The Occasionally Dishonest Casino Narrated by: Shoko Asei Alexander Eng

Basic Probability Basic probability can be used to predict simple, isolated events, such as the likelihood a tossed coin will land on heads/tails The occasionally dishonest casino concept can be used to assess the likelihood of a certain sequence of events occurring

Loaded Dice A casino’s use of a fair die most of the time, but occasional switch to a loaded die A loaded die displays preference for landing on a particular face(s) This is difficult to detect because of the low probability of it appearing

Emissions Model of a casino where two dice are rolled ◦ One is fair with all faces being equally probable P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 ◦ The other is loaded where the number “6” accumulates 1/2 of the probability distribution of the faces of this die P(1) = P(2) = P(3) = P(4) = P(5) = 1/10 P(6) = 1/2 ◦ These are emission probabilities

State Transitions The changes of state are called transitions ◦ Exchange of fair/loaded dice ◦ Stays or changes at each point in time ◦ The probability of the outcome of a roll is different for each state The casino may switch dice before each roll ◦ A fair to a loaded die with probability of 0.05 ◦ A loaded to a fair die with probability of 0.1 ◦ These are transition probabilities

Bayes’ Rule A concept for finding the probability of related series of events It relates the probability of: ◦ Event A conditional to event B ◦ Event B conditional to event A ◦ Both probabilities not necessarily the same

Markov Chain Models A Markov chain model (MCM) is a sequence of states whose probabilities at a time interval depend only upon the value preceding it It is based on the Markov assumption, which states that “the probability of a future observation given past and present observations depends only on the present”

Hidden Markov Models A Hidden Markov model (HMM) is a statistical model with unknown parameters ◦ Transitions between different states are nondeterministic with known probabilities ◦ Extension of a Markov chain model The system has observable parameters from which hidden parameters can be explained

The Occasionally Dishonest Casino: An Example Hidden Markov model structure ◦ Emission and transition probabilities are known ◦ Sequence observed is “656” ◦ Usage of fair and/or loaded die is unknown Find path, or sequence of states, that most likely produced the observed sequence

The Occasionally Dishonest Casino: An Example P(656|FFL) = Probability of “FFL” path given “656” sequence = P(F) * P(6|F) * P(F  F) * P(5|F) * P(F  L) * P(6|L) Emission Probabilities P(6|F) = 1/6 P(5|F) = 1/6 P(6|L) = 1/2 Transition Probabilities P(F  F) = 0.95 P(F  L) = 0.05 P(F) = P(L) = Probability of starting with fair/loaded die 1 st toss2 nd toss3 rd toss

The Occasionally Dishonest Casino: An Example PathSequence Calculation 1FFF P(F)*P(6|F)*P(FF)*P(5|F)*P(FF)*P(6|F) 2LFF P(L)*P(6|L)*P(LF)*P(5|F)*P(FF)*P(6|F) 3FLF P(F)*P(6|F)*P(FL)*P(5|L)*P(LF)*P(6|F) 4FFL P(F)*P(6|F)*P(FF)*P(5|F)*P(FL)*P(6|L) 5LLF P(L)*P(6|L)*P(LL)*P(5|L)*P(LF)*P(6|F) 6LFL P(L)*P(6|L)*P(LF)*P(5|F)*P(FL)*P(6|L) 7FLL P(F)*P(6|F)*P(FL)*P(5|L)*P(LL)*P(6|L) 8LLL P(L)*P(6|L)*P(LL)*P(5|L)*P(LL)*P(6|L)

The Occasionally Dishonest Casino: An Example PathSequence Calculation Probability 1FFF(1/2)*(1/6)*(0.95)*(1/6)*(0.95)*(1/6) LFF(1/2)*(1/2)*(0.10)*(1/6)*(0.95)*(1/6) FLF(1/2)*(1/6)*(0.05)*(1/10)*(0.10)*(1/6) FFL(1/2)*(1/2)*(0.95)*(1/6)*(0.05)*(1/2) LLF(1/2)*(1/2)*(0.90)*(1/10)*(0.10)*(1/6) LFL(1/2)*(1/2)*(0.10)*(1/6)*(0.05)*(1/2) FLL(1/2)*(1/6)*(0.05)*(1/10)*(0.90)*(1/2) LLL(1/2)*(1/2)*(0.90)*(1/10)*(0.90)*(1/2)

The Occasionally Dishonest Casino: An Example Most likely path comprises greatest portion of the probability distribution Three consecutive tosses of a loaded die most likely produced the sequence “656” 8LLL(1/2)*(1/2)*(0.90)*(1/10)*(0.90)*(1/2)

A Final Note The occasionally dishonest casino concept is applicable to many systems Commonly used in bioinformatics to model DNA or protein sequences ◦ Consider a twenty-sided die with a different amino acid representing each face…

Snake Eyes!