Outline Probabilities Conditional probabilities Bayes’ theorem Distributions • Discrete • Continuous Advanced Herd Management Distribution functions Probabilities and distributions Sampling from distributions • Estimation • Hypotheses Anders Ringgaard Kristensen • Confidence intervals Slide 1 Slide 2 Probabilities: Basic concepts Interpretations of probabilities The probability concept is used in daily language. What do we mean when we say: At least 3 different interpretations are observed: • The probability of the outcome ”5” when rolling a dice • A “frequentist” interpretation: is 1/6? • The probability expresses how frequent we will observe a • The probability that cow no. 543 is pregnant is 0.40? given outcome if exactly the same experiment is • The probability that USA will attack North Korea within repeated a “large” number of times. The value is rather objective. 5 years is 0.05? • An objective belief interpretation: • The probability expresses our belief in a certain (unobservable) state or event. The belief may be based on an underlying frequentist interpretation of similar cases and thus be rather objective. • A subjective belief interpretation: • The probability expresses our belief in a certain unobservable (or not yet observed) event. Slide 3 Slide 4 ”Experiments” Example of experiment An experiment may be anything creating an outcome we can observe. The ���������������� is the set of all possible Rolling a dice: outcomes. • The sample space is � = {1, 2, 3, 4, 5, 6} • Examples of events: An ������ �� is a subset of �� i.e. �� ⊆ � • � 1 = {1} Two events � 1 and � 2 are called �������� , if they • � 2 = {1, 5} have no common outcomes, i.e. if � 1 � � 2 = ∅ • � 3 = {4, 5, 6} • Since � 1 � � 3 = ∅ , � 1 and � 3 are disjoint. • � 1 and � 2 are ��� disjoint, because � 1 ∩ � 2 = {1} Slide 5 Slide 6 1
A simplified definition Example: Rolling a dice Let � be the sample space of an experiment. A probability distribution P on � is a function, Like before: � = {1, 2, 3, 4, 5, 6} so that A valid probability function on S is, for � ⊆ � : • P( � ) = 1. • P( � ) = | � |/6 where | � | is the size of � (i.e. the number of elements it contains) • For any event �� ∈ �� 0 ≤ P( � ) ≤ 1 • P({1}) = P({2}) = P({3}) = P({4}) = P({5}) = • For any two disjoint events � 1 and � 2 , P({6}) = 1/6 • P(A 1 ∪ A 2 ) = P(A 1 ) + P(A 2 ) • P({1, 5}) = 2/6 = 1/3 • P({1, 2, 3}) = 3/6 = 1/2 Notice, that many other valid probability functions could be defined (even though the one above is the only one that makes sense from a frequentist point of view). Slide 7 Slide 8 Conditional probabilities Independence Let � and � be two events, where P( � ) > 0 The ����������� probability of � given �� is If two events � and � are independent, then P( � � � ) = P( � )P( � ). written as P( � | � ), and it is by definition Example: Rolling two dices • � = {(1, 1), (1, 2),…, (1, 6),…, (6, 6)} • For any �� ⊆ � : P( � ) = | � |/36 • � = {(6, 1), (6, 2), …, (6, 6)} ⇒ P( � ) = 6/36 = 1/6 • � = {(1, 6), (2, 6), …, (6, 6)} ⇒ P( � ) = 6/36 = 1/6 • � � � = {(6, 6)} and P( � � ���� (1/6)(1/6) = 1/36 Slide 9 Slide 10 Example: Rolling a dice Conditional sum rule Again, let � = {1, 2, 3, 4, 5, 6}, and P( � ) = | � |/6. Let � 1 , � 2 , … � � be pair wise disjoint events Define � = {1, 2, 3}, and A = {2}. so that Then � � � = {2}, and Let �� be an event so that P( � ) > 0. Then The logical result: If you know the outcome is 1, 2 or 3, it is reasonable to assume that all 3 values are equally probable. Slide 11 Slide 12 2
