Download probability - Instructure

Probability distributions and
likelihood
Readings
• Ecological detective:
– Chapter 3 Probability distributions
• Wikipedia (seriously!)
– e.g. Beta distribution, lognormal distribution, etc.
Overview (three lectures)
• Probability vs. likelihood
• Probability distributions: binomial, poisson,
normal, lognormal, negative binomial, beta,
gamma, multinomial
• Likelihood profile
• The concept of support
• Model selection, likelihood ratio, AIC
• Robustness
• Contradictory data
Probability
Likelihood
If I flip a fair coin 10 times,
what is the probability of it
landing heads up every time?
I flipped a coin 10 times and
obtained 10 heads. What is the
likelihood that the coin is fair?
Given the fixed parameter (p
= 0.5), what is the probability
of different outcomes?
Given the fixed outcomes
(data), what is the likelihood of
different parameter values?
Probabilities add up to 1.
Likelihoods do not add up to 1.
Hypotheses (parameter values)
are compared using likelihood
values (higher = better).
Probability
Area under curve
between 5 and 10
0.12
Height of curve at x = 10
Height of curve at x = 14
0.12
0.10
Probability density
Probability density
Likelihood
0.08
0.06
0.04
0.02
0.00
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
20
25
30
Values of x
What is the probability that 5 ≤ x ≤ 10
given a normal distribution with µ =
13 and σ = 4? Answer: 0.204
What is the probability that –1000 ≤
x ≤ 1000 given a normal distribution
with µ = 13 and σ = 4? Answer: 1.000
0
5
10
15
20
25
30
Values of x
What is the likelihood that µ = 13 and
σ = 4 if you observed a value of
(a) x = 10 (answer: the likelihood is
0.075)
(b) x = 14 (answer: the likelihood is
0.097)
Conclusion: if the observed value was
14, it is more likely that the
parameters are µ = 13 and σ = 4,
because 0.097 is higher than 0.075.
We use the same (in this case normal) probability
distribution function for both probability and likelihood!
Probability density
0.12
Likelihood is the
height, probability
is the area
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
Values of x
20
25
30
Common probability distributions
• Discrete: binomial, Poisson, negative binomial,
multinomial
• Continuous: normal, lognormal, beta, gamma,
(negative binomial)
• For all of these the Excel sheet gives the
likelihood at each value of x, given the
parameters of the distribution
7 Statistical distributions.xlsx
Many distributions defined here, including Excel
functions and functions defined directly in the
spreadsheet
SD and CV (all distributions)
standard deviation (SD)  variance
SD
coefficient of variation (CV) 
mean
Binomial probability distribution
Number of trials
N  k
N k
Pr{ Z  k}    p 1  p 
k
Number of successes
mean(Z )  Np
Probability of success [0,1]
var(Z )  Np(p  1)
N 
N!
 
 k  k!(N  k)!
The “factorial term”
How many ways are there of selecting k
objects from among N objects
Example: probability of getting k = 5 heads when flipping a coin N =
10 times, if the coin is fair (p = 0.5). Note: known number of trials.
7 Statistical distributions.xlsx, sheet Binomial
Poisson probability distribution
Pr{ Z  k} 
e
mean(Z )  
var(Z )  
k

k!
Expected number of events
Number of events
Example: On average there are λ = 9.4 fatal traffic accidents in
Washington State every week. What is the probability that there
would be k = 0 in a week? (Note: rare event out of large number
of possible events.)
7 Statistical distributions.xlsx, sheet Poisson
Limitations of Poisson
• Has only one parameter, which is both the mean
and the variance
• We often have discrete count data, but in real-life
data the variance is often larger than predicted by
the Poisson
Thus we often use the negative
binomial
• Closely related to the Poisson and binomial
• One extra parameter related to the variance
• VERY useful
Standard negative binomial
Number of failures
(k  r  1)!
r k
Pr{ Z  k} 
1  p  p
(r  1)! k!
Probability of a success
pr
Number of successes
mean(Z ) 
1 p
Squint a lot and this looks
pr
var(Z ) 
2
kind of like a binomial
(1  p)
Example: a factory makes widgets successfully with probability p.
How many successful widgets have been made when r = 3 failed
widgets have been made? The distribution predicts the
probability of k = 0, 1, 2, … successful widgets being made.
7 Statistical distributions.xlsx, sheet Neg binomial (std)
Ecological usefulness?
• Great for factories, but almost no ecological
problems can be thought of as successes or
failures in this way
• Great for factory production problems
• But we want a function with parameters for
– Mean
– Overdispersion (increased variance = increased chance
of extreme events)
• Integer events are rare in nature, we want to deal
with real numbers
Practitioner’s negative binomial
Gamma function (factorial that
accepts non-integers, see later)
Overdispersion parameter
 1

(  k)  
Pr{ Z  k} 
 1

1
( )(k  1)     
1
mean(Z )  
var(Z )    
1
  
 1

   
k
Predicted mean
2
As θ increases, variance increases, hence “overdispersion”
As θ → , var(Z) → 
As θ → 0, var(Z) = λ, just like a Poisson!
Example: our data contain observations k, with mean λ and
variance greater than λ. Find the value of overdispersion θ that
best accounts for this increased variance.
7 Statistical distributions.xlsx, sheet Neg binomial (practitioner)
Weird facts about the practitioner’s
negative binomial
• When θ → 0 this doesn’t just smell like a Poisson,
and act like a Poisson, it is the Poisson
• By replacing the factorials with gamma functions,
the r and k can be real numbers not just integers
• What on earth is a gamma function???
Gamma function Γ()
For background information only: this is a generalized factorial
function that accepts real numbers not just integers

