PostgreSQL: improve the performance when counting the distinct - postgresql

Question

I am currently working on improving the performance of our db. And I need some help from you.
I have a table and its index like this
CREATE TABLE public.ar
(
id integer NOT NULL DEFAULT nextval('id_seq'::regclass),
user_id integer NOT NULL,
duration double precision,
is_idle boolean NOT NULL,
activity_id integer NOT NULL,
device_id integer NOT NULL,
calendar_id integer,
on_taskness integer,
week_id integer,
some_other_column_below,
CONSTRAINT id_ PRIMARY KEY (id),
CONSTRAINT a_unique_key UNIQUE (user_id, device_id, start_time_local, start_time_utc, end_time_local, end_time_utc)
)
CREATE INDEX ar_idx
ON public.ar USING btree
(week_id, calendar_id, user_id, activity_id, duration, on_taskness, is_idle)
TABLESPACE pg_default;
Then I am trying to run a query like this
EXPLAIN ANALYZE
SELECT COUNT(*)
FROM (
SELECT ar.user_id
FROM ar
WHERE ar.user_id = ANY(array[some_data]) -- data size is 352
AND ROUND(ar.duration) >0 AND ar.is_idle = false
AND ar.week_id = ANY(ARRAY[some_data]) -- data size is 37
AND ar.calendar_id = ANY(array[some_data]) -- data size is 16716
GROUP by ar.user_id
) tmp;
And below is the explain result
Aggregate (cost=31389954.72..31389954.73 rows=1 width=8) (actual time=252020.695..252020.695 rows=1 loops=1)
-> Group (cost=31389032.69..31389922.37 rows=2588 width=4) (actual time=251089.270..252020.659 rows=351 loops=1)
Group Key: ar.user_id
-> Sort (cost=31389032.69..31389477.53 rows=177935 width=4) (actual time=251089.268..251776.202 rows=6993358 loops=1)
Sort Key: ar.user_id
Sort Method: external merge Disk: 95672kB
-> Bitmap Heap Scan on ar (cost=609015.18..31371079.88 rows=177935 width=4) (actual time=1670.413..248939.440 rows=6993358 loops=1)
Recheck Cond: ((week_id = ANY ('{some_data}'::integer[])) AND (user_id = ANY ('{some_data}'::integer[])))
Rows Removed by Index Recheck: 2081028
Filter: ((NOT is_idle) AND (round(duration) > '0'::double precision) AND (calendar_id = ANY ('{some_data}'::integer[])))
Rows Removed by Filter: 534017
Heap Blocks: exact=29551 lossy=313127
-> BitmapAnd (cost=609015.18..609015.18 rows=1357521 width=0) (actual time=1666.334..1666.334 rows=0 loops=1)
-> Bitmap Index Scan on test_index_only_scan_idx (cost=0.00..272396.77 rows=6970353 width=0) (actual time=614.366..614.366 rows=7269830 loops=1)
Index Cond: ((week_id = ANY ('{some_data}'::integer[])) AND (is_idle = false))
-> Bitmap Index Scan on unique_key (cost=0.00..336529.20 rows=9948573 width=0) (actual time=1041.999..1041.999 rows=14959355 loops=1)
Index Cond: (user_id = ANY ('{some_data}'::integer[]))
Planning time: 25.563 ms
Execution time: 252029.237 ms
I used distinct as well, and the result is the same.
So my questions are below.
The ar_idx contains user_id, but when searching for rows, why does it use the unique_key instead of the index I created?
I thought group by will not do the sort(that is why I did not choose distinct), but why does the sort happen in the explain analyze?
The running time is pretty long(more than 4 minutes). How do I make it faster? Is the index wrong? Or anything else I can do.
Be advised, the ar table contains 51585203 rows.
Any help will be appreciated. Thx.
---------------------------update--------------------------
After I created this index, everything goes really fast now. I don't understand why, anyone can explain this to me?
CREATE INDEX ar_1_idx
ON public.ar USING btree
(calendar_id, user_id)
TABLESPACE pg_default;
And I changed the old index to
CREATE INDEX ar_idx
ON public.ar USING btree
(week_id, calendar, user_id, activity_id, duration, on_taskness, start_time_local, end_time_local) WHERE is_idle IS FALSE
TABLESPACE pg_default;
-----updated analyze results-----------
Aggregate (cost=31216435.97..31216435.98 rows=1 width=8) (actual time=13206.941..13206.941 rows=1 loops=1)
Buffers: shared hit=25940518 read=430315, temp read=31079 written=31079
-> Group (cost=31215436.80..31216403.88 rows=2567 width=4) (actual time=12239.336..13206.894 rows=351 loops=1)
Group Key: ar.user_id
Buffers: shared hit=25940518 read=430315, temp read=31079 written=31079
-> Sort (cost=31215436.80..31215920.34 rows=193417 width=4) (actual time=12239.334..12932.801 rows=6993358 loops=1)
Sort Key: ar.user_id
Sort Method: external merge Disk: 95664kB
Buffers: shared hit=25940518 read=430315, temp read=31079 written=31079
-> Index Scan using ar_1_idx on activity_report ar (cost=0.56..31195807.48 rows=193417 width=4) (actual time=0.275..10387.051 rows=6993358 loops=1)
Index Cond: ((calendar_id = ANY ('{some_data}'::integer[])) AND (user_id = ANY ('{some_data}'::integer[])))
Filter: ((NOT is_idle) AND (round(duration) > '0'::double precision) AND (week_id = ANY ('{some_data}'::integer[])))
Rows Removed by Filter: 590705
Buffers: shared hit=25940518 read=430315
Planning time: 25.577 ms
Execution time: 13217.611 ms