Seeing that some of the particles created in high-energy colliders live for really short times, so much that often it is hard to observe them directly, physicists look at the expected way particles decay. How do we predict that a hypothesized particle would decay in a particular way without already observing it do so? For instance, the higgs boson was detected by its decay pattern, not by direct observation: something like, "it quacks like a Higgs, it walks like a Higgs, thus, it must be a higgs." On the other hand, Beta decay was first observed, then explained. Is there a way physicists do so? How can we be sure that the particle that decayed was the one we were looking for and not some other one decaying in a similar fashion?
When beta decay was observed , (as gamma and alpha too) there was no theory, just exceptional unexpected data. The theory was built up step by step. First defining the "strength" of the interaction for each observation, to start with the average lifetime. This clarified into the three fundamental forces .
In the beginning, when I was in graduate school, the interactions were not clear, one thought that beta decay went through the four Fermi interaction. Before the quark model turned into the standard model there were the by ways of the Regge poles to explain the behavior of strong interactions. ( I used them to interpret the data in my thesis back in 1970's).
All these theories made predictions, which were successful, but it was only with the standard model of particle physics that a predictive theoretical model stabilized for so long and with such a wide scope.
How can we be sure that the particle that decayed was the one we were looking for and not some other one decaying in a similar fashion?
As the other answer says, if the conservation laws for energy and spin and the rest of quantum numbers allow a decay, there will be a calculable probability for the decay to happen.
When a new resonance is found, as the Higgs boson at the LHC, one checks its width, decay modes, spin, etc, and this will go on with new data. If there is no contradiction then the identification is accepted.
In GUTS theories the proton can decay. The lifetime is predicted to be of order 10^32 years and there exist experiments trying to measure proton decays, at the moment setting limits from lack of observations.
Supersymmetric particles are sought at the LHC. At the moment no new, except the Higgs, resonances have been seen. If a new resonance is seen, then the game of what it is will start, checking decay modes, quantum numbers etc.
The general basic idea is "if it's not forbidden, there's a chance it will happen". You can use QFT (specifically the Standard Model) to define the rules and calculate probabilities of particular processes. When a particle is hypothesised, so are (some of) its properties, including how it interacts.
Bear in mind that we're dealing with probabilities and distributions. You can never predict the decay of any one specific particle, just like you can't predict the roll of a specific dice. Inversely, you (usually) can't say that a specific event with the right final state is the particle you're looking for. You have to build up a distribution of a large number of events and find an excess consistent with the properties of the particle.