EDIT 2:
Yesterday, an article was posted by the Pokemon Go GamePress team that had a reasonable estimate on how the medal catch rate bonus works. The working hypothesis is that the +1, +2, and +3 bonuses act as multipliers on the catch chance for an encounter, meaning that the Final Catch Rate (see above) is multiplied by 1.10 for a bronze medal, 1.20 for a silver medal, and 1.30 for a gold medal.
We also have statistical evidence suggesting that the bonus provided by gold medals is 1.3; applied in a similar matter as other multipliers in the GUCT formula.
It is worth noting that this is based on 1147 data points, but 1053 of them are for the gold medal bonus. The results show that the estimated value for the +3 bonus is 1.339 +/- 0.031
, but based on how the catch mechanics work, it seems to be a safe bet that the real value is strictly 1.30.
Bear in mind the following disclaimer:
This is clearly just a reasonable theory, not a statistical conclusion, but we consider it likely enough that we'll go forth with those multipliers as we continue our research in this area.