Setup:

The height or vertical coordinate) of an object in free fall is described by an equation of the form

x(t)=?₀+?₁t+?₂t²,

where?₀, ?₁, and ?₂ are some parameters, andtstands for time.

At certain timesY₁,...,Y_n, we make noisy observationst₁,...,t_n, respectively, of the height of the object. Based on these observations, we would like to estimate the object's vertical trajectory.

We consider the special case where there is only one unknown parameter. We assume that?₀ (the height of the object at time zero) is a known constant. We also assume that?₂ (which is related to the acceleration of the object) is known. We view?1as the realized value of a continuous random variable?₁.

The observed height at timet_IisY₁=?₀+?₁t₁ +?₂t_i²+W_i,

i=1,...,n,

whereW_i models the observation noise.

We assume that?₁?N(0,1),

W₁,...W_n?N(0,?²), and that all these random variables are independent.

In this case, finding the MAP estimate of?₁involves the minimization of

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