this
Smoother
The smoother determines the state estimates x at time indices k = 1 to k = nk that minimize the following objective:
	           T  -1
	+ (x  - xi)  Pi (x  - xi)
	    1             1

	   nk 
	  ----              T  -1
	+ >    (z  - h (x ))  R  (z  - h (x )) 
	  ----   k    k  k     k   k    k  k
	  k = 1

	  nk-1 
	  ----              T   -1
	+ >    (x   - g (x ))  Q  (x    - g (x )) 
	  ----   k+1   k  k     k   k+1    k  k
	  k = 1
where
Notation Meaning
nk total number of time points
k time index
xi initial state estimate
Pi initial covariance
x State vector
z Measurement vector
h Measurement function
g Transition function
R Measurement variance
Q Transition variance
Filter
The filter does not use data that comes after a time index for estimating the state value at that time index. To be specific, it determines the state estimate x at time index k = N by minimizing the following objective as a function of the state value at time indices k = 1 to k = N:
	           T  -1
	+ (x  - xi)  Pi (x  - xi)
	    1             1

	   N 
	  ----              T  -1
	+ >    (z  - h (x ))  R  (z  - h (x )) 
	  ----   k    k  k     k   k    k  k
	  k = 1

	   N-1 
	  ----              T   -1
	+ >    (x   - g (x ))  Q  (x    - g (x )) 
	  ----   k+1   k  k     k   k+1    k  k
	  k = 1