Probability distributions and likelihood

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Probability distributions and likelihood
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Readings

• Ecological Detective
• Chapter 3 Probability distributions
• Chapter 7 Likelihood

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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

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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

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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
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Factorial term
You may remember the concept of N things taken k
at a time - then again you may not
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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
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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

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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

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The negative binomialoutcomes discrete,
continuous observations
R is the expected number of observations k is a
parameter related to variance
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The normal distributioncontinuous distribution
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This is the familiar bell shaped curve
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Quiz But what is the Y axiswhat units?
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The Y axis is the first derivative of the
cumulative probability distribution
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The log normal distribution
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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

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The beta distribution
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Shapes of the beta
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Summary by nature of trials and observations
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Moving from probability distributions to
likelihood
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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
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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
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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
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Rescale to max1
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Log likelihoods
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Multiple observations

• If observations are independent then

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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

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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

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Multiple observations

• Assume we go out twice more, capture 100 animals
  each time, and 3 and then 4 are captured

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Combining all data
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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

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The concept of support

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

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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.
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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
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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

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What is c?
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Contaminated data
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Fit with robust estimation
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Demonstrate robustness in excel

• likelihood lecture workbook.xls

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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

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The infamous northern cod
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What they say about r
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Likelihoods for contradictory data
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Combined likelihood
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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