Wordpress on Sub Directory with Nginx using Proxy Pass with No Input File error

You should be able to specify just index index.php;.

You should also change

location / { try_files $uri $uri/ /blog/index.php?$query_string; }

to

location / { try_files $uri $uri/ /blog/index.php?$args; }

and change:

fastcgi_split_path_info ^(.+\.php)(/.+)$;
fastcgi_pass unix:/var/run/php/php5.6-fpm.sock;

to:

fastcgi_split_path_info ^(/blog)(/.*)$;
fastcgi_pass php;

This is coming from the WordPress documentation as well as the NGINX documentation.