Fitting Linear Models with Bayes

Our Etherpad

https://etherpad.wikimedia.org/p/607-bayes-2018

Bayesian Inference

Estimate probability of a parameter
State degree of believe in specific parameter values
Evaluate probability of hypothesis given the data
Incorporate prior knowledge

Today

Markov Chain Monte-Carlo Sampling
Fitting and evaluating linear models fit with MCMC
Modifying our Prior

Bayes Theorem So Far

\(p(\theta | X) = \frac{p(X | \theta)P(\theta)}{\displaystyle \sum_{i=0}^{j} p(X | \theta_{i})p(\theta_{i})}\)

Algebraically Solvable

What About Continuous Variables

\(p(\theta | X) = \frac{p(X | \theta)P(\theta)}{\int p(X | \theta)p(\theta)d\theta}\)

Analytically Solveable for Conjugate Priors…for linear models

But Where are we going?

\(p(\theta | X) = \frac{\int p(X | \theta)P(\theta|\eta)p(\eta)d\eta}{\int\int p(X | \theta)p(\theta)d\theta d\eta}\)

Non-gaussian models, hierarchical models, and more: need numerical integration approach

Markov Chain Monte Carlo Sampling (MCMC)

Solution: Sampling from the Posterior

We don’t need to integrate/sample everything
What if we tried candidate values of \(\theta\) to get p(X|\(\theta\))
If we had a clever way of searching parameter space…
Maybe, spending more time sampling more likely values?
Enter the ATOM BOMB!

Markov Chain Monte Carlo Sampling (MCMC)

If we cannot analytically solve a distribution, we can still simulate from it:

Chose a set of starting values X at t=0
Chose a random set of parameters, Y, based on X
Calculate an acceptance ratio, \(\alpha\), based on P(Y)/P(X)
If \(\alpha \ge 1\) X(t+1) = Y
Otherwise, select a uniform random number between 0 and 1, U
If \(U \le \alpha\), X(t+1) = Y. Otherwise, X(t+1) = X.
Rinse and repeat

(Note, this is the Metropolis-Hastings Algorithm - there are others)

King Markov and His Islands (sensu Richard McElreath)

King Markov and His Islands

How to move around Islands

Flip a coin. Heads, move. Tails, stay.

Figure out the number of people on each island.

Assign a probability, p = next island / (total population)

Choose a random number. If number < p, move.

Rinse and repeat

Metropolis MCMC in Action: 10 Weeks

Population = Island Number

Metropolis MCMC in Action: 50 Weeks

Population = Island Number

Metropolis MCMC in Action: 1000 Weeks

Population = Island Number

Metropolis MCMC For Models

Each island is a set of parameter choices
Each “population” is a posterior density
The path is a ‘chain’

Markov Chain Monte Carlo Sampling (MCMC)

This is a time series. To use it for inference to sample from the final stationary distribution:

Discard a ’burn in’ set of samples
‘Thin’ your chain or use other methods to reduce temporal autocorrelation
Examine chain for convergence on your posterior distribution
Evaluate multiple chains to ensure convergence to a single distribution

Markov Chain Monte Carlo Sampling (MCMC)

Multiple Chains to Check Convergence and Improve Answer

MCMC In Practice for Models

from http://www.cnblogs.com/Nietzsche/p/4255948.html

MCMC is not a Panacea

How can MCMC Fail?

MCMC (particularly Metropolis) can get stuck
Start values can still be important
Particularly a problem with many parameters which are correlated
One way we try and assess is fit with many chains and make sure they converge

MCMC Algorithms

Metropolis MCMC inefficient
Many algorithms to come up with clever proposed moves to speed up
Gibbs sampling used for BUGS, JAGS, etc.
- Still has same problems as Metropilis
Or… Abandon search and use more deterministic sampling
- Hamiltonian MCMC

King Hamilton and His BatBoat (also from McElreath)

King Hamilton and His BatBoat

Boat passes by all of the island, back and forth
Boat slows to see people in porportion to how many folk
We sample position through time, more positions in areas where boat is slow

Metropolis versus Hamiltonian

Neal 2011, http://www.cs.utoronto.ca/~radford/ham-mcmc.abstract.html

Metropolis versus Hamiltonian

Neal 2011, http://www.cs.utoronto.ca/~radford/ham-mcmc.abstract.html

Metropolis versus Hamiltonian

Neal 2011, http://www.cs.utoronto.ca/~radford/ham-mcmc.abstract.html

Implementing HMC via Stan

A few libraries in R use Stan
rstanarm is from the Stan team
- several pre-compiled models
brms is constantly expanding
- Models compiled, but, infinitely flexible

Linear Modeling with Bayes

brms

Powerful package that fits bayesian models using Hamiltonian Monte-Carlo (reduced autocorrelation in chain)
Available tools to tweak priors
Flexible in error distribution
Can accomodate random effects, autocorrelation, etc
Uses STAN to fit models, but same syntax as base R models
Constantly updated for more weird models (post an issue, I give it 2 days)
Compiles model into C++ code, then samples
For maximum flexibility, rethinking or rstan

Bayesian Pufferfish

Pufferfish are toxic/harmful to predators
Batesian mimics gain protection from predation
Evolved response to appearance?
Researchers tested with mimics varying in toxic pufferfish resemblance

Does Resembling a Pufferfish Reduce Predator Visits?

The Steps of Statistical Modeling

What is your question?
What model of the world matches your question?
Build a model
Evaluate model assumptions
Evaluate model results
Visualize

Implementing the Puffer Model in brms

puffer_brms <- brm(predators ~ resemblance, 
                   family = gaussian(link = "identity"),
                   data = puffer)

What were the priors?

prior_summary(puffer_brms)

                 prior     class        coef group resp dpar nlpar bound
1                              b                                        
2                              b resemblance                            
3 student_t(3, 10, 10) Intercept                                        
4  student_t(3, 0, 10)     sigma

Note that these are flat (slope) or weakly informative!

