Defining multiple categories for showing latest posts

This is incorrect formatting of a PHP array:

'cat' => '4', '5', '6', '7'

Use category__in and pass an array of category IDs:

$args = array(
    'posts_per_page' => 4,
    'category__in' => array( 4, 5, 6, 7 )
);

See WP_Query for all valid parameters and how to format them correctly.