R Notebook

Let’s load the dinosaur data from the rethinking package. This is
mass vs age for different species of dinosaurs.

library(rethinking)
data("Dinosaurs")

d<-Dinosaurs
head(d)

plot(mass~age,col=species,data=d)

Model 1 Basic linear regression model. Mass = intercept + beta*age

myLM<-lm(mass~age,data=d)
summary(myLM)

Call:
lm(formula = mass ~ age, data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-7792.3 -2314.9  -336.6   704.0 17389.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  -2660.2     1758.2  -1.513   0.1407   
age            713.2      242.2   2.944   0.0062 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5058 on 30 degrees of freedom
Multiple R-squared:  0.2242,    Adjusted R-squared:  0.1983 
F-statistic: 8.669 on 1 and 30 DF,  p-value: 0.006196

AIC(myLM)

[1] 640.5808

Model 2 Linear regression model with multiple different intercepts for each species.
Mass = different intercept for each species + beta*age

myLM2<-lm(mass~age+species,data=d)
summary(myLM2)

Call:
lm(formula = mass ~ age + species, data = d)

Residuals:
     Min       1Q   Median       3Q      Max 
-10449.7   -828.2     -0.9   1021.6   8954.2 

Coefficients:
                                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)                          8784.1     2546.3   3.450  0.00200 ** 
age                                   512.6      203.3   2.521  0.01843 *  
speciesMaiasaura peeblesorum       -10162.9     2770.4  -3.668  0.00115 ** 
speciesMassospondylus carinatus    -12973.7     2008.4  -6.460 9.16e-07 ***
speciesPsittacosaurus mongoliensis -11851.7     2198.5  -5.391 1.36e-05 ***
speciesShuvuuia deserti             -9808.4     2958.0  -3.316  0.00279 ** 
speciesSyntarsus rhodesiensis      -11077.2     2371.2  -4.672 8.73e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3319 on 25 degrees of freedom
Multiple R-squared:  0.7216,    Adjusted R-squared:  0.6548 
F-statistic:  10.8 on 6 and 25 DF,  p-value: 6.165e-06

AIC(myLM2)

[1] 617.7831

Model 3 Linear regression model with multiple different slopes for each species. This represents an interaction where the effect of age depends on species. Mass = intercept + different beta for each species*age

myLM3<-lm(mass~species:age,data=d)
summary(myLM3)

Call:
lm(formula = mass ~ species:age, data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-6363.2  -217.7   179.2   544.6  3155.4 

Coefficients:
                                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)                            -1084.00     733.92  -1.477    0.152    
speciesApatosaurus excelsus:age         1668.93     104.06  16.039 1.14e-14 ***
speciesMaiasaura peeblesorum:age         384.84     249.35   1.543    0.135    
speciesMassospondylus carinatus:age      116.12      91.49   1.269    0.216    
speciesPsittacosaurus mongoliensis:age   164.13     146.35   1.121    0.273    
speciesShuvuuia deserti:age              465.06     533.78   0.871    0.392    
speciesSyntarsus rhodesiensis:age        213.62     197.63   1.081    0.290    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1614 on 25 degrees of freedom
Multiple R-squared:  0.9342,    Adjusted R-squared:  0.9184 
F-statistic: 59.15 on 6 and 25 DF,  p-value: 1.461e-13

AIC(myLM3)

[1] 571.6325

Model 4 Linear regression model with multiple different intercepts and slopes for each species.
Mass = different intercept for each species + different beta for each species*age

myLM4<-lm(mass~species*age,data=d)
summary(myLM4)

Call:
lm(formula = mass ~ species * age, data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-1810.2    -6.7    -0.1     4.0  3872.0 

