There are a number of possible approaches to determining if you need the posts to support the beam above.
1. The radical approach. You knock the posts out, and if the beam falls down, you know you needed the posts. Not recommended.
2. The mathematical approach. You determine the size, span, species, and grade of the beam above. You determine the design load on the beam. You calculate the maximum bending moment and deflection of the beam if the posts are removed, and you determine if the beam is strong enough and stiff enough to hold up the roof without posts. Requires some knowledge of structural mechanics, and requires you to remove the facing so you can measure the beam.
3. A reasonable alternative. Install a temporary screw type jack to hold up the beam. Remove one of the posts, allowing the load to transfer to the jack. Slowly lower the jack. If the beam follows the jack down, the post is clearly load bearing. If the beam does not move as the jack is lowered, you will be able to see space between the top of the jack and the beam, and you will know that the post is not load bearing.
This is an internet chat forum, and if you leave your question up long enough, you will get a wide range of responses, ranging from "what are you waiting for, that post is clearly non-structural" to "that post is clearly carrying load". Since no one on this forum has visited your site, and you have not met anyone on this forum, you are likely not going to be able to figure out whether that post is really load bearing from a range of internet chat forum opinions. My recommendation is to ASSUME that the post is load bearing, although clearly poorly attached. You may want to use the temporary jack technique to be sure, unless you are confident in your structural analysis. Unless you know FOR CERTAIN that the post is not load bearing, I would install new posts using proper connection brackets top and bottom. The post looks architecturally appropriate, just badly installed.