MCMC diagnostics - Convergence and Posterior Behavior

These are all checks of the model fitting process. If they fail, either tweak the MCMC process (e.g., nsims), or your model is incompatible with the data

All assumptions of linear regression hold - and then there are a few checks for MCMC fits

Good ole’ QQ

But - often use simulated posterior estimates

Observed v. Average of Simulated Fitted Value

Do Simulated Posterios of the Data Fit our Observations?

Blue = Distribution of our observations, Lines = Simulations

Does Distribution of Sample Estimates Match Distribution of Simulated Sample Estimates?

Does Distribution of Error Match Distribution of Simulated Error Estimates?

How sensitive is our simulation to leaving out one value?

If >1, inspect that point!

These are all data and error generating process checks. If they fail, you may need to respecify your model (or your prior)

Finally - the Parameters

So…. what did we get?

 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: predators ~ resemblance 
   Data: puffer (Number of observations: 20) 
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup samples = 4000

Population-Level Effects: 
            Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept       2.00      1.65    -1.08     5.35       3168 1.00
resemblance     2.97      0.62     1.66     4.19       3252 1.00

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma     3.26      0.59     2.36     4.67       2873 1.00

Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample 
is a crude measure of effective sample size, and Rhat is the potential 
scale reduction factor on split chains (at convergence, Rhat = 1).

Note Rhat values - must = 1!!!

Or, just look at your Credible Interval

For 80% Credible Interval:

                       10%        90%
b_Intercept    -0.07782262   4.061401
b_resemblance   2.19418257   3.732649
sigma           2.59563720   4.026133
lp__          -57.97553042 -54.865521

What is the weight of the tail less that 0?

Weight of Intercept ≤ 0? 0.1085

Weight of Slope ≤ 0? 0

Talking about Uncertainty the IPCC Way

{width = “80%”}

Compare to LM

Bayesian Fit

term	estimate	std.error	lower	upper
b_Intercept	2.00	1.65	-0.66	4.69
b_resemblance	2.97	0.62	1.90	3.98
sigma	3.26	0.59	2.47	4.31
lp__	-56.14	1.34	-58.76	-54.74

LM Fit

term	estimate	std.error	statistic	p.value
(Intercept)	1.92	1.51	1.28	0.22
resemblance	2.99	0.57	5.23	0.00

Visualize the mean fit…

And the distributions of fits

Add the distribution of predictions

OK, ok, but what about the prior…

See your posterior relative to your prior

                 prior     class        coef group resp dpar nlpar bound
1                              b                                        
2                              b resemblance                            
3 student_t(3, 10, 10) Intercept                                        
4  student_t(3, 0, 10)     sigma

Adding a stronger prior

  puffer_brms_prior <- brm(predators ~ resemblance, 
                   family = gaussian(link = "identity"),
                   data = puffer,
                   prior = c(prior(normal(3, 1), class = Intercept),
                             prior(normal(1, 0.5), class = b),
                             prior(uniform(0, 10), class = sigma)))

  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
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  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
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  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
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      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:3:
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  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
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  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:5:
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  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core.hpp:14:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Geometry:58:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core.hpp:14:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Dense:7:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Eigenvalues:58:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:96:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Sparse:26:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/SparseCore:66:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:96:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Sparse:27:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/OrderingMethods:71:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
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  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:96:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Sparse:29:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/SparseCholesky:43:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:96:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Sparse:32:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/SparseQR:35:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:96:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/Sparse:33:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/IterativeLinearSolvers:46:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
      #pragma clang diagnostic pop
                               ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core.hpp:44:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:13: warning: unused function 'set_zero_all_adjoints' [-Wunused-function]
  static void set_zero_all_adjoints() {
              ^
  In file included from file2676668b5a19.cpp:8:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math.hpp:4:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
  In file included from /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat.hpp:70:
  /Library/Frameworks/R.framework/Versions/3.5/Resources/library/StanHeaders/include/stan/math/prim/mat/fun/autocorrelation.hpp:18:8: warning: function 'fft_next_good_size' is not needed and will not be emitted [-Wunneeded-internal-declaration]
  size_t fft_next_good_size(size_t N) {
         ^
  15 warnings generated.
  ld: warning: text-based stub file /System/Library/Frameworks//CoreFoundation.framework/CoreFoundation.tbd and library file /System/Library/Frameworks//CoreFoundation.framework/CoreFoundation are out of sync. Falling back to library file for linking.

Did it blend?

rhat(puffer_brms_prior)

  b_Intercept b_resemblance         sigma          lp__ 
     1.000209      1.001566      1.002379      1.000968

Did it blend?

Compare results!

Weak Prior:

                    2.5%      97.5%
b_Intercept    -1.084108   5.347033
b_resemblance   1.662425   4.190118
sigma           2.362067   4.672798
lp__          -59.750270 -54.677054

Strong Prior:

                       2.5%      97.5%
  b_Intercept    -0.3301055   4.755686
  b_resemblance   0.5639942   2.357556
  sigma           3.2480414   7.916630
  lp__          -69.9967159 -65.487324

Compare results!

Red = Weak Priors, Blue = Strong Prior

In Conclusion…Bayes

Yes, it’s more fidly and there are more assumptions
BUT - you can now talk in the language or probability
AND - drawing from the posterior makes visualization and understanding of uncertainty a snap
Inherently recognizes “All models are wrong, some are useful”