Coefficients:
                                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)                             -9809.0     1448.5  -6.772 1.38e-06 ***
speciesMaiasaura peeblesorum             9682.6     1919.4   5.045 6.20e-05 ***
speciesMassospondylus carinatus          9739.3     1645.7   5.918 8.67e-06 ***
speciesPsittacosaurus mongoliensis       9797.4     1906.0   5.140 4.98e-05 ***
speciesShuvuuia deserti                  9808.7     2145.9   4.571 0.000185 ***
speciesSyntarsus rhodesiensis            9808.1     1876.2   5.228 4.08e-05 ***
age                                      2469.8      142.4  17.347 1.60e-13 ***
speciesMaiasaura peeblesorum:age        -2270.3      311.5  -7.289 4.76e-07 ***
speciesMassospondylus carinatus:age     -2451.6      165.3 -14.827 2.98e-12 ***
speciesPsittacosaurus mongoliensis:age  -2466.5      242.2 -10.186 2.32e-09 ***
speciesShuvuuia deserti:age             -2469.2      746.6  -3.307 0.003518 ** 
speciesSyntarsus rhodesiensis:age       -2466.5      285.8  -8.631 3.54e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1037 on 20 degrees of freedom
Multiple R-squared:  0.9783,    Adjusted R-squared:  0.9663 
F-statistic: 81.88 on 11 and 20 DF,  p-value: 3.77e-14

AIC(myLM4)

[1] 546.163

Repeating the models above except fitting them using a Bayesian model

Bayesian Model 1

bayes_LM2 <- ulam(
alist(
mass ~ dnorm( mu , sigma ) ,
mu <- a + bA * age ,
a ~ dnorm( 0 , 5 ) ,
bA ~ dnorm( 0 , 5 ) ,
sigma ~ dexp( 1 )
) , data = d, chains = 1, log_lik = TRUE)

Compiling Stan program...

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In file included from /var/folders/4f/_h6ql5191cl7bc390kh2xd380000gn/T/RtmpL5vLvZ/model-918f5445b523.hpp:1:
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In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor.hpp:14:
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In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/ode_rk45.hpp:9:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint.hpp:76:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint/integrate/observer_collection.hpp:23:
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In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/function_base.hpp:21:
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In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index/stl_type_index.hpp:47:
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:132:33: warnin

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g: 'unary_function<const std::error_category *, unsigned long>' is deprecated [-Wdeprecated-declarations]
        struct hash_base : std::unary_function<T, std::size_t> {};
                                ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:692:18: note: in instantiation of template class 'boost::hash_detail::hash_base<const std::error_category *>' requested here
        : public boost::hash_detail::hash_base<T*>
                 ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:420:24: note: in instantiation of template class 'boost::hash<const std::error_category *>' requested here
        boost::hash<T> hasher;
                       ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:551:9: note: in instantiation of function template specialization 'boost::hash_combine<const std::error_category *>' requested here
        hash_combine(seed, &v.category());
        ^

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/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__functional/unary_function.h:23:29: note: 'unary_function<const std::error_category *, unsigned long>' has been explicitly marked deprecated here
struct _LIBCPP_TEMPLATE_VIS _LIBCPP_DEPRECATED_IN_CXX11 unary_function
                            ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:825:41: note: expanded from macro '_LIBCPP_DEPRECATED_IN_CXX11'
#    define _LIBCPP_DEPRECATED_IN_CXX11 _LIBCPP_DEPRECATED
                                        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:810:49: note: expanded from macro '_LIBCPP_DEPRECATED'
#      define _LIBCPP_DEPRECATED __attribute__((deprecated))
                                                ^

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Running MCMC with 1 chain, with 1 thread(s) per chain...

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Chain 1 finished in 0.1 seconds.

precis(myLM3)

WAIC(myLM3)

Bayesian Model 2

bayes_LM2 <- ulam(
alist(
mass ~ dnorm( mu , sigma ) ,
mu <- a[sp_id] + bA* age ,
a[sp_id] ~ dnorm( 0 , 5 ) ,
bA ~ dnorm( 0 , 5) ,
sigma ~ dexp( 1 )
) , data = d, chains = 1, log_lik = TRUE)

Compiling Stan program...

