Algorithmic Foundations of Computational Biology: Course 5

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Algorithmic Foundations of Computational Biology
Course 5 -

• Statistical Significance in Bioinformatics
• Statistics
• Probability Theory

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SIGNIFICANT SIMILARITYFOR TWO DNA SEQUENCES

• ACTACCGCGTAAATTCTAAC
• ACACTTACGTTAACCCGGGA

Size of sequences 20 Number of matches 8
If the sequences were generated at random with 4
letters A, C, G, T, having equal probability of
occurrence at any position, then the two
sequences should agree at about ¼ of their
positions. 20/45. But we observe 8
agreements! Is this significant ?
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WHAT ARE THE ASSUMPTIONS ?

• How unlikely is this outcome if the sequences
  were generated at random ?
• Assumption Equal probabilities for A, C, G, T at
  any site
• Assumption Independence of all A, C, G, T
  involved
• Clearly in our case, something other than chance
  is going on!!!

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STATISTICS

• Optimal methods for analyzing data generated by a
  random process
• What to measure ?

• ACTACCGCGTAAATTCTAAC
• ACACTTACGTTAACCCGGGT

8 3
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ACCURACY OF ASSUMPTIONS

• The probability calculated based on the
  assumptions about data (equal probability at any
  site and independence)
• Accuracy of conclusions of statistical analysis
  depends on the accuracy of assumptions made

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SIMPLIFYING ASSUMPTIONS

• We need to make simplifying assumptions, even
  when they do not hold.
• Required by the complex computations involved

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RANDOM VARIABLES

• A discrete random variable is a numerical
  quantity that in some experiment that involves
  randomness takes one value from some discrete set
  of values
• Rolling a two six-sided dice, the random variable
  X sum of the two outcomes
• Toss of a fair coin, the random variable
• Y number of tosses until the first head
  appears

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Number of Matches

• the number of matches among two random DNA
  sequences of length 20 is a random variable,
  denoted Y
• The observed value of Y in our example, denoted
  y, equals 8

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PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE

• Is the set of values that this random variable
  can take together with their associated
  probabilities
• Example. Toss a fair coin twice. Let X be the
  random variable, X the number of heads
  obtained

Values of Y 0 1 2 Probabilities .25
.5 .25
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INDEPENDENCE

• A central concept in probability and statistics
• Two or more events are independent if the outcome
  of one event does not affect in any way any other
  event
• Discrete random variables are independent if the
  value of one does not affect in any way the
  probabilities associated with the possible values
  of any other random variable

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Examples

• Different rolls of a die are independent
• Different tosses of coin are independent

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The BERNOULLI Random Variable

• A Bernoulli trial is a single trial with two
  outcomes, called success and failure
• The probability of success is denoted p and the
  probability of failure is q 1-p
• The Bernoulli random variable is
• Y number of successes
• obtained in this trial

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Bernoulli Probability Distribution
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The BINOMIAL Distribution

• A Binomial random variable is the number of
  successes in a fixed number of n of independent
  Bernoulli trials with the same probability of
  success for each trial
• The number of heads in some fixed number of
  tosses of a coin is an example of a binomial
  random variable

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ASSUMPTIONS the 4 conditions

• Each trial must result in one of two possible
  outcomes success or failure
• Trails must be independent
• The probability of success must be the same on
  all trials
• The number n of trials must be fixed in advance
  not determined by the outcomes of the trials

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The BINOMIAL Probability Distribution

• The Binomial random variable is the variable
• Y number of successes in n trials

n choose y, also known as the Binomial
coefficient
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Observations

• Bernoulli distribution is a special case of the
  Binomial distribution (when n1)
• p is often an unknown parameter

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Careful when using Binomial distribution

• Are the 4 conditions satisfied ?
• When comparing two DNA sequences our question
  about whether 8 matches are due to chance or not
  is based on the assumption that the number of
  matches follow a Binomial distribution
• Success is the event that two nucleotides in
  corresponding positions in the two sequences
  match

• ACTACCGCGTAAATTCTAAC
• ACACTTACGTTAACCCGGGT

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Careful (cont)

• It is not necessarily true that the probability
  of success is the same at all sites
• It is not necessarily true that independence
  holds population genetics shows that
  nucleotides frequencies at close sites tend to
  evolve in dependent fashion leading to dependence
  of observing a success at very close sites
• Thus 2 of the 4 conditions for a Binomial
  distribution do not hold for our pair of DNA
  sequences comparison

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SIMPLIFICATIONS ARE A MUST

• Still it might be desirable to make these
• incorrect assumptions as approximations
• Constructing models implies making simplifying
  assumptions about the process generating the data

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The UNIFORM Distribution

• The simplest probability distribution
• A uniformly distributed random variable Y takes
  values
• 1,2,,m each with same probability

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The GEOMETRIC Distribution

• Suppose a sequence of independent Bernoulli
  trials is performed, each having probability of
  success p
• The geometric distributed random variable is the
  variable Y the number of trials before but not
  including the first failure
• The possible values of the random variable
• are 1,2,3 .

