Questions tagged [posterior]

Refers to the probability distribution of parameters conditioned on data in Bayesian statistics.

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Notation for conditional density

Are $p(\mu \mid \sigma)$ and $p(\mu ; \sigma)$ equivalent? I've seen the notation $p(b_i \mid T_i, \delta_i, y_i ; \theta)$ used to represent the posterior distribution for $b_i$. I am assuming that ...
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Shrinkage effects in a hierarchical model

I am working on the chimpanzees dataset from Richard McElreath's text, "Statistical Rethinking", edition 2. I have built 2 simple models, one a fixed effects model and the other a hierarchical model. ...
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What distributions are conjugate to themselves, besides the normal?

I know the normal distribution is conjugate to itself; are there others? Is there some sort of intuition behind why a given distribution would be conjugate to itself?
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Estimate random effects for a new individual with a linear mixed effects model

Consider repeated observations $\mathcal{Y} = (y_{i,j})_{i,j}$ obtained for $p$ individuals ($1 \leq i \leq p$), at different time points $t_{i,j}$ $(1 \leq j \leq n_i$). The "random slope and ...
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What do these equations on Bayesian regression (MAP) from Chapter 3.3 in PRML by Bishop mean?

This was taken from Ch 3.3 on Bayesian Linear Regression from Pattern Recognition in Machine Learning by Bishop. Apparently the posterior can be described by eq 3.49. Eq 3.48 represents the prior ...
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Explanation of Equation 5.3 from Gaussian Processes for Machine Learning

I am currently reading through C. E. Rasmussen & C. K. I. Williams' Gaussian Processes for Machine Learning and was going through chapter 5. I could not exactly understand the derivation of ...
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Range of integration for joint and conditional densities

Did I mess up the range of integration in my solution to the following problem ? Consider an experiment for which, conditioned on $\theta,$ the density of $X$ is \begin{align*} f_{\theta}(x) = \...
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Bayesian Linear Regression, trouble with posterior. Variance equal identity

I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $\pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...
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What is the conceptual difference between posterior and likelihood? [duplicate]

I have trouble discerning conceptually between these two notions. I am aware of their formal relations, proprieties and what not, but I just can't wrap my head around what they "mean", if that even ...
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Is there a derivation for the Posterior Predictive Distribution?

I came across this term in the deep learning book: $p(x_{m+1}|x_1 ... x_m) = \int p(x_{m+1}|\theta)p(\theta|x_1 ... x_m)d\theta$ After some research I find that this term is the definition of the ...
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What is the posterior mean of $\mu$ given a randomly stopped i.i.d. observations from a Normal

Let's imagine I have a machine giving me an independent random number from a normal distribution $N(\mu,1)$ whenever I push a button. I have a stopping rule to decide how many samples to collect - I ...
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Residual Analysis

I have just reproduced an example regarding a regression model for fibre strength data. The data consisted of tensile strengths of silicon carbide fibre at four different lengths. From the data, a ...
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meaning of posterior distribution and credible internal

In Bayesian method, we can get a posterior distribution of a parameter. Now I want to do some simulation to know if the posterior distribution is the same as the true distribution. For example, mean ...