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In file included from /var/folders/4f/_h6ql5191cl7bc390kh2xd380000gn/T/RtmpL5vLvZ/model-918f40531545.hpp:1:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/src/stan/model/model_header.hpp:4:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math.hpp:19:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev.hpp:10:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev/fun.hpp:198:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor.hpp:14:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/integrate_ode_rk45.hpp:6:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/ode_rk45.hpp:9:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint.hpp:76:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint/integrate/observer_collection.hpp:23:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function.hpp:30:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/detail/prologue.hpp:17:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/function_base.hpp:21:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index.hpp:29:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index/stl_type_index.hpp:47:
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:132:33: warnin

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g: 'unary_function<const std::error_category *, unsigned long>' is deprecated [-Wdeprecated-declarations]
        struct hash_base : std::unary_function<T, std::size_t> {};
                                ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:692:18: note: in instantiation of template class 'boost::hash_detail::hash_base<const std::error_category *>' requested here
        : public boost::hash_detail::hash_base<T*>
                 ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:420:24: note: in instantiation of template class 'boost::hash<const std::error_category *>' requested here
        boost::hash<T> hasher;
                       ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:551:9: note: in instantiation of function template specialization 'boost::hash_combine<const std::error_category *>' requested here
        hash_combine(seed, &v.category());
        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__functional/unary_function.h:23:29: note: 'unary_function<const std::error_category *, unsigned long>' has been explicitly marked deprecated here
struct _LIBCPP_TEMPLATE_VIS _LIBCPP_DEPRECATED_IN_CXX11 unary_function
                            ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:825:41: note: expanded from macro '_LIBCPP_DEPRECATED_IN_CXX11'

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#    define _LIBCPP_DEPRECATED_IN_CXX11 _LIBCPP_DEPRECATED
                                        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:810:49: note: expanded from macro '_LIBCPP_DEPRECATED'
#      define _LIBCPP_DEPRECATED __attribute__((deprecated))
                                                ^

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Running MCMC with 1 chain, with 1 thread(s) per chain...

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precis(bayes_LM2,depth=2)

WAIC(bayes_LM2)

Bayesian Model 3

bayes_LM3 <- ulam(
alist(
mass ~ dnorm( mu , sigma ) ,
mu <- a + bA[sp_id] * age ,
a ~ dnorm( 0 , 5 ) ,
bA[sp_id] ~ dnorm( 0 , 5) ,
sigma ~ dexp( 1 )
) , data = d, chains = 1, log_lik = TRUE)

Compiling Stan program...

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In file included from /var/folders/4f/_h6ql5191cl7bc390kh2xd380000gn/T/RtmpL5vLvZ/model-918f181826ea.hpp:1:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/src/stan/model/model_header.hpp:4:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math.hpp:19:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev.hpp:10:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev/fun.hpp:198:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor.hpp:14:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/integrate_ode_rk45.hpp:6:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/ode_rk45.hpp:9:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint.hpp:76:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint/integrate/observer_collection.hpp:23:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function.hpp:30:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/detail/prologue.hpp:17:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/function_base.hpp:21:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index.hpp:29:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index/stl_type_index.hpp:47:
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:132:33: warnin

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g: 'unary_function<const std::error_category *, unsigned long>' is deprecated [-Wdeprecated-declarations]
        struct hash_base : std::unary_function<T, std::size_t> {};
                                ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:692:18: note: in instantiation of template class 'boost::hash_detail::hash_base<const std::error_category *>' requested here
        : public boost::hash_detail::hash_base<T*>
                 ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:420:24: note: in instantiation of template class 'boost::hash<const std::error_category *>' requested here
        boost::hash<T> hasher;
                       ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:551:9: note: in instantiation of function template specialization 'boost::hash_combine<const std::error_category *>' requested here
        hash_combine(seed, &v.category());
        ^

