We have completed a high-contrast direct imaging survey for giant planets around 57 debris disk stars as part of the Gemini NICI Planet-Finding Campaign. We achieved median H-band contrasts of 12.4 mag at 0.5 and 14.1 mag at 1 separation. Follow-up observations of the 66 candidates with projected separation < 500 AU show that all of them are background objects. To establish statistical constraints on the underlying giant planet population based on our imaging data, we have developed a new Bayesian formalism that incorporates (1) non-detections, (2) single-epoch candidates, (3) astrometric and (4) photometric information, and (5) the possibility of multiple planets per star to constrain the planet population. Our formalism allows us to include in our analysis the previously known Beta Pictoris and the HR 8799 planets. Our results show at 95% confidence that <13% of debris disk stars have a >5MJup planet beyond 80 AU, and <21% of debris disk stars have a >3MJup planet outside of 40 AU, based on hot-start evolutionary models. We model the population of directly-imaged planets as d^2N/dMda ~ m^alpha a^beta, where m is planet mass and a is orbital semi-major axis (with a maximum value of amax). We find that beta < -0.8 and/or alpha > 1.7. Likewise, we find that beta < -0.8 and/or amax < 200 AU. If we ignore the Beta Pic and HR 8799 planets (should they belong to a rare and distinct group), we find that < 20% of debris disk stars have a > 3MJup planet beyond 10 AU, and beta < -0.8 and/or alpha < -1.5. Our Bayesian constraints are not strong enough to reveal any dependence of the planet frequency on stellar host mass. Studies of transition disks have suggested that about 20% of stars are undergoing planet formation; our non-detections at large separations show that planets with orbital separation > 40 AU and planet masses > 3 MJup do not carve the central holes in these disks.
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