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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27 views

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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1 answer
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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 ...
0 votes
1 answer
222 views

How do you calculate the probability of seeing the same image in a set of 5 when the set is drawn randomly?

Given a webpage that displays 5 images drawn at random And given a pool of 70 images And assuming each image has an equally likely chance of being drawn And assuming if an image is drawn for 1 of the ...
2 votes
1 answer
674 views

Conditional Probability on Disease

A man living in a country where only 1 out of 1000 people has the virus A. There is a test available that gives a positive result 5% of the time when the patient does not have virus A and a negative ...
-1 votes
1 answer
52 views

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 ...
1 vote
1 answer
727 views

In this Bayesian network, where does this posterior probability come from?

I'm reading Building Intelligent Interactive Tutors (Woolf, 2009) on student models for ITSs. On page 261, the author presents an example for a simple Bayesian network ($S \rightarrow E$), where $S$ ...
1 vote
0 answers
48 views

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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15 views

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 $t$, ...
1 vote
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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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1 answer
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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 ...
64 votes
3 answers
40k views

A generalization of the Law of Iterated Expectations

I recently came across this identity: $$E \left[ E \left(Y|X,Z \right) |X \right] =E \left[Y | X \right]$$ I am of course familiar with the simpler version of that rule, namely that $E \left[ E \...
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0 answers
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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 ...
2 votes
1 answer
499 views

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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2 answers
269 views

Derivation of the formula for the probability of a class, given conditionally independent attributes

The following is a formula that finds the posterior probability of a class (i.e. yes or no) given four conditionally independent attributes: $$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|...
2 votes
2 answers
238 views

Bayes theorem question

A car accident occurs and of the 5 witnesses, 4 of them saw a green car, and 1 of them saw a yellow car. In the universe, 85% of cars are yellow and 15% of cars are green. Witnesses report ...
0 votes
1 answer
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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 :)
1 vote
1 answer
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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) &...
4 votes
3 answers
256 views

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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0 answers
18 views

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 ...
40 votes
15 answers
10k views

What is the intuition behind the formula for conditional probability?

The formula for the conditional probability of $\text{A}$ happening given that $\text{B}$ has happened is:$$ P\left(\text{A}~\middle|~\text{B}\right)=\frac{P\left(\text{A} \cap \text{B}\right)}{P\left(...
0 votes
1 answer
47 views

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, ...
2 votes
2 answers
69 views

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 ...
1 vote
0 answers
22 views

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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0 answers
12 views

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{...
1 vote
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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 ...
11 votes
2 answers
1k views

Conditional expectation of a truncated RV derivation, gumbel distribution (logistic difference)

I have two random variables which are independent and identically distributed, i.e. $\epsilon_{1}, \epsilon_{0} \overset{\text{iid}}{\sim} \text{Gumbel}(\mu,\beta)$: $$F(\epsilon) = \exp(-\exp(-\frac{...
1 vote
1 answer
40 views

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 ...
0 votes
1 answer
16 views

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 ...
1 vote
2 answers
87 views

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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0 answers
168 views

Multinomial conditional logistic regression

I have a dependent variable that comes with 3 categories A, B, and C, and I want to fit a multinomial logistic regression for an exposure variable, along with some confounding factors. In order to ...
1 vote
0 answers
59 views

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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0 answers
23 views

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 ...
1 vote
1 answer
21 views

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 ...
6 votes
1 answer
2k views

Causal Markov condition simple explanation

I am trying to explain in simple words the Causal Markov condition to establish probabilistic causation. The definition from Hausman and Woodward (1999) is the following: Let G be a causal graph ...
8 votes
2 answers
364 views

Sum of sample given a priori knowledge of its maximum

Given a sample of discrete random variables $X_1, X_2, \ldots, X_n \sim F$, I am looking to calculate the distribution given by the probability mass function: $$P\left(\sum_{i=1}^n X_i = x~\middle|~\...
0 votes
0 answers
11 views

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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0 answers
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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 \...
1 vote
1 answer
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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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0 answers
31 views

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 ...
1 vote
1 answer
69 views

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 ...
1 vote
1 answer
94 views

Statistical Models that "Exploit" Distributional Knowledge of the Predictor Variables

I am trying to see if there any Statistical Models that (better) "Exploit" Distributional Knowledge of the Predictor Variables. For example, I feel that is a common misconception (e.g Where ...
0 votes
0 answers
33 views

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 ...
20 votes
3 answers
70k views

How can I calculate the conditional probability of several events?

Could you inform me please, how can I calculate conditioned probability of several events? for example: P (A | B, C, D) - ? I know, that: P (A | B) = P (A $\cap$ B) / P (B) But, unfortunately, I ...
0 votes
1 answer
68 views

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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0 answers
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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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11 views

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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1 answer
265 views

Is it possible to calculate conditional PD / unconditional PD from Hazard Rate?

I'm just wondering that can I convert hazard rate to probability of default? Suppose I have the lifetime table data as per below: Time Total Default Non-Default At Risk 0 - - - 356,335 1 5,587 1,...

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