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/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__functional/unary_function.h:23:29: note: 'unary_function<const std::error_category *, unsigned long>' has been explicitly marked deprecated here
struct _LIBCPP_TEMPLATE_VIS _LIBCPP_DEPRECATED_IN_CXX11 unary_function
                            ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:825:41: note: expanded from macro '_LIBCPP_DEPRECATED_IN_CXX11'
#    define _LIBCPP_DEPRECATED_IN_CXX11 _LIBCPP_DEPRECATED
                                        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:810:49: note: expanded from macro '_LIBCPP_DEPRECATED'
#      define _LIBCPP_DEPRECATED __attribute__((deprecated))
                                                ^

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1 warning generated.

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precis(bayes_LM3,depth=2)

WAIC(bayes_LM3)

Bayesian Model 4

bayes_LM4 <- ulam(
alist(
mass ~ dnorm( mu , sigma ) ,
mu <- a[sp_id] + bA[sp_id] * age ,
a[sp_id] ~ dnorm( 0 , 5) ,
bA[sp_id]  ~ dnorm( 0 , 5 ) ,
sigma ~ dexp( 1 )
) , data = d, chains = 1, log_lik = TRUE)

Compiling Stan program...

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In file included from /var/folders/4f/_h6ql5191cl7bc390kh2xd380000gn/T/RtmpL5vLvZ/model-918f542b7636.hpp:1:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/src/stan/model/model_header.hpp:4:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math.hpp:19:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev.hpp:10:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/rev/fun.hpp:198:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor.hpp:14:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/integrate_ode_rk45.hpp:6:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/stan/math/prim/functor/ode_rk45.hpp:9:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint.hpp:76:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/numeric/odeint/integrate/observer_collection.hpp:23:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function.hpp:30:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/detail/prologue.hpp:17:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/function/function_base.hpp:21:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index.hpp:29:
In file included from /Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/type_index/stl_type_index.hpp:47:
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:132:33: warnin

\

g: 'unary_function<const std::error_category *, unsigned long>' is deprecated [-Wdeprecated-declarations]
        struct hash_base : std::unary_function<T, std::size_t> {};
                                ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:692:18: note: in instantiation of template class 'boost::hash_detail::hash_base<const std::error_category *>' requested here
        : public boost::hash_detail::hash_base<T*>
                 ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:420:24: note: in instantiation of template class 'boost::hash<const std::error_category *>' requested here
        boost::hash<T> hasher;
                       ^
/Users/brianbeckage/.cmdstan/cmdstan-2.31.0/stan/lib/stan_math/lib/boost_1.78.0/boost/container_hash/hash.hpp:551:9: note: in instantiation of function template specialization 'boost::hash_combine<const std::error_category *>' requested here
        hash_combine(seed, &v.category());
        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__functional/unary_function.h:23:29: note: 'unary_function<const std::error_category *, unsigned long>' has been explicitly marked deprecated here
struct _LIBCPP_TEMPLATE_VIS _LIBCPP_DEPRECATED_IN_CXX11 unary_function
                            ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:825:41: note: expanded from macro '_LIBCPP_DEPRECATED_IN_CXX11'

|

#    define _LIBCPP_DEPRECATED_IN_CXX11 _LIBCPP_DEPRECATED
                                        ^
/Library/Developer/CommandLineTools/SDKs/MacOSX.sdk/usr/include/c++/v1/__config:810:49: note: expanded from macro '_LIBCPP_DEPRECATED'
#      define _LIBCPP_DEPRECATED __attribute__((deprecated))
                                                ^

/

-

\

|

/

-

\

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/

1 warning generated.

-

\

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/

 
Running MCMC with 1 chain, with 1 thread(s) per chain...

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Chain 1 finished in 0.2 seconds.

precis(bayes_LM4,depth=2)

WAIC(bayes_LM4)

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