X2. I want to look at their dependency (joint probability) at some quantile q. Theoretically, in the ...
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Variational Inference Mean-Field Gaussian
I am new to variational inference and got very confused about some basic ideas. We want to use the mean-field gaussian family to approximate a complicated high-dimensional distribution. I want to ...
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Expectation of transition in a Markov process
I have the following transition matrix for a Discrete Markov process with 3 states, say states A, B and C:
\begin{bmatrix}0.985992 & 0.0134092 & 0.000599272\\
0.0265225 & ...
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How do you get P(A|C) from P(A, B|C)?
Is there a way to calculate P(A|C) from P(A, B|C)? I'm guessing that the following will only work when A and B are conditionally independent: P(A, B|C) = P(A|C)P(B|C).
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Joint gaussian conditional on its sum greater than a value
Let us consider $X\sim\mathcal{N}(\mu,\Sigma)$ being a $d$-dimensional multivariate Gaussian random variable. I know that it is possible to calculate the distribution of $X|S=s$, where $S$ is the sum ...
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How to estimate joint probability or conditional probability using marginal ones?
I have 2 datasets:
The 1st one gives us the probability that $m$ events occur on $n$ observations ($m$ columns for $n$ rows)
The 2nd one tells us if the event occurred (1 if occurred, 0 else ; always ...
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Conditional exponential family implies joint exponential family
Suppose $p(X_j | X_1,\ldots,X_{j-1})$ comes from some known exponential family for every $j=1,\ldots,k$. Does it follow that the joint distribution comes from some (possibly different) exponential ...
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How to quantify the asymmetry of a probabilistic dependency?
Let $X$ and $Y$ be dependent random variables. Is there a typical way to quantify how much more/less knowing $X$ tells you about $Y$ than knowing $Y$ tells you about $X$?
For example, if $Y=X^2$, then ...
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