Against a bridge camera or high-end compact, the arguments are straightforward: a larger sensor gives better image quality, and interchangeable lenses give more flexibility.

For dSLR vs. one of the mirrorless alternatives, it's more complicated, but there's two broad reasons here as well: technology maturity and system maturity.

Under technical maturity:

1. Contrast-detect autofocus as used in most compact and mirrorless cameras is just not as fast as phase-detect. This will change as processor power and algorithms improve, but phase-detect has an intrinsic advantage because the AF sensors tell the system "go this direction to get the correct focus", whereas contrast detect has to seek back and forth to see which way is better. Perhaps in the future, a hybrid solution will be preferred, or perhaps contrast-detect (which already has an accuracy advantage) will become fast enough that it just won't matter.
2. Electronic viewfinders have come a long way, but they still don't beat optics. This is an area that will definitely change: I'm very comfortable saying that in ten years, electronics will beat optics in every way. Five years, maybe. Right now, though, advantage to through-the-lens optics.

System maturity:

1. Fewer choices for lenses or for TTL-automated flashes. The systems are relatively new and being built up from scratch. You can use lens adapters in many cases, but usually with reduced functionality (and without the advantages of smaller size). If a compact mirrorless system has what you want (and they do cover the basics), this might not hold you back, but a more-established system — even including relatively small ones like 4/3rds (in the non-micro form) — offers more flexibility.
2. One can't help but notice that the two 800-pound gorillas, Canon and Nikon, haven't thrown their hats in the ring. (Gorillas wear hats, right?) That's not a problem, per se, but it might be a reason to keep watching.

Update: Rumor has Nikon announcing a mirrorless interchangeable lens camera Very Soon Now. This looks to be an interesting offering, because, like the Pentax Q, it uses a tiny compact-camera-sized sensor. This basically fits under all of the caveats above (except the gorilla one, of course), with the addition of the sensor size concerns. Time will tell on quality, but it looks like these will basically fall under "expensive toy designed for the Japanese market" rather than competition to dSLRs.