Understanding Python Code

[subhabangalore at gmail.com]

Dear Group,

I have a Python code taken from Wikipedia.("http://en.wikipedia.org/wiki/Forward%E2%80%93backward_algorithm")

The code is pasted below. 

>>> states = ('Healthy', 'Fever')
>>> end_state = 'E'
>>> observations = ('normal', 'cold', 'dizzy')
>>> start_probability = {'Healthy': 0.6, 'Fever': 0.4}
>>> transition_probability = {
   'Healthy' : {'Healthy': 0.69, 'Fever': 0.3, 'E': 0.01},
   'Fever' : {'Healthy': 0.4, 'Fever': 0.59, 'E': 0.01},
   }
>>> emission_probability = {
   'Healthy' : {'normal': 0.5, 'cold': 0.4, 'dizzy': 0.1},
   'Fever' : {'normal': 0.1, 'cold': 0.3, 'dizzy': 0.6},
   }
>>> def fwd_bkw(x, states, a_0, a, e, end_st):
	
    L = len(x)
    print "$$1",L
 
    fwd = []
    f_prev = {}
    # forward part of the algorithm
    for i, x_i in enumerate(x):
	    print "$$2",i,x_i
            f_curr = {}
            for st in states:
		    if i == 0:
			    # base case for the forward part
                            prev_f_sum = a_0[st]
                            print "$$3",prev_f_sum
                    else:
			    prev_f_sum = sum(f_prev[k]*a[k][st] for k in states) ##? 
			    print "$$4",prev_f_sum
 
                    f_curr[st] = e[st][x_i] * prev_f_sum
                    print "$$5",f_curr[st]
 
            fwd.append(f_curr)
            f_prev = f_curr
            print "$$6",f_prev
 
    p_fwd = sum(f_curr[k]*a[k][end_st] for k in states)
    print "FORWARD IS:",p_fwd

 
    bkw = []
    b_prev = {}
    # backward part of the algorithm
    for i, x_i_plus in enumerate(reversed(x[1:]+(None,))):
	    print "##1:",i,x_i_plus
            b_curr = {}
            for st in states:
		    if i == 0:
			    # base case for backward part
                            b_curr[st] = a[st][end_st]
                            print "##2:",b_curr[st]
                    else:
			    b_curr[st] = sum(a[st][l]*e[l][x_i_plus]*b_prev[l] for l in states) ##?
			    print "##3:",b_curr
            bkw.insert(0,b_curr)
            b_prev = b_curr
            print "##4:",b_prev
 
    p_bkw = sum(a_0[l] * e[l][x[0]] * b_curr[l] for l in states)
    print "BACKWARD IS:",p_bkw
 
    # merging the two parts
    posterior = []
    for i in range(L):
        posterior.append({st: fwd[i][st]*bkw[i][st]/p_fwd for st in states})
 
    assert p_fwd == p_bkw
    return fwd, bkw, posterior

>>> def example():
    return fwd_bkw(observations,
                   states,
                   start_probability,
                   transition_probability,
                   emission_probability,
                   end_state)

>>> for line in example():
    print(' '.join(map(str, line)))

    
$$1 3
$$2 0 normal
$$3 0.6
$$5 0.3
$$3 0.4
$$5 0.04
$$6 {'Healthy': 0.3, 'Fever': 0.04000000000000001}
$$2 1 cold
$$4 0.223
$$5 0.0892
$$4 0.1136
$$5 0.03408
$$6 {'Healthy': 0.0892, 'Fever': 0.03408}
$$2 2 dizzy
$$4 0.07518
$$5 0.007518
$$4 0.0468672
$$5 0.02812032
$$6 {'Healthy': 0.007518, 'Fever': 0.028120319999999997}
FORWARD IS: 0.0003563832
##1: 0 None
##2: 0.01
##2: 0.01
##4: {'Healthy': 0.01, 'Fever': 0.01}
##1: 1 dizzy
##3: {'Healthy': 0.00249}
##3: {'Healthy': 0.00249, 'Fever': 0.00394}
##4: {'Healthy': 0.00249, 'Fever': 0.00394}
##1: 2 cold
##3: {'Healthy': 0.0010418399999999998}
##3: {'Healthy': 0.0010418399999999998, 'Fever': 0.00109578}
##4: {'Healthy': 0.0010418399999999998, 'Fever': 0.00109578}
BACKWARD IS: 0.0003563832
{'Healthy': 0.3, 'Fever': 0.04000000000000001} {'Healthy': 0.0892, 'Fever': 0.03408} {'Healthy': 0.007518, 'Fever': 0.028120319999999997}
{'Healthy': 0.0010418399999999998, 'Fever': 0.00109578} {'Healthy': 0.00249, 'Fever': 0.00394} {'Healthy': 0.01, 'Fever': 0.01}
{'Healthy': 0.8770110375573259, 'Fever': 0.1229889624426741} {'Healthy': 0.623228030950954, 'Fever': 0.3767719690490461} {'Healthy': 0.2109527048413057, 'Fever': 0.7890472951586943}
>>> 

As I was trying to understand it. [It is a Machine Learning topic, which has no connection with the question here, 
I am just trying to put the Python confusion.]

Generally I understood the code and to understand it in a better way,
I had put one print after the places I am having questions.

But two question still remains. I have put the places of question with "##?" mark.

The questions are,
i) prev_f_sum = sum(f_prev[k]*a[k][st] for k in states)
here f_prev is called, 
f_prev is assigned to  f_curr ["f_prev = f_curr"]
f_curr[st]  is again being calculated as, ["f_curr[st] = e[st][x_i] * prev_f_sum"] which again calls "prev_f_sum"

I am slightly confused which one would be first calculated and how to proceed next?

ii) The similar aspect happens again,

b_curr[st] = sum(a[st][l]*e[l][x_i_plus]*b_prev[l] for l in states)
here, b_prev is used, which is defined in
b_prev = b_curr

If any one of the esteemed members may kindly guide me to understand
this code.
Apology for any indentation error, etc. 

Thanking in Advance,
Regards,
Subhabrata Banerjee.