Bayes’ theorem Sum rule: Dice example Define the 3 disjoint events � 1 = {1, 2}, � 2 = Let A 1 , A 2 , … A n be pair wise disjoint events so {3, 4}, � 3 = {5, 6} that Thus � 1 ∪ � 2 ∪ � 3 = � Define ���� {1, 3, 5} (we know that P( � ) = ½) P( � | � 1 ) = P( � � � 1 )/P( � 1 ) = (1/6)/(1/3) = ½ Let B be an event so that P( B ) > 0. Then P( � | � 2 ) = P( � � � 2 )/P( � 2 ) = (1/6)/(1/3) = ½ P( � | � 3 ) = P( � � � 3 )/P( � 3 ) = (1/6)/(1/3) = ½ Thus Bayes’ theorem is extremely important in all kinds of reasoning under uncertainty. Updating of belief. Slide 13 Slide 14 Updating of belief, I Updating of belief, II In a dairy herd, the conception rate is known to be 0.40. Now, let us assume that the farmer observes the cow, and concludes, Define M as the event ”mating” for a cow. that it is not in heat. Define Π + as the event ”pregnant” for the same cow, and Π - as the Thus, we have observed the event H - and we would like to know the event ”not pregnant”. probability, that the cow is pregnant, i.e. we wish to calculate P( Π + | H - ) Thus P( Π + | M ) = 0.40 is a conditional probability. Given that the cow has been mated, the probability of pregnancy is 0.40. We apply Bayes’ theorem: Correspondingly, P( Π - | M ) = 0.60 After 3 weeks the farmer observes the cow for heat. The farmer’s heat detection rate is 0.55. Define H + as the event that the farmer detects heat. Thus, P( H + | Π - ) = We know all probabilities in the formula, and get 0.55, and P( H - | Π - ) = 0.45 There is a slight risk that the farmer erroneously observes a pregnant cow to be in heat. We assume, that P( H + | Π + ) = 0.01 Notice, that all probabilities are figures that makes sense and are In other words, our belief in the event ”pregnant” increases from 0.40 to estimated on a routine basis (except P( H + | Π + ) which is a guess) 0.59 based on a negative heat observation result Slide 15 Slide 16 Discrete distributions Summary of probabilities In some cases the probability is defined by a certain function defined over the Probabilities may be interpreted sample space. • As frequencies In those cases, we say that the outcome • As objective or subjective beliefs in certain events The belief interpretation enables us to represent uncertain is drawn from a standard distribution. knowledge in a concise way. There exist standard distributions for Bayes’ theorem lets us update our belief (knowledge) as new many natural phenomena. observations are done. If the sample space is a countable set, we denote the corresponding distribution as discrete. Slide 17 Slide 18 3
Discrete distributions The binomial distribution I If �� is the random variable representing Consider an experiment with binary outcomes: the outcome, the expected value of a Success (s) or failure (f) discrete distribution is defined as • Mating of a sow → Pregnant (s), not pregnant (f) • Tossing a coin → Heads (s), tails (f) • Testing for a disease → Present (s), not present (f) Assume that the probability of success is � and The variance is defined as that the experiment is repeated �� times. Let � be the total number of successes observed in the � experiments. We shall look at two important discrete The sample space of the compound � experiments distributions: is � = {0, 1, 2, …, � } • The binomial distribution The ��������������� � is then said to be ���������� • The Poisson distribution. distributed with parameters � and � . Slide 19 Slide 20 The binomial distribution II The binomial distribution III The probability function P( � = � ) is (by The mean (expected value) of a binomial objective frequentist interpretation) distribution is simply E( � ) = �� . given by The variance is Var( � ) = �� (1M � ) The binomial distribution is one of the where most frequently used distribution for natural phenomena. is the �������������������� which may be calculated or looked up in a table. Slide 21 Slide 22 The Poisson distribution I The binomial distribution IV If a certain phenomenon occurs a random with a Three binomial distributions with n = 10 constant intensity (but independently of each others) the total number of occurrences � in a 0,35 0,3 time interval of a given length (or in a space of 0,25 a given area) is Poisson distributed with 0,2 0,2 P( k ) 0,5 parameter λ 0,15 0,8 0,1 Examples: 0,05 • Number of (nonMinfectious) disease cases per 0 month 0 1 2 3 4 5 6 7 8 9 10 k • Number of feeding system failures per year • Number of labor incidents per year Three binomial distributions, where ���� 10, and � = 0.2, 0.5 and 0.8, respectively. Slide 23 Slide 24 4
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