I am writing a program that attempts to duplicate the algorithm discussed at the beginning of this article,
http://www-stat.stanford.edu/~cgates/PERSI/papers/MCMCRev.pdf
F is a function from char to char. Assume that Pl(f) is a 'plausibility' measure of that function. The algorithm is:
Starting with a preliminary guess at the function, say f, and a then new function f* --
- Compute Pl(f).
- Change to f* by making a random transposition of the values f assigns to two symbols.
- Compute Pl(f*); if this is larger than Pl(f), accept f*.
- If not, flip a Pl(f)/Pl(f*) coin; if it comes up heads, accept f*.
- If the coin toss comes up tails, stay at f.
I am implementing this using the following code. I'm using c# but tried to make it somewhat more simplified for everyone. If there is a better forum for this please let me know.
var current_f = Initial(); // current accepted function f
var current_Pl_f = InitialPl(); // current plausibility of accepted function f
for (int i = 0; i < 10000; i++)
{
var candidate_f = Transpose(current_f); // create a candidate function
var candidate_Pl_f = ComputePl(candidate_f); // compute its plausibility
if (candidate_Pl_f > current_Pl_f) // candidate Pl has improved
{
current_f = candidate_f; // accept the candidate
current_Pl_f = candidate_Pl_f;
}
else // otherwise flip a coin
{
int flip = Flip();
if (flip == 1) // heads
{
current_f = candidate_f; // accept it anyway
current_Pl_f = candidate_Pl_f;
}
else if (flip == 0) // tails
{
// what to do here ?
}
}
}
My question is basically whether this looks like the optimal approach to implementing that algorithm. IT seems like I may be getting stuck in some local maxima / local minima despite implementing this method.
EDIT - Here is the essentially whats behind Transpose() method. I use a dictionary / hash table of type << char, char >> that the candidate function(s) use to look any given char -> char transformation. So the transpose method simply swaps two values in the dictionary that dictates the behavior of the function.
private Dictionary<char, char> Transpose(Dictionary<char, char> map, params int[] indices)
{
foreach (var index in indices)
{
char target_val = map.ElementAt(index).Value; // get the value at the index
char target_key = map.ElementAt(index).Key; // get the key at the index
int _rand = _random.Next(map.Count); // get a random key (char) to swap with
char rand_key = map.ElementAt(_rand).Key;
char source_val = map[rand_key]; // the value that currently is used by the source of the swap
map[target_key] = source_val; // make the swap
map[rand_key] = target_val;
}
return map;
}
Keep in mind that a candidate function that uses the underlying dictionary is basically just:
public char GetChar(char in, Dictionary<char, char> theMap)
{
return theMap[char];
}
And this is the function that computes Pl(f):
public decimal ComputePl(Func<char, char> candidate, string encrypted, decimal[][] _matrix)
{
decimal product = default(decimal);
for (int i = 0; i < encrypted.Length; i++)
{
int j = i + 1;
if (j >= encrypted.Length)
{
break;
}
char a = candidate(encrypted[i]);
char b = candidate(encrypted[j]);
int _a = GetIndex(_alphabet, a); // _alphabet is just a string/char[] of all avl chars
int _b = GetIndex(_alphabet, b);
decimal _freq = _matrix[_a][_b];
if (product == default(decimal))
{
product = _freq;
}
else
{
product = product * _freq;
}
}
return product;
}