Review of probability. Nuno Vasconcelos UCSD

Transcription

1 Review of probability Nuno Vasconcelos UCSD

2 robability probability is the language to deal with processes that are non-deterministic examples: if I flip a coin 00 times how many can I expect to see heads? what is the weather going to be like tomorrow? are my stocks going to be up or down? am I in front of a classroom or is this just a picture of it?

3 Sample space the most important concept is that of a sample space our process defines a set of events these are the outcomes or states of the process example: we roll a pair of dice call the value on the up face at the n th toss x n note that possible events such as odd number on second throw two sixes x = and x = 6 can all be expressed as combinations of the sample space events x 6 6 x

4 Sample space is the list of possible events that satisfies the following properties: finest grain: all possible distinguishable events are listed separately mutually exclusive: if one event happens the other does not (if x = 5 it cannot be anything else) collectively exhaustive: any possible outcome can be expressed as unions of sample space events x 6 6 x mutually exclusive property simplifies the calculation of the probability of complex events collectively exhaustive means that there is no possible outcome to which we cannot assign a probability

5 robability measure probability of an event: number expressing the chance that the event will be the outcome of the process probability measure: satisfies three axioms (A) 0 for any event A (universal event) = if A B = then (A+B) = (A) + (B) x 6 e.g. (x 0) = (x even U x odd) = (x even)+ (x odd) 6 x

6 robability measure the last axiom combined with the mutually exclusive property of the sample set allows us to easily assign probabilities to all possible events back to our dice example: suppose that the probability of any pair (x x ) is /36 x we can compute probabilities of all union events 6 (x odd) = 8x/36 = (U) = 36x/36 = (two sixes) = /36 (x = and x = 6) = /36 6 x

7 robability measure note that there are many ways to define the universal event U x 6 e.g. A = {x odd} B = {x even} U = A U B on the other hand U = () U () U (3) U U (66) 6 x the fact that the sample space is finest grain exhaustive and mutually exclusive and the measure axioms make the whole procedure consistent

8 Random variables random variable is a function that assigns a real value to each sample space event we have already seen one such function: (x x ) = /36 for all (x x ) notation: specify both the random variable and the value that it takes in your probability statements we do this by specifying the random variable as subscript and the value as argument (x x ) = /36 means rob[=(x x )] = /36 without this probability statements can be hopelessly confusing

9 Random variables two types of random variables: discrete and continuous really means what types of values the RV can take if it can take only one of a finite set of possibilities we call it discrete this is the dice example we saw there are only 36 possibilities x 6 6 x

10 Random variables if it can take values in a real interval we say that the random variable is continuous e.g. consider the sample space of weather temperature we know that it could be any number between -50 and 50 degrees random variable T [-5050] note that the extremes do not have to be very precise we can just say that (T < -45 o ) = 0 most probability notions apply equal well to discrete and continuous random variables

11 Discrete RV for a discrete RV the probability assignments given by a probability mass function (MF) this can be thought of as a normalized histogram satisfies the following properties α 0 a ( a ) ( a ) = a examplefor the random variable {3 0} where = i if the grade of student z on class is between 5i and 5(i+) we see that (4) = α

12 Continuous RV for a continuous RV the probability assignments are given by a probability density function (DF) this is just a continuous function satisfies the following properties 0 ( a ) ( a ) da a = examplefor the Gaussian random variable of mean µ and variance σ ( a) exp π σ ( a µ ) σ =

13 Discrete vs continuous RVs in general the same up to replacing summations by integrals note that DF means density of probability this is probability per unit the probability of a particular event is always zero (unless there is a discontinuity) we can only talk about r( a b) = ( t) dt note also that DFs are not upper bounded b a e.g. Gaussian goes to Dirac when variance goes to zero

14 Multiple random variables frequently we have problems with multiple random variables e.g. when in the doctor you are mostly a collection of random variables x : temperature x : blood pressure x 3 : weight x 4 : cough we can summarize this as a vector = (x x n ) of n random variables (x x n ) is the joint probability distribution

15 Marginalization important notion for multiple random variables is marginalization e.g. having a cold does not depend on blood pressure and weight all that matters are fever and cough that is we need to know 4 (ab) we marginalize with respect to a subset of variables (in this case and 4 ) this is done by summing (or integrating) the others out (cold )? = ) ( ) ( dx dx x x x x x x = ) ( ) ( x x x x x x x x

16 Conditional probability another very important notion: so far doctor has 4 (fevercough) still does not allow a diagnostic for this we need a new variable Y with two states Y {sick not sick} doctor measures fever and cough levels these are no longer unknowns or even random quantities the question of interest is what is the probability that patient is sick given the measured values of fever and cough? this is exactly the definition of conditional probability what is the probability that Y takes a given value given observations for Y ( sick 98 high) 4 Y ( sick cough)?

