# Questions tagged [marginal-distribution]

The marginal distribution refers to the probability distribution of a subset of variables contained in a joint distribution.

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### Marginal probability expressed as posterior expectation [closed]

I'm reading Probabilistic Machine Learning: Advanced Topics by Kevin Murphy and fail to see how, on page 339, plugging $g(\boldsymbol{\theta}) = p(\theta_1=\theta_1^*\vert\boldsymbol{\theta}_{2:D})$ ...
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### MCMC seems very sensible to the evidence

currently starting to study bayesian ML, and specifically MCMC, in order to compute the posterior: $$P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}$$ Now, I see how the acceptance ratio makes sense ...
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### Marginalising and taking the log of the likelihood

Say you have a likelihood function $L(X|\theta_1,\theta_2)$ and you marginalise out $\theta_1$ and then take the log of the marginalised likelihood function. Would this be the same as taking the log ...
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### Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
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### Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)

In the Core Statistics by Simon Wood it says: "If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
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### Copula to ensure one team wins and the other loses (Bernoulli margins)

Team $1$ has a historical win percentage of $p_1$. Team $2$ has a historical win percentage of $p_2$. The upcoming game features team $1$ against team $2$ and cannot end in a tie (one team wins, and ...
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### Incorrect reasoning on sampling from marginals

Suppose we want to sample $X$ from a density $p(x) = e^{f(x)}$. My thought was that maybe we can introduce an auxiliary variable $Y$ such that $p(y|x) \propto e^{-\|(x,y)\|^2/2 - f(x)}$. Then the ...
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### Conditional distribution $f(x|y)$ if $X$ and $Y$ are independent

Suppose we have two randome variables $X$ and $Y$, with joint distribution $f(x,y)$. $X$ and $Y$ are independent if and only if the marginal distribution of $X$ is the same as the conditional ...
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### Doing empirical Bayes with improper prior - marginals that do not exist?

I am considering a Bayesian linear model for which the prior is not proper. The model is as usual $y = X \theta + w$ where $w \sim N(0, \sigma^2)$, and $\theta, \sigma^2$ are unknown. The distribution ...
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### Existence of Distribution with Given Multivariate Marginals

Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$. Fix $k<n$. ...
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### Regression models with uncertainty as T-Distribution [duplicate]

Are there any distributions which represent uncertainty about $\hat y$ as a $t$ distribution. In other words: $p(y\mid x, ~\text{model}) \sim \mathbf T(\hat y, \text{df}, \sigma)$ where $\bf T$ is a ...
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### Gaussian as Markov kernels

Let the one-step transition dynamics of a system be given as \begin{equation*} \mathbf{P}[X_{t+1} | X_t] = N(f_\theta(X_t), I) \end{equation*} that is the Normal with mean a function of $X_t$ ...
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### In the case of the beta-binomial model, how to compute the marginal likelihood?

I have a question about Bayesian statistics. In a book called PML, I came across the following description of how to calculate the marginal likelihood of a beta-binomial distribution. In the case of ...
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### Completing the square and marginalizing a multivariate Gaussian [closed]

Edit: This question has been closed for being unrelated although I see similar questions posted here with the same objective, yet not with enough detailed answers or not exactly what I am looking for (...
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### Finding regression equations from joint distribution

Let $X$ and $Y$ be two random variables with joint probability density function $f(x, y) = 1$ if $− y < x < y, 0 <y< 1$ and $0$ elsewhere. Find the regression equation of $Y$ on $X$ and ...
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