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Probability distributions and likelihood

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Consider a discrete outcome - a coin is heads or tails, an animal (or plant) ... The infamous northern cod. Likelihood. 48. What they say about r. Likelihood. 49 ... – PowerPoint PPT presentation

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Title: Probability distributions and likelihood


1
Probability distributions and likelihood
2
Readings
  • Ecological Detective
  • Chapter 3 Probability distributions
  • Chapter 7 Likelihood

3
Overview
  • Probability distributions - binomial, poisson,
    normal, lognormal, negative binomial, beta
  • Likelihood
  • Likelihood profile
  • The concept of support
  • Model Selection Likelihood Ratio, AIC
  • Robustness - contradictory data

4
The binomial distributiondiscrete outcomes
discrete trials
  • Consider a discrete outcome - a coin is heads or
    tails, an animal (or plant) lives or dies
  • We examine a fixed number of such events - a
    number of flips of the coin, a certain number of
    animals that may or may not survive

5
The binomial formula
Z is the observed number of outcomes N is the
number of trials p is the probability of the
event happening on a given trial
6
Factorial term
You may remember the concept of N things taken k
at a time - then again you may not
7
The Poissonoutcomes discrete, continuous number
of observations
r is the expected number of events can be defined
as r t, r is a rate and t is the time
8
Limitations of Poisson
  • Has only one parameter, which is both the mean
    and the variance
  • We often have discrete count data, but want the
    variance to be estimable or at least larger than
    Poisson

9
Thus we often use the negative binomial
  • Also discrete outcomes with continuous
    observations
  • Is derived from the Poisson where the rate
    parameter is a random variable

10
The negative binomialoutcomes discrete,
continuous observations
R is the expected number of observations k is a
parameter related to variance
11
The normal distributioncontinuous distribution
12
This is the familiar bell shaped curve
13
Quiz But what is the Y axiswhat units?
14
The Y axis is the first derivative of the
cumulative probability distribution
15
The log normal distribution
16
Key notes re lognormal distribution
  • Since x is a constant, when calculating
    likelihoods we often drop the 1/x term
  • If s.d. is fixed, then the entire first term is a
    constant (also true in the normal) and can be
    ignored
  • expected value of lognormal is not the mean

17
The beta distribution
18
Shapes of the beta
19
Summary by nature of trials and observations
20
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21
Moving from probability distributions to
likelihood
22
Probability
The probability observing data Yi given parameter
p. If Y is poisson distributed, then in one unit
of time the probability of observing k events is
23
When using data, the data are known and the
hypothesis (parameter) is unknown. Thus we ask,
given the data how likely are alternative
hypotheses.
Note that now the subscript is on the
hypothesis! In probability the hypothesis is
known and the data unknown, in likelihood the
data are known and the hypothesis unknown. We
assume that likelihood is proportional to
probability
24
The probability of all outcomes for a given
hypothesis must sum to 1.0. This is not true for
likelihood, the likelihood of all hypotheses for
a given outcome will not be 1.0. Assuming a
Poisson model, and we had k4
25
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26
Rescale to max1
27
Log likelihoods
28
Multiple observations
  • If observations are independent then

29
Mark recapture example
  • We tagged 100 fish
  • Went back a few days later (after mixing etc)
  • And recaptured 100 fish
  • 5 were tagged.
  • We use Poisson distribution to explore the
    likelihood of different population sizes

30
What we need
  • Data is number marked, number recaptured, and
    tags recaptured
  • tagged is marked/population size
  • expected recoveries is tagged recaptured
  • expected recoveries is r of the Poisson

31
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32
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33
Multiple observations
  • Assume we go out twice more, capture 100 animals
    each time, and 3 and then 4 are captured

34
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35
Combining all data
36
The likelihood profile
  • Fix the parameter of interest at discrete values
    and find the maximum likelihood by searching over
    all other parameters
  • In the bad old days when people reported
    confidence intervals, you can use the likelihood
    profile to calculate a confidence interval
  • add demo from logistic model using macro

37
The concept of support
  • Edwards 1972, Likelihood
  • Think of the relative likelihood as the amount of
    support the data offer for the hypothesis

38
The lognormal distribution
Lindley, D.V. 1965. Introduction to probability
statistics from a Bayesian viewpoint. Part 1.
Probability. Cambridge U. Press. 259
p. Lognormal distribution page 143.
39
Readings on robustness and contradictory data
Robustness Numerical Recipes pp
539 Contradictory data Schnute, J. T. and R.
Hilborn. 1993. Analysis of contradictory data
sources in fish stock assessment. Canadian
Journal of Fisheries and Aquatic Sciences 50
1916-1923
40
Robustness
  • In the real world, assumptions are not always met
  • For instance, data may be mis-recorded, the wrong
    animal may be measured, the instrument may have
    failed, or some major assumption may have been
    wrong
  • Outliers exist

41
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42
What is c?
43
Contaminated data
44
Fit with robust estimation
45
Demonstrate robustness in excel
  • likelihood lecture workbook.xls

46
Contradictory data
  • We often have two independent measures of
    something, that disagree
  • The problem here is not that an individual data
    point is contaminated, but that the data set
    isnt measuring what we hope

47
The infamous northern cod
48
What they say about r
49
Likelihoods for contradictory data
50
Combined likelihood
51
Challenges in likelihood
  • All probability statements are based on the
    assumptions of the models
  • We normally do not admit that either data are
    contaminated, or data sets are not reflecting
    what we think they are
  • Thus we almost certainly overestimate the
    confidence in our analysis
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