17 Conditional probability note the very important difference between conditional and joint probability joint probability is an hypothetical question with respect to all variables what is the probability that you will be sick and cough a lot? Y ( sick cough)?

18 Conditional probability conditional probability means that you know the values of some variables what is the probability that you are sick given that you cough a lot? Y ( sick cough)? given is the key word here conditional probability is very important because it allows us to structure our thinking shows up again and again in design of intelligent systems

19 Conditional probability fortunately it is easy to compute we simply normalize the joint by the probability of what we know ( sick 98) = Y Y (98) note that this makes sense since ( sick98) and by the marginalization equation the definition of conditional probability just makes these two statements coherent Y sick 98) + Y ( not sick 98) = ( Y ( sick98) + ( 98) (98) Y not sick = simply says that given what we know we still have a valid probability measure universal event {sick} U {not sick} still probability after observation

20 The chain rule of probability is an important consequence of the definition of conditional probability note that from this definition ( y x ) Y ( y x) ( ) Y = x more generally it has the form ( x x... xn ) =... ( x x... x )... n n n (... ) x x x n 3 n... ( xn xn ) ( x ) n n n n combination with marginalization allows us to make hard probability questions simple

21 The chain rule of probability e.g. what is the probability that you will be sick and have 04 o of fever? Y ( sick04) = ( 04) (04) Y sick breaks down a hard question (prob of sick and 04) into two easier questions rob (sick 04): everyone knows that this is close to one Y ( sick 04) =! You have a cold!

22 The chain rule of probability e.g. what is the probability that you will be sick and have 04 o of fever? sick04) = ( sick 04) (04) ( Y Y rob(04): still hard but easier than (sick04) since we know only have one random variable (temperature) does not depend on sickness it is just the question what is the probability that someone will have 04 o? gather a number of people measure their temperatures and make an histogram that everyone can use after that

23 The chain rule of probability in fact the chain rule is so handy that most times we use it to compute probabilities e.g. sick) = ( sick t dt ( Y Y ) = sick t) ( t dt ( Y ) in this way we can get away with knowing (marginalization) (t) which we may know because it was needed for some other problem Y (sick t) we can ask a doctor or approximate with rule of thumb Y ( sick t) t > 0 98 < t < 0 t < 98

24 Independence another fundamental concept for multiple variables two variables are independent if the joint is the product of the marginals note: implies that ( ) ( ) ( ) a b a b = ( ) a b ( a b) = ( a) ( b) = which captures the intuitive notion: if is independent of knowing does not change the probability of e.g. knowing that it is sunny does not change the probability that it will rain in three months

25 Independence extremely useful in the design of intelligent systems frequently knowing makes Y independent of Z e.g. consider the shivering symptom: if you have temperature you sometimes shiver it is a symptom of having a cold but once you measure the temperature the two become independent Y ( sick98 shiver) = ( sick 98 shiver) S Y S shiver 98) (98) S ( = ( sick 98) shiver 98) Y S (98) ( simplifies considerably the estimation of the probabilities

26 Independence useful property: if you add two independent random variables their probability distributions convolve i.e. if z = x + y and xy are independent then ( z) ( z)* ( z) Z = where * is the convolution operator for discrete random variables for continuous random variables ( z) = ( k) ( z k) Z k Z ( z) = ( t) Y ( z t) dt y Y

27 Moments important properties of random variables summarize the distribution important moments mean: µ = E[x] variance: σ = E[(x-µ) ] various distributions are completely specified by a small number of moments σ µ mean variance µ = k σ = k discrete ( k) k ( k) (k-µ) σ continuous µ = ( k) k dk = ( k) (k - µ ) dk

28 Mean µ = E[x] is the center of mass of the distribution mean µ = discrete k ( k) k continuous µ = ( k) k dk is a linear quantity if Z = + Y then E[Z] = E[] + E[Y] this does not require any special relation between and Y always holds other moments are the mean of powers of n th order moment is E[ n ] n th central moments is E[(-µ) n ] σ µ

29 Variance σ = E[(x-µ) ] measures the dispersion around the mean it is the second central moment discrete continuous variance σ = k ( k) (k-µ) σ = ( k) (k - µ ) dk in general not linear if Z = + Y then Var[Z] = Var[] + Var[Y] only holds if and Y are independent it is related to nd order moment by σ = E = E [( ) ] [ ] x µ = E x xµ + µ [ ] [ ] [ ] x E x µ + µ = E x µ σ µ

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