Bayesian Synthesis of a Pathogen Growth Model

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Bayesian Synthesis of a Pathogen Growth Model

• Mark Powell, USDA/ORACBA, Washington, DC
• Mark Tamplin, USDA/ARS, Wyndmoor, PA
• Bradley Marks, Mich. St. Univ., E. Lansing, MI
• International Association of Food Protection
• New Orleans, LA , August 10-13, 2003

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Bayesian Synthesis

• Bayesian synthesis is proposed as one means of
  developing and evaluating predictive microbiology
  models.
• Motivated by empirical example.
• Apply Baranyi growth model to data from two
  studies on the growth of Listeria monocytogenes
  Buchanan et al. (1989) and Pin et al. (2001) -
  5ºC, pH 7, population growth after 240 h.

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Bayesian Synthesis
Prior on the Model Inputs
Model
Observed Data on Output
Stated Prior Model Output
Synthesizing Algorithm
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Bayesian Synthesis
Baranyi
Prior on the Model Inputs
Model
Observed Data on Output
Stated Prior Model Output
Synthesizing Algorithm
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Bayesian Synthesis
Baranyi
FAO
Prior on the Model Inputs
Model
Observed Data on Output
Ross
Stated Prior Model Output
Synthesizing Algorithm
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Bayesian Synthesis
Baranyi
FAO
Prior on the Model Inputs
Model
Observed Data on Output
Ross
Stated Prior Model Output
Buchanan
Synthesizing Algorithm
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Bayesian Synthesis
Baranyi
FAO
Prior on the Model Inputs
Model
Observed Data on Output
Ross
Pin
Stated Prior Model Output
Buchanan
Synthesizing Algorithm
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Bayesian Synthesis
Baranyi
FAO
Prior on the Model Inputs
Model
Observed Data on Output
Ross
Pin
Stated Prior Model Output
Buchanan
Synthesizing Algorithm
Raftery
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1. Baranyi Growth Model

• Originally introduced by Baranyi and Roberts
  (1994). Parameters have an intuitive biological
  interpretation.

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2. Prior on Inputs

• y0 uniform(2,4)
• ymax lognormal(9,1)
• mu lognormal(0.0239, 0.0239)
• FAO 1999, representative GT 29 h for L.
  monocytogenes at 5C, 7pH)

• tlag (116, 94)
• Ross (1999), distribution for tlag has peak _at_ 4-6
  GT, 95th percentile _at_ 10-15 GT.
• Draw thousands of independent samples from these
  distributions, run the combinations through
  Baranyi model to obtain ...

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Implied Prior on Output
Implied Prior on Output g240 normal(1.3422,
1.2631)
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3. Stated Prior on OutputBuchanan et al (1989)
Stated Prior on Output g240 normal(3.9362,
0.9770)
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4. Observed Data on OutputPin (2001)
16 Trials g240 mean 3.2592 stdev 0.6876
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5. Bayesian Synthesis Algorithm

• Compute density of the 5,000 model predictions on
  the implied prior
• Compute density of 5,000 model predictions on the
  stated prior
• Compute the likelihood of the 5,000 model
  predictions given the observed data
• Compute importance sampling weights
• Sample values from the joint input and implied
  output distribution with probabilities
  proportional to the sampling weights

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Importance Sampling Weights
Note if stated density implied density, then
the sampling weights are just the likelihood of
the model predictions, given the observed data
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Importance Sampling
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Importance Sampling
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Importance Sampling
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Importance Sampling
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Importance Sampling
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Importance Sampling
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Results Posterior Distribution for Baranyi Model
Parameters
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Results Posterior Distribution for Predicted
Growth
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Thanks

• Questions?