# Bayesian updating logNormal distribution

I have a question regarding statistical updating. Basically I have a probability density function of a random variable X and, at each time step, I obtain a new sample $x_i$ belonging to this distribution. I know that X ~ logNormal($\mu,\sigma$). I am looking for a way to update this probability density function as a new sample $x_i$ is available a each time step. I am considering to use Bayesian analysis, but still not sure how to set-up the solution. I thank you all in advance for your hints!!! I am new in the platform and I really appreciate to start being part of it.

• The likelihood function (logNormal) can be updated each time you obtain a new x. To do that, it suffices to estimate mu and sigma based on the most up to date sample. No? Oct 6 '16 at 9:42
• Thanks @ocram. You mean compute a new $\mu$ and $\sigma$ as a new sample $x_i$ is available? Let's say I can define prior for the LogNormal distribution. I want to update the posterior when $x_i$ is available. Does it make sense?
– Lcol
Oct 6 '16 at 10:11
• Note that you define a prior distribution on the parameters. You may find it easier to work on the log scale and take logs of your observations as they arrive reducing it to a problem of updating the distribution on parameters for a normal instead Oct 6 '16 at 10:13
• Not clear for me what you want to update. The likelihood ? The prior? In the latter case, your previous posterior becomes your current prior. The prior combined with the new data point gives the updated posterior. Which in turns becomes your new prior. Oct 6 '16 at 10:18
• @ocram yes I that's exaclty what I meant. Update the old distribution (prior) by a more updated one (posterior). So, as far as I understand from you I can proceed as follows: I get new data sample $\log(x_i)$ and compute a new mean value $\mu$. This will update my prior knowledge on $\mu$ ~ Normal?
– Lcol
Oct 6 '16 at 10:32

There is no difference between observing a normal sample $(X_1,...,X_n)$ and a log-normal sample $(Y_1,....,Y_n)$ as one can be turned into the other by a $\log$ or $\exp$ transform. You may hence use the ultra-classic Gaussian model framework (see e.g. Chapter 2 of our book!) for log-normal samples.