Near Abbey Pond VT in Summer 2016:

1. Data and Context
2. Scientific Question
3. Models
4. Results
5. Conclusion

• Collaborative work with Prof. David Allen in Biology
• 2008 and 2014 census of all trees Edwin S. George Reserve at the University of Michigan
  • 22634 trees alive on both dates
  • 34 different species

Important variable in forest ecology. Surrogate for age, shadow cast, tree height/size, basal area/biomass.

Random sample of 6 trees:

Convert DBH's to outcome variable: annual growth

Color represents top 15 species (in terms of biomass):

Observed annual growth:

1. Data and Context
2. Scientific Question
3. Models
4. Results
5. Conclusion

Can we model the annual growth of a tree while accounting for

• Basic information about it: species, current DBH, etc.
• Information about its nearby competitors
• In particular:
  • the number of competitors
  • the DBH/size/biomass of the competitors
  • nearness of the competitors
  • the species of the competitors

This translates to a statistical model that needs to incorporate:

• Population parameters \(\mu_i\): true unknown annual growth for species \(i=1, \ldots, 34\).
• Relationship parameter \(\lambda_{ij}\): true unknown competitive effect of species \(i\) on species \(j\)
• Sample data \(\vec{y}_{i}\): observed annual growth for all trees of species \(i\)
• The point of statistics: using sample data to estimate unknown parameters

Effect of American Basswoods on growth of Witch Hazels via \(\lambda_{AB, WH}\). Aid or impede?

American Basswood	Witch Hazel

We display 4 (of 34) estimates of \(\mu_i\) in red: sample mean \(\overline{y}_i\).

1. Data and Context
2. Scientific Question
3. Models
   1. Canham '04
   2. Bayesian Models in General
   3. Our Bayesian Model
4. Results
5. Conclusion

At the root, model assumes:

\[ \mbox{observed annual growth} \sim \mbox{Normal}\left(\mu_i, \sigma^2\right) \]

where \(\mu_i\) incorporates all

• species \(i\) specific information
• all species \(i\) vs \(j\) competitive information

Example: Say \(\mu_{AB}=0.20\), the observed growth of 1000 American Basswood trees might be:

• Model is complicated:
  • Lots of parameters to estimate: \(\lambda_{ij}\), \(\alpha\), \(\beta\), \(\gamma\), \(a_1\), \(a_2\), \(a_3\), etc.
  • Who says this functional form is even right?
• Practical problem: maximum likelihood solver crashes for certain species!
• Biggie: Distribution of sample sizes is uneven

Small sample size for certain species…

… but also for pairs of species. How can we estimate \(\lambda_{ij}\)?

1. Data and Context
2. Scientific Question
3. Models
   1. Canham '04
   2. Bayesian Models In General
   3. Our Bayesian Model
4. Results
5. Conclusion

To infer about \(\mu\) based on observations \(\vec{y}\):

• Prior \(\pi(\mu)\): prior belief about \(\mu\)
• Data generating mechanism: \(p(\vec{y}|\mu)\)
• Posterior \(\pi(\mu | \vec{y})\): updated belief about \(\mu\)
• Put it together: \[\pi(\mu | \vec{y}) \propto p(\vec{y}| \mu) \times \pi(\mu)\]

Who cares? See blackboard

• If \(n_i\) is small, more weight is given to the prior mean
• If \(n_i\) is large, more weight is given to the data (via sample mean)

• \(\mu_{oak}\) is a hyperparameter; a parameter to rule them all
• Using this, we can borrow information from other similar species when the sample size is small
• If the sample size is large, less borrowing
• Bayesian methods lend themselves well to this hierarchical structuring

We display 4 (of 34) estimates \(\overline{y}_i\) of \(\mu_i\) in red.

In blue we mark the estimate of the hyperparameter \(\mu\):

1. Data and Context
2. Scientific Question
3. Models
   1. Canham '04
   2. Bayesian Models In General
   3. Our Bayesian Model
4. Results
5. Conclusion

The model for the observed annual growth of a specific tree, so far, incorporates:

1. DBH of tree itself
2. Biomass of competitors: competition for resources
3. Number of competitors: "
4. i.e. we don't distinguish the species of the competitors yet

For species \(i\), we model

\[ \mbox{observed annual growth} \sim \mbox{Normal}\left(\mu_i, \sigma^2\right) \]

• where \[ \mu_{i} = \beta_{0,i} + \beta_{1, i} \mbox{dbh} + \beta_{2,i}\mbox{biomass} + \beta_{3,i}\mbox{num competitors} \]
• Each of the parameters \((\beta_{0,i}, \beta_{1, i}, \beta_{2,i}, \beta_{3,i})\) come from distributions with their own hyperparameters \((\beta_{0}, \beta_{1}, \beta_{2}, \beta_{3})\).

1. Data and Context
2. Scientific Question
3. Models
   1. Canham '04
   2. Bayesian Models In General
   3. Our Bayesian Model
4. Results
5. Conclusion

We focus only on two species for now:

1. Data and Context
2. Scientific Question
3. Models
   1. Canham '04
   2. Bayesian Models In General
   3. Our Bayesian Model
4. Results
5. Conclusion

• Implement this for all species at once and get all estimates.
• Understand methods for comparing how well different models fit the data.
• Think of clever ways to build hierarchy.
• We only considered growth models. What about birth/death?

Even with small \(n_i\), using Bayesian Hierarchical models can incorporate inherent contextual structure in the data to nevertheless obtain reliable estimates of all parameters.

There are in general two ways to derive a distribution:

Analytically (Pen & Paper)	Via Simulation

• In many cases, \(\pi(\mu|\vec{y})\) is too hard to derive by hand, so we rely on Markov Chain Monte Carlo methods to simulate draws from this distribution.

So say the unknown distribution \(\pi(\mu|\vec{y})\) is this. Without knowing the true shape of this distribution…

… we simulate random samples using MCMC and approximate all quantities.