Basically a new build-failure-analyzer.xml configuration file with default settings is written when "Save" is clicked in Jenkins global configuration (http://jenkins_ulr/configure).

I also observed that when installing build-failure-analyzer first time, then restart Jenkins, still there is no build-failure-analyzer.xml in JENKINS_HOME.
If I change a global setting and click save, the file appears.
If I add a failure cause, and and use it, it stays and is visible in build-failure-analyzer.xml.
And the failure cause is still there after restarting Jenkins.
But if I change the global configuration, the modifed file is overwritten with default settings, which means that I effectively loose all my failure causes.

My BFA global settings are in the screenshot, and I notice alse that it defaults to enable the gerrit-trigger notifications, although that plugin is not installed.

Reproduce:
Install Jenkins, install all mentioned plugins except BFA
Install BFA, restart Jenkins, Notice there is no BFA configuration file
Change a global setting and save. Notice BFA configuration file now exist
Add a failure cause - do what ever, the failure cause stays
Change a global setting and save. Notice BFA configuration file still exists, but the failure cause is gone.