(z)   e t t z 1dt when z is a real number
0
(z  1)  z(z) one of its properties
(z)  (z  1)! when z is an integer
Excel: does not have a gamma
function but has a ln of gamma
function (GAMMALN)
( 1  k)
( 1 )(k  1)
  ( 1  k)  
 exp ln 

1

(

)

(
k

1)

 
 exp ln ( 1  k)  ln ( 1 )  ln (k  1)
Multinomial probability distribution
Total number observed
Observed number
in category k
n!
x1 x2
xk
Pr{ X i  xi } 
p1 p2 ...pk
x1 ! x2 !...xk !
Model-predicted proportion
mean( X i )  npi
in category k
var( X i )  npi (1  pi )
Example: fitting a model to proportions at age (or proportions
at length) data. Model produces predicted proportions pi and
data gives observed numbers xi in each category. Total
numbers sampled = n = x1 + x2+ … + xk
7 Statistical distributions.xlsx, sheet Multinomial
Probability density
0.14
0.12
Predicted values
0.10
Data (n=100)
0.08
0.06
0.04
0.02
0.00
Probability density
0
2
4
6
8
10 12 14 16 18 20 22 24 26
Values of x
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
Predicted values
Data (n=10000)
0
2
4
6
8
10 12 14 16 18 20 22 24 26
Values of x
7 Statistical distributions.xlsx, sheet Multinomial
Unrealism of multinomial
(and other distributions too!)
• Assumes every sampling event is completely
independent
• But there is much correlation in reality
– Same trawl, area, time of day, day of year, gender, etc.
• Real data never ever fit a multinomial this well
• Later lectures will introduce the concept of “effective
sample size” neff, which will be smaller than reported
sample size n.
Normal
distribution
Probability density
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
20
25
30
Values of x
2

x   

1
f ( x) 
exp  

2
2
2 2


mean(x)  
var(x)   2
7 Statistical distributions.xlsx, sheet Normal
Lognormal
distribution
Probability density
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
20
25
30
Values of x
2

ln x  ln   

1 1
f ( x) 
exp  

2
2 x
2
2


2 
mean(x)   exp  
 2 
var(x)   2 exp( 2 )  1 exp( 2 )
7 Statistical distributions.xlsx, sheet Lognormal
Lognormal: key notes
• 0<x<
• Mean(x) is not µ
• If we want the mean to be µ, then replace the model
parameter with:
 *   exp(
2
)
2
• Used widely for abundance and biomass
Probability density
3.5
Beta distribution
3.0
0.5,0.5
2.5
2.0
1.5
1.0
0.5
0.0
0
0.2
0.4
0.6
0.8
1
Values of x
1.2
Probability density
(   )  1
 1
f (x) 
x 1  x 
( )( )

mean(x) 
1.0
0.8
1,1
0.6
0.4
0.2
0.0
0
0.2
0.4
0.6
0.8
1
Values of x
1.4
Probability density
 

var(x) 
(   )2 (    1)
(   )
1
Note:
is often written as
( )( )
B( ,  )
1.2
1.0
0.8
0.6
1.3,1.3
0.4
0.2
0.0
0
0.2
0.4
0.6
0.8
1
0.8
1
Values of x
Probability density
2.5
2.0
1.5
1.0
4,4
0.5
0.0
0
0.2
3.0
0.5,2
5
4
3
2
1
2.5
0.6
8
Probability density
6
0.4
Values of x
9
7
Probability density
Probability density
8
2,6
2.0
1.5
1.0
0.5
7
6
5
50,50
4
3
2
1
0
0
0.2
0.4
0.6
Values of x
0.8
1
0.0
0
0
0.2
0.4
0.6
Values of x
0.8
1
0
0.2
0.4
0.6
0.8
1
Values of x
7 Statistical distributions.xlsx, sheet Beta
Beta: key notes
• Values confined to be 0 < x < 1
• Can mimic almost any shape within those bounds
• Although bounded, can change the bounds by
multiplying / dividing x values
• E.g. survival parameters
Probability density
0.25
Gamma distribution


 1   x
f (x) 
x e
( )

mean(x) 


var(x)  2

0.20
0.15
0.10
4, 1
0.05
0.00
0
2
4
6
8
10
8
10
8
10
8
10
Values of x
0.50
Probability density
0.45
0.40
0.35
4, 2
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
2
Probability density
4
6
Values of x
0.40
0.35
0.30
1.1, 0.5
0.25
0.20
0.15
0.10
0.05
0.00
0
0.30
2
0.25
Probability density
Probability density
0.0007
0.20
0.15
0.10
60, 5
0.05
4
6
Values of x
0.9, 0.0001
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
0.00
0
5
10
15
Values of x
20
25
0
2
4
6
Values of x
7 Statistical distributions.xlsx, sheet Gamma
Gamma: key notes
• 0≤x<
• Somewhat like an exponential, lognormal, or normal
• Flexibility without being bounded like the beta
distribution
• E.g. salmon arrival numbers plotted over time
• Excel function beta.dist() assumes parameters α*
= α and β* =1/β