# Deriving the posterior density for a lognormal likelihood and Jeffreys's prior

The likelihood function of a lognormal distribution is:

$$f(x; \mu, \sigma) \propto \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right )$$

and Jeffreys's Prior is:

$$p(\mu,\sigma) \propto \frac{1}{\sigma^2}$$

so combining the two gives:

$$f(\mu,\sigma^2|x)= \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right ) \cdot \sigma^{-2}$$

I know that the posterior density for $$\sigma^2$$ is inverse Gamma distributed, so I have to calculate

$$f(\sigma^2|x) = \int f(\mu,\sigma^2|x)~\mathrm d\mu$$

but I have no clue where to start here.

After Glen_b's comment I give it a shot:

\begin{align}f(\mu,\sigma^2|x)&= \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right ) \cdot \sigma^{-2} \\&= \sigma^{-n-2} \prod_{i=1}^n \frac{1}{x_i} \exp \left ( - \frac{1}{2\sigma^2} \sum_{i=1}^n (\ln x_i - \mu ) \right) \end{align}

but I cannot see this going anywhere.

Another idea I got is to define $$y_i=\ln(x_i)$$, then $$y$$ is normal distributed. So

\begin{align}f(\mu,\sigma^2 |y) &= \left [ \prod_{i=1}^n \frac{1}{\sqrt{2 \pi}} \cdot \frac{1}{\sigma} \exp \left ( - \frac{1}{2 \sigma^2} (y_i - \mu)^2 \right ) \right ] \cdot \frac{1}{\sigma^2}\\& \propto \sigma^{-n-2} \cdot \exp \left ( - \frac{1}{2 \sigma^2} \sum_{i=1}^n (y_i - \bar y)^2 + n(\bar y - \mu)^2 \right ) \\& = \sigma^{-n-2} \cdot \exp \left ( - \frac{1}{2 \sigma^2} ( (n-1)s^2 + n(\bar y - \mu)^2 ) \right ) \\&= \sigma^{-n-2} \cdot \exp \left ( - \frac{1}{2 \sigma^2} ( (n-1)s^2 \right ) \exp \left (n(\bar y - \mu)^2 ) \right ) \end{align}

then integrate

$$\sigma^{-n-2} \cdot \exp \left ( - \frac{1}{2 \sigma^2} ( (n-1)s^2 \right ) \int \exp \left ( - \frac{1}{2 \sigma^2} n(\bar y - \mu)^2 ) \right ) ~\mathrm d \mu;$$

by the method you suggested, I get:

$$\int \exp \left ( - \frac{1}{2 \sigma^2} n(\bar y - \mu)^2 ) \right ) ~\mathrm d \mu = \sqrt{\frac{2\pi \sigma^2}{n}}$$

So:

$$\propto (\sigma^2)^{-(n+1)/2} \exp \left ( - \frac{1}{2 \sigma^2} ( (n-1)s^2 \right )$$

which is indeed inverse Gamma distributed.

But I am unsure if this is correct, it's also the same result as I get for a normal likelihood.

I found this in the literature (without any further explanantion):

• In your first line of mathematics (the likelihood), don't drop the $\sigma$ term in the constant. Commented Sep 5, 2014 at 10:39
• That's Sir Harold Jeffreys, so Jeffreys prior, Jeffreys' prior and Jeffreys's prior are all defensible, but Jeffrey's is wrong. He preferred the last form. Commented Sep 5, 2014 at 12:36
• Now, when you combine the two, keep those $\sigma$ terms in. Commented Sep 5, 2014 at 13:26
• The thing you found in the literature is a posterior for $\theta=(\mu,\sigma)$. Commented Sep 6, 2014 at 8:05

Note that - regarded as a function in $\mu$ - what you have is proportional to a normal density.

So step 1 is to complete the square in $\mu$ that's in the exponent, pull out the front of the integral any superfluous constants, and then multiply the term in the integral by the constant required to make it integrate to 1. Then divide out in front of the integral by the same constant (so you don't change the value of the overall expression).

Since you have a density in the integral, replace the term in the integral by 1.

You're left with a function of $\sigma$ (one that has notionally replaced $\mu$ with something akin to an estimate of it).

Now see the density for an inverse gamma here:

$$f(x; \alpha, \beta)= \frac{\beta^\alpha}{\Gamma(\alpha)}x^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right)$$

(in this case, using a shape-scale parameterization).

Assuming you have the prior correct (I haven't checked that) --

you seek a posterior density for $\sigma^2$. Note that your function from after the integration can be written in the form $c\cdot(\sigma^2)^{-\text{something}}\cdot\exp(-\text{something-else}/\sigma^2)$.

So you have an expression proportional to an inverse gamma density in $\sigma^2$. (Since it must be a density, supply the required constant needed to make it integrate to 1.)

• You didn't really need to change $\ln(x)$ to $y$. It's observed data so those are just constants. It's $\mu$ that's the variable. You completed the square. Note that there's a term in $\mu$ and one not in $\mu$. The next step from where you got to is already in my answer. Commented Sep 5, 2014 at 15:58
• I updated my post again (I kept the y for simplicity) Commented Sep 6, 2014 at 7:55
• how do I derive the result in the literature? Commented Sep 6, 2014 at 8:43
• Through pretty much the same approach as above, but you don't integrate $\mu$ out, you just pull out the terms. Commented Sep 6, 2014 at 9:04