# What does $d$ mean in this notation of the “usual noninformative prior of $\mu_i$ and $\sigma_i$?”

Samiuddin, (1976) states:

or, typset with $\LaTeX$ as originally posted

We start with the usual noninformative prior distribution of $\mu_i$ and $\sigma_i (i = 1,2,\ldots, k)$

$$\pi(\mu_1, \mu_2, \ldots, \mu_k; \sigma_1, \sigma_2, \ldots, \sigma_k) \propto d\mu_1d\mu_2\ldots d\mu_k \frac{d\sigma_1d\sigma_2\ldots d\sigma_k} {\sigma_1\sigma_2\ldots \sigma_k}$$

What does this notation mean?

Samiuddin, M. 1976. Bayesian Test of Homogeneity of Variance. Journal of the American Statistical Assoc. Vol. 71, No. 354

• More than half the problem is that you have not correctly transcribed this expression! The right hand side is $d\mu_1 d\mu_2 \cdots d\mu_k \frac{d\sigma_1 d\sigma_2 \cdots d\sigma_k}{\sigma_1 \sigma_2 \cdots \sigma_k}$: the $d$'s are differentials, not $d$'s with subscripts. See formula (2.1) at jstor.org/pss/2285344 . – whuber Mar 29 '11 at 6:07
• @whuber thanks for pointing that out. The paper was pre-$\LaTeX$ and the Greek letters are just a bit smaller than the Latin ones. – David LeBauer Mar 29 '11 at 14:10
• fixed. This makes me really appreciate Donald Knuth. – David LeBauer Mar 29 '11 at 14:16
• This is a somewhat abusive notation as the lhs should be instead$$\pi(d\mu_1\cdots d\mu_kd\sigma_1\cdots d\sigma_k)$$ – Xi'an Mar 6 '16 at 14:39

This is shorthand notation for a "differential" of the mean and variance parameters. The longhand version goes:

$$p(\mu\in[\mu_1,\mu_1+d\mu_1)|I)\propto d\mu_1$$

This indicates a uniform probability with respect to $\mu$. A more familiar notation is:

$$p(\mu|I)\propto 1$$

It comes from the "proper" derivation of a PDF from a CDF.

$$lim_{dy\rightarrow 0}P(Y\in[y,y+dy))=f(y)dy$$

EDIT: I initially wrote this answer in a hasty fashion, and so had a bit of unclear notation myself. In my example, I only had a 1-dimension variable $\mu_1$, and all the above relate to a 1-dimensional random variable. I think the statistical physics literature ("maxent" people) uses this notation (but not entirely sure) - Edwin Jaynes, Larry Bretthorst, Stephen Gull, and others. I've never seen it explained in any more detail than what I have given.

And second is that $I$ stands for "prior information", not an identity matrix. This is just a good habit to express $I$ explicitly as part of your assumptions - so that you don't forget that 1) they are there, and 2) you answer depends on the prior information just as much it depends on the data.

• thanks for explaining this to me, but I am still confused - in your notation, is $\mu$ a vector and $I$ an identity matrix? – David LeBauer Mar 27 '11 at 23:37
• can you point me to a reference? I have Gelman's Bayesian Data Analysis and Carlin and Louis "Bayes and Empirical Bayes Methods" handy but can't relate their explanations to your equations 1 and 2. I understand that how to calculate the PDF from the CDF, but not how this gets you from to eqn 1 or from 1 to 2. – David LeBauer Mar 27 '11 at 23:50
• It's the multivariate analog of the usual 'noninformative' prior; a more succint/modern way to write it would be $p(\mu, \sigma)\propto \prod_{i=1}^k 1/\sigma_i$ (with $\mu$ and $\sigma$ k-vectors). Like @probabilityislogic says, the $d$'s are differentials and come from the proper definition of a pdf. I think they ought to be typeset $d\mu_i$. Pop in a one for them and you should be fine :) You're unlikely to see that notation in either of those texts, I think. – JMS Mar 28 '11 at 2:13