# Presentation: 066122

## 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

## 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.

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- Synthesizing
- Algorithm

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- Baranyi

- Synthesizing
- Algorithm

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- FAO

- Baranyi

- Ross

- Synthesizing
- Algorithm

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- FAO

- Buchanan

- Baranyi

- Ross

- Synthesizing
- Algorithm

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- FAO

- Buchanan

- Baranyi

- Pin

- Ross

- Synthesizing
- Algorithm

## Bayesian Synthesis

- Prior on the
- Model Inputs

- Stated Prior
- Model Output

- Observed
- Data on
- Output

- Model

- FAO

- Buchanan

- Baranyi

- Pin

- Ross

- Synthesizing
- Algorithm

- Raftery

## 1. Baranyi Growth Model

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

## 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 5°C, 7pH)

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

## Implied Prior on Output

- Implied Prior on Output:
*g**240* ~ normal(1.3422, 1.2631)

## 3. Stated Prior on Output:Buchanan et al (1989)

- Stated Prior on Output:
*g**240* ~ normal(3.9362, 0.9770)

## 4. Observed Data on Output:Pin (2001)

- 16 Trials: g240
- mean = 3.2592
- stdev = 0.6876

## Slide 14

## 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

## 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

## Importance Sampling

## Importance Sampling

## Importance Sampling

## Importance Sampling

## Importance Sampling

## Importance Sampling

## Results: Posterior Distribution for Baranyi Model Parameters 

## Results: Posterior Distribution for Predicted Growth

## Thanks

- Questions?