Fitting models to data

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Fitting models to data II(The Basics of
Maximum Likelihood Estimation)

• Fish 458, Lecture 9

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The Principle of ML Estimation

• We wish to select the values for the parameters
  so that the probability that the model generated
  (is responsible for) the data is a high as
  possible.
• Taken another way if we have two candidate sets
  of parameters and the probability that one
  generated the data is ten times the other, we
  would naturally prefer the former.
• OK, so how to we define this probability.

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The Likelihood Function

• What we need to compute is the likelihood
  function
• If we have a discrete set of hypotheses / set of
  parameter vectors, then

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A First Example

• We observe Y6 and know that the observation
  process is based on the equation
• Given Y6, the likelihood function is normal

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A First Example - II
Y4
Y6
Note the parameter and not the data we are
given the data
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Multiple Data Sources

• If we have multiple data sources (CPUE and survey
  data for Cape Hake), we can establish a
  likelihood for each data source. The likelihood
  for the two data sources combined is the product
  of the likelihoods for each data source
• Note We often work with the logarithm of the
  likelihood function, i.e.

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Likelihood Estimation

• Identify the questions.
• Identity the data sources.
• Select alternative models.
• Select appropriate likelihood functions for each
  data source.
• Find the values for the parameters that maximize
  the likelihood function (hence Maximum Likelihood
  Estimation).

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Finding the Maximum Likelihood Estimates
The best estimate is 6, because this value of ?
leads to the maximum likelihood
9
Therefore.
We need to know which probability density
functions to use for which data types.

• The probability distributions encountered most
  commonly are
• Normal / multivariate normal
• t
• Log-normal
• Poisson
• Negative binomial
• Beta
• Binomial / multinomial

You need to know when to use each distribution
and its functional form (up to any normalizing
constants).
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The Normal and t-distributions

• The density functions for the normal and
  t-distributions are
• ? is the mean
• ? is the standard deviation ( for the t)
• k is the degrees of freedom.
• We use these distributions when the data are the
  sum of terms. The t-distribution allows account
  to be taken of small sample sizes (?lt30).

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The Normal and t-distributions
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Key Point with Normal Likelihood
Let us say we wish to fit the model
assuming normally distributed errors, i.e.
The likelihood function is therefore
Taking logarithms and multiplying by -1 gives
This is implies that if you assume
normally-distributed errors, the answers will be
identical to those from least squares.
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Time for an Example!

• We wish to fit the Dynamic Schaefer model to the
  bowhead census data.
• q is assumed to be 1 here because the surveys
  provide absolute indices of abundance.
• We have information on the trend in abundance
  from 1978-93 (increase of 3.2 per annum (SD
  0.76) based on 8 data points).
• We have an estimate of abundance for 1993 of 7800
  (SD 564).

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How to Deal with this Example!

• The model
• The likelihood function is the product of a
  normal likelihood (for the abundance estimate)
  and a t-likelihood (for the trend). Ignoring
  constants independent of the model parameters
• We take logs, multiply by minus one and minimize
  to find the estimates for K and r.
• Note that we can ignore any constants why?
• The t-distribution is chosen for the slope why?

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The Outcome
B19937710 Slope78-932.95
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The Lognormal distribution

• The density function
• ? is the median (not the mean)
• ? is the standard deviation of the logarithm
  (approximately the coefficient of variation of
  x).
• The lognormal distribution is used extensively in
  fisheries assessments because x is always larger
  than zero this is true for most data sources
  (CPUE, survey indices, estimates of death rates,
  etc.)

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The Multivariate Normal-I

• The density function
• is the vector of means.
• is the variance-covariance matrix.
• d is the length of the vector.
• This isnt nearly as bad as it looks.

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The Multivariate Normal-II

• We use the multivariate normal when the data
  points are correlated (e.g. surveys with common
  correction factors). For example for bowheads

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Readings

• Hilborn and Mangel (1997) Chapter 7
• Haddon (2001), Chapter 4