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The GEOMETRIC Distribution (cont)

• The probability of several independent events is
  the product of their probabilities
• For Y y, there must be y successes followed by
  one failure
• The length of a successful run

• ACTACCGCGTAAATTCTAAC
• ACACTTACGTTAACCCGGGT

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The NEGATIVE BINOMIAL Distribution

• A sequence of independent Bernoulli trials each
  with a probability p of success
• The Binomial distribution has n such trials with
  n fixed in advance, and the random variable is
  the number of successes in these n random trials
• In the Generalized Geometric distribution, the
  number of successes is fixed in advance, at some
  value m, and the random variable is N the number
  of trials up to and including this m success
• N is said to have the negative binomial
  distribution

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The NEGATIVE BINOMIAL Distribution (cont)

• The probability that Nn is the probability that
  the first n-1 trials result in exactly m-1
  successes and n-m failures and the trial n
  results in success

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PROBABILITY THEORY

• Probability measures uncertainty
• Experiments are performed involving chance or
  randomness they are things that can be repeated.
• Suppose you roll a pair of dice once.
• you get a pair of numbers (a,b) such that
• a 1,,6 and b 1,,6
• (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
• (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
• (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
• (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
• (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),
• (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

Sample Space
Outcomes
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PROBABILITY THEORY (cont)

• The things that we measure are called events
• Rolling a 7 (1,6), (2,5), (3,4),
  (4,3),(5,2),(6,1)
• We say that the experiment of rolling out a pair
  of dice give rise to a Sample Space S which is
  just the 36 outcomes possible, and an event is
  just a set of some of these outcomes.

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PROBABILITY THEORY (cont)

• Tossing a coin twice
• Outcome example H,T
• Sample Space SH,H, H,T,T,H, T,T
• Event A at least one Head occurs
• A H,H, H,T,T,H

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PROBABILITY THEORY (cont)

• Sample space provides a mathematical model of
  real-life situations for which it is supposed to
  be an abstraction
• Mathematical analyses can only be performed on
  the abstract objects of the sample space and not
  on real-life situation itself
• Since the abstraction resemble the real world you
  may think that the mathematical relationships you
  found have something to do with the real world
• You can perform now scientific experiments to
  check out the real world situation

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PROBABILITY THEORY (cont)

• If you were successful, the mathematical model
  helped you decipher the real world you will
  know this because the results of your experiments
  are consistent with the mathematical
  relationships your obtained from the model
• It could, of course, also happen that your
  mathematical model was too simple, or otherwise
  in error and did not give a true picture of the
  real world. In such a case, the mathematical
  relationships, while true for the model, cannot
  be verified by the laboratory experiments. We
  then need another better model.

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PROBABILITY THEORY (cont)

• The Sample Space constructed to model a real life
  situation is a figment of the imagination of the
  observer of that situation, it depends on what
  the observers thinks is important. It is not in
  general unique, and it depends on the subjective
  interpretation of what is the relevant
  information.

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Tyche, or Fortuna, the Goddess of Probability

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PROBABILITY THEORY (cont)

• Consider the Sample Space S, say with the 36
  outcomes of rolling a pair of dice.
• To each of the outcome in the sample space
  associate a number between 0 and 1 such that the
  sum of these numbers over all outcomes is equal
  to 1.
• The number associated with a particular outcome
  is called the probability of the outcome, and the
  entire assignment of probabilities to outcomes is
  called a probability distribution on S.

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PROBABILITY THEORY (cont)

• We now define the probability for any event A in
  the sample space S.
• If A is the empty set, P(A)0.
• If
  then
• So given the probability distribution on S we can
  figure out the probabilities of all events in S.

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PROBABILITY SPACE

• The sample space with its probability
  distribution is called a probability space

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The Car and Goat Problem

• Monty Hall, the master of ceremonies at the
  Lets Make a Deal game show confronts you wit
  three closed doors, one of which hides the car of
  your dreams. Behind each of the other two doors,
  however, is standing a smelly goat. You will
  choose a door and win whatever is behind it.
• You decide on a door, and announce your choice.
• Your host opens then one of the other two doors
  and reveals a goat.
• He then ask you whether you would like to switch
  your choice to the unopend door that you did not
  at first choose.
• Is it in your advantage to switch ??????

Monty Halls game show
Lets Make a Deal