Hive - over (partition by ...) with a column not in group by - hive

Question

Is it possible to do something like:
select
avg(count(distinct user_id))
over (partition by some_date) as average_users_per_day
from user_activity
group by user_type
(notably, the partition by column, some_date, is not in the group by columns)
The idea I'm going for is something like: the average users per day by user type.
I know how to do it using subqueries (see below), but I'd like to know if there is a nice way using only over (partition by ...) and group by.
Notes:
From reading this answer, my understanding (correct me if I'm wrong) is that the following query:
select
avg(count(distinct a)) over (partition by b)
from foo
group by b
can be expanded equivalently to:
select
avg(count_distinct_a)
from (
select
b,
count(distinct a) as count_distinct_a
from foo
group by b
)
group by b
And from that, I can tweak it a bit to achieve what I want:
select
avg(count_distinct_user_id) as average_users_per_day
from (
select
user_type,
count(distinct user_id) as count_distinct_user_id
from user_activity
group by user_type, some_date
)
group by user_type
(notably, the inner group by user_type, some_date differs from the outer group by user_type)
I'd like to be able to tell the partition by-group by interaction to use a "sub-group-by" for the windowing part. Please let me know if my understanding of partition by/group by is completely off.
EDIT: Some sample data and desired output.
Source table:
+---------+-----------+-----------+
| user_id | user_type | some_date |
+---------+-----------+-----------+
| 1 | a | 1 |
| 1 | a | 2 |
| 2 | a | 1 |
| 3 | a | 2 |
| 3 | a | 2 |
| 4 | b | 2 |
| 5 | b | 1 |
| 5 | b | 3 |
| 5 | b | 3 |
| 6 | c | 1 |
| 7 | c | 1 |
| 8 | c | 4 |
| 9 | c | 2 |
| 9 | c | 3 |
| 9 | c | 4 |
+---------+-----------+-----------+
Sample intermediate table (for reasoning with):
+-----------+-----------+---------------------+
| user_type | some_date | distinct_user_count |
+-----------+-----------+---------------------+
| a | 1 | 2 |
| a | 2 | 2 |
| b | 1 | 1 |
| b | 2 | 1 |
| b | 3 | 1 |
| c | 1 | 2 |
| c | 2 | 1 |
| c | 3 | 1 |
| c | 4 | 2 |
+-----------+-----------+---------------------+
SQL is: select user_type, some_date, count(distinct user_id) from user_activity group by user_type, some_date.
Desired result:
+-----------+---------------------+
| user_type | average_daily_users |
+-----------+---------------------+
| a | 2 |
| b | 1 |
| c | 1.5 |
+-----------+---------------------+