The Occasionally Dishonest The Occasionally Dishonest CasinoCasino

Narrated by:Shoko AseiAlexander Eng

Basic ProbabilityBasic ProbabilityBasic 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 DiceLoaded DiceA casino’s use of a fair die most of the time,

but occasional switch to a loaded dieA loaded die displays preference for landing on

a particular face(s)This is difficult to detect because of

the low probability of it appearing

EmissionsEmissions

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 dieP(1) = P(2) = P(3) = P(4) = P(5) = 1/10P(6) = 1/2

◦ These are emission probabilities

State TransitionsState 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 stateThe 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’ RuleBayes’ 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 ModelsMarkov 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 ModelsHidden Markov ModelsA 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 The Occasionally Dishonest Casino: An ExampleCasino: 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 The Occasionally Dishonest Casino: An ExampleCasino: An Example

P(656|FFL) = Probability of “FFL” path given “656” sequence = P(F) * P(6|F) * P(FF) * P(5|F) * P(FL) * P(6|L)

Emission ProbabilitiesP(6|F) = 1/6P(5|F) = 1/6P(6|L) = 1/2

Transition ProbabilitiesP(FF) = 0.95P(FL) = 0.05

P(F) = P(L) = Probability of starting with fair/loaded die

1st toss 2nd toss 3rd toss

The Occasionally Dishonest The Occasionally Dishonest Casino: An ExampleCasino: An Example

Path Sequence Calculation

1 FFF P(F)*P(6|F)*P(FF)*P(5|F)*P(FF)*P(6|F)

2 LFF P(L)*P(6|L)*P(LF)*P(5|F)*P(FF)*P(6|F)

3 FLF P(F)*P(6|F)*P(FL)*P(5|L)*P(LF)*P(6|F)

4 FFL P(F)*P(6|F)*P(FF)*P(5|F)*P(FL)*P(6|L)

5 LLF P(L)*P(6|L)*P(LL)*P(5|L)*P(LF)*P(6|F)

6 LFL P(L)*P(6|L)*P(LF)*P(5|F)*P(FL)*P(6|L)

7 FLL P(F)*P(6|F)*P(FL)*P(5|L)*P(LL)*P(6|L)

8 LLL P(L)*P(6|L)*P(LL)*P(5|L)*P(LL)*P(6|L)

The Occasionally Dishonest The Occasionally Dishonest Casino: An ExampleCasino: An ExamplePath Sequence Calculation Probability

1 FFF (1/2)*(1/6)*(0.95)*(1/6)*(0.95)*(1/6) 0.002089

2 LFF (1/2)*(1/2)*(0.10)*(1/6)*(0.95)*(1/6) 0.000668

3 FLF (1/2)*(1/6)*(0.05)*(1/10)*(0.10)*(1/6) 0.000007

4 FFL (1/2)*(1/2)*(0.95)*(1/6)*(0.05)*(1/2) 0.000990

5 LLF (1/2)*(1/2)*(0.90)*(1/10)*(0.10)*(1/6) 0.000375

6 LFL (1/2)*(1/2)*(0.10)*(1/6)*(0.05)*(1/2) 0.000104

7 FLL (1/2)*(1/6)*(0.05)*(1/10)*(0.90)*(1/2) 0.000188

8 LLL (1/2)*(1/2)*(0.90)*(1/10)*(0.90)*(1/2) 0.010125

The Occasionally Dishonest The Occasionally Dishonest Casino: An ExampleCasino: 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”

8 LLL (1/2)*(1/2)*(0.90)*(1/10)*(0.90)*(1/2) 0.010125

A Final NoteA Final NoteThe occasionally dishonest casino

concept is applicable to many systemsCommonly used in bioinformatics to

model DNA or protein sequences◦ Consider a twenty-sided die with a

different amino acid representing each face…

Snake Eyes!Snake Eyes!