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Variance of marginal posterior distribution

Here is a counter-example. Staying in your example of the standard gaussian model and its gamma-normal conjugate prior: let's do a change of variable to $\sigma^2$ instead of $1/\sigma^2$. We still ...
Guillaume Dehaene's user avatar
1 vote

Variance of marginal posterior distribution

While this is probably true for most relevant examples an easy counterexample exist in that $\theta$ and $\phi$ can be independent in both prior and likelihood. I have constructed a realistic example ...
Lukas Lohse's user avatar
  • 2,862
4 votes
Accepted

Can I include a variable related to the outcome variable into statistical analysis?

The Problem My interpretation of the problem is that you want to know how the difference in the number of contacts depends on the demographic variables you listed. What you have tried now is to take ...
Frans Rodenburg's user avatar
0 votes

Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances?

If the $X_i$ are positively correlated, then the variance of the empirical mean is going to be bigger than in the independent case. If the $X_i$ are negatively correlated, then the variance of the ...
Guillaume Dehaene's user avatar
1 vote
Accepted

Why can the standard error of the weighted mean be smaller than the standard errors of the individual measurements?

The short answer, as @LulY pointed out in the comments, is that you now have a larger combined sample size. Take the standard error of the mean for example: $$\mathrm{SE} = \frac{s}{\sqrt{n}}$$ Larger ...
Frans Rodenburg's user avatar
1 vote

Relation between pairwise distance sum and sum of distance to mean (gap statistic)

A simpler way, without losing generality, is to take the center of gravity as the origin of the space (even if it means doing a translation of the cluster).: $$ \mu = \frac{1}{n}\sum\limits _{j =1}^{n}...
glegoux's user avatar
  • 111
0 votes

Under what conditions are there pairwise monotonic relationships between mean, variance, and (positive) skewness of a lower-bounded distribution?

It turns out there is a lower bound on the skewness of any strictly positive data set having given mean and sd: $$ g_1 > \sigma/\mu - \mu/\sigma. $$ This doesn't seem entirely consistent with your ...
David C. Norris's user avatar
1 vote

Didactic example of mean-variance dependency in linear models

I like the idea of showing the consequences of not respecting the mean-variance relationship. However, I would say that there are more issues than just that if you attempt to model counts with an ...
Frans Rodenburg's user avatar
10 votes
Accepted

What does it mean for observations to be uncorrelated and have constant variance?

These are assumptions made for certain models to ensure certain properties, like valid test statistics. There's a great overview here. The key word here is assumption. These need not hold up in real ...
Frans Rodenburg's user avatar
7 votes

What does it mean for observations to be uncorrelated and have constant variance?

$y_i$'s are not just real numbers. They are random variable. Specifically, the simplest linear model assumes $$y_i = x_i^T\beta+\varepsilon_i,\quad \varepsilon_i\overset{iid}{\sim}\mathcal{N}(0,\sigma^...
Voyager's user avatar
  • 305
10 votes

What does it mean for observations to be uncorrelated and have constant variance?

Random variables VS observations. Strictly speaking, there are random variables (which take values in $\mathbb{R}$) and realizations of these random variables (which are elements of $\mathbb{R}$). ...
Idontgetit's user avatar

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