Statistical population modelling for census support

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# Statistical population modelling for census support
## Conclusive remarks
### Edith Darin

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# Introduction
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# What have we learned so far?

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# Bayesian philosophy

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#### A story

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The prior

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#### Observed data

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The likelihood

#### Updated information
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The posterior

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# Example

*Suppose you have a globe representing the planet that you can 
hold in the hand*

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Question: how much of the surface is covered by water?

Experiment:
1. Toss the globe in the air
2. Catch it back
3. Record the nature of the surface under your right finger

Scenario example: WLWWLWWWLWLL
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![](day5_presentation_files/pic/day5_globe.PNG)
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.footnote[*Statistical Rethinking, A Bayesian Course with Examples in R and Stan* by Richard McElreath]

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# Example

.center[<img src="day5_presentation_files/pic/day5_water.PNG" alt="drawing" width="350"/>
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.footnote[*Statistical Rethinking, A Bayesian Course with Examples in R and Stan* by Richard McElreath]

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# MCMC estimation process

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### Chains

### Initialisation

### Warmup

### Iterations

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# Population model
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### Population counts are discrete, positive

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![](day5_presentation_files/pic/day5_dag1.png)
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`$$population \sim Poisson( \lambda )$$`
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### Population counts have overdispersion

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![](day5_presentation_files/pic/day5_dag2.png)
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`$$pop \sim Poisson( pop\_density * settled) \\$$`
`$$pop\_density \sim Lognormal( \mu, \sigma)$$`
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### Population counts varie by region and type

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![](day5_presentation_files/pic/day5_dag3.png)
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`$$pop \sim Poisson( pop\_density * settled)$$`

`$$pop\_density \sim Lognormal( \alpha_{t,r}, \sigma)$$`

$$\alpha_{t,r} \sim Normal(\alpha_t, \nu_t) $$
 
`$$\alpha_t \sim Normal( \alpha, \nu) \\$$`
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### Population counts varie locally

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![](day5_presentation_files/pic/day5_dag4.png)
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.pull-right[
`$$pop \sim Poisson( pop\_density * settled)$$`

`$$pop\_density \sim Lognormal( \mu, \sigma)$$`

`$$\mu = \alpha_{t,r} + \beta X \\[10pt]$$`

`$$\alpha_{t,r} \sim Normal(\alpha_t, \nu_t)$$`
 
`$$\alpha_t \sim Normal( \alpha, \nu)$$`
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### Local variations varie by settlement type

.pull-left[
![](day5_presentation_files/pic/day5_dag5.png)
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.pull-right[
`$$pop \sim Poisson( pop\_density * settled)$$`

`$$pop\_density \sim Lognormal( \mu, \sigma)$$`

`$$\mu = \alpha_{t,r} + \beta_t X \\[10pt]$$`

`$$\alpha_{t,r} \sim Normal(\alpha_t, \nu_t)$$`
 
`$$\alpha_t \sim Normal( \alpha, \nu)$$`
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# What remains to be learned?

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# Do your own model!

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# Practice

1. Defining models
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2. Defining priors
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3. Collecting covariates
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4. Structuring the hierarchy

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# Add additional submodels

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# Modelling

#### Hierarchical structure on the variance term

.pull-left[![](day5_presentation_files/pic/day5_variance.PNG)]

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`$$pop \sim Poisson( pop\_density * settled)$$`
`$$pop\_density  \sim Lognormal( \mu, \sigma_t)$$`

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# Modelling

#### Missing households: **Measurement error model** on population count

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![](day5_presentation_files/pic/day5_missing.png)
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.pull-right[$$ N \sim Binomial (pop, \theta) $$
`$$pop \sim Poisson( pop\_density * settled)$$`
`$$pop\_density \sim Lognormal( \mu, \sigma)$$`

WorldPop.  2020. Bottom-up  gridded  population  estimates  for  Zambia. 
https://dx.doi.org/10.5258/SOTON/WP00662]

Other potential measurement error models: covariates, settlement...
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# Modelling

#### Complex sampling design: **Weighted likelihood**

`$$pop_i \sim Poisson( pop\_density_i * settled_i)$$`
`$$pop\_density_i  \sim Lognormal( \mu, \tau_i)$$`
$$ \tau_i = \frac{w_i}{\sigma}$$

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# Modelling

#### Age and sex structure

.pull-left[
<img src="day5_presentation_files/pic/day5_agesex3.png" alt="drawing" width="300"/>
<img src="day5_presentation_files/pic/day5_agesex4.png" alt="drawing" width="300"/>

]
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`$$pop \sim Poisson( pop\_density * settled)$$`

`$$pop\_density  \sim Lognormal( \mu, \sigma)$$`

$$ pop_g \sim Multinomial( pop, \gamma) $$
$$ \gamma \sim Dirichlet(rep(1,g))$$]

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# Gain programming expertise

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# Programming

- Vectorisation

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- Reparametrisation