# Parameter estimation with differential equation?

I have an unknown function in an ODE as follows:

dx/dt=Q(t)-a*x(t),

a is a function of time and unknown. I plan to consider a as an unknown values at different time points. Then I use MCMC to estimate the unknown parameters by having measurements, x(t). My question is if there is any advantage in using smoothing spline compared to interpolation? or Gaussian process? Which one do you recommend?

• Since $a(t)$ isn't observed, it's a latent function. I don't see how interpolation or smoothing splines could be used to model it, because they must be fit to observed values. Since you're taking a Bayesian approach, you need to define a prior over latent functions. Gaussian processes could be used here. Splines could work as well (by defining a probability distribution over the parameters). Or various other distributions over time series. The choice essentially reflects your a priori beliefs about the structure of $a(t)$ (and perhaps approximations you're willing to make for tractability). – user20160 Oct 19 '20 at 20:37
• thank you for your reply. I plan to use t=[t1 t2 ... t8], a=[p1 p2 ... p8], where pis are unknown. For each pi , I have a uniform prior. At each MCMC iteration, pis are sampled and a model is fitted to the data (using GP or smoothing spline) and the ODE will be solved. By this strategy, the unknown parameters pis will be estimated. My question is for model fitting which strategy works better while I have no idea about the dynamics of a(t). Thanks – Bita Labibi Oct 19 '20 at 20:42