No such feature is provided.
The standard filter that is built into the activity feed plugin is not very intuitive.
In what way is it not intuitive? Suggestions are certainly welcomed.
This way, the filter will always be visible.
The filter bar shows exactly what you've filtered by. Is there a specific reason you want all the inputs always visible?
I want to make a filter in the module of links based on different contact types (Friend, Co Worker, Family, etc.)
Filters can be applied by GET. So you can create an HTML module and add buttons that link to your CB Activity page and force a filter to be applied. Example as follows.
index.php?option=com_comprofiler&view=pluginclass&plugin=cbactivity&action=activity&stream=1&filter=FILTER_ID_HERE
Replace FILTER_ID_HERE with the ID of the filter you want to apply. A filters ID is just its row number. So for example in your case "Connections" is just a filter ID of 1. All of the searches work this way. So for example if you want a URL to mention someone it'd just be as follows.
index.php?option=com_comprofiler&view=pluginclass&plugin=cbactivity&action=activity&stream=1&mentioned=USERNAME_HERE
Due to this you could effectively make your own search module entirely with an HTML form.