From 3bc8dfd63b1e6dfec1698f099d237e1d9006d143 Mon Sep 17 00:00:00 2001 From: Felipe Gonzalez Date: Tue, 14 May 2024 23:49:15 -0600 Subject: [PATCH] =?UTF-8?q?Agregar=20secci=C3=B3n=20con=20m=C3=A1s=20de=20?= =?UTF-8?q?flujo=20bayesiano?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- notas/15-aplicaciones-series.qmd | 2 +- notas/16-mas-flujo-bayesiano.qmd | 492 ++++++++++++++++++ notas/_quarto.yml | 1 + notas/figuras/betancourt-flujo.png | Bin 0 -> 423087 bytes notas/figuras/inf-cal-betancourt.png | Bin 0 -> 170949 bytes notas/src/flujo-mb/1-simular-ensemble.stan | 15 + notas/src/flujo-mb/2-ajustar-modelo.stan | 13 + .../2-modelo-poisson-cero-inflado.stan | 35 ++ notas/src/flujo-mb/2-modelo-poisson.stan | 21 + notas/src/flujo-mb/2-simular-ensemble.stan | 20 + 10 files changed, 598 insertions(+), 1 deletion(-) create mode 100644 notas/16-mas-flujo-bayesiano.qmd create mode 100644 notas/figuras/betancourt-flujo.png create mode 100644 notas/figuras/inf-cal-betancourt.png create mode 100644 notas/src/flujo-mb/1-simular-ensemble.stan create mode 100644 notas/src/flujo-mb/2-ajustar-modelo.stan create mode 100644 notas/src/flujo-mb/2-modelo-poisson-cero-inflado.stan create mode 100644 notas/src/flujo-mb/2-modelo-poisson.stan create mode 100644 notas/src/flujo-mb/2-simular-ensemble.stan diff --git a/notas/15-aplicaciones-series.qmd b/notas/15-aplicaciones-series.qmd index 500619e..fc10dbc 100644 --- a/notas/15-aplicaciones-series.qmd +++ b/notas/15-aplicaciones-series.qmd @@ -991,7 +991,7 @@ ajuste_ent <- bsts(drivers ~ kms + rear + PetrolPrice, ```{r} plot(ajuste_ent) ``` -Checamos predicciones validando un periodo de validación hasta +Checamos predicciones con un periodo de validación hasta antes de la intervención: ```{r} diff --git a/notas/16-mas-flujo-bayesiano.qmd b/notas/16-mas-flujo-bayesiano.qmd new file mode 100644 index 0000000..9b5ec89 --- /dev/null +++ b/notas/16-mas-flujo-bayesiano.qmd @@ -0,0 +1,492 @@ +# Más de flujo bayesiano + +```{r} +#| include: false +library(tidyverse) +library(kableExtra) +library(DiagrammeR) +ggplot2::theme_set(ggplot2::theme_light()) +``` + + +En esta parte veremos más detalles del flujo Bayesiano de trabajo, partiendo del +plantwamiento de [M. Betancourt](https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow) + +Veamos el diagrama que se propone en este artículo: + +![Flujo bayesiano, Betancourt](./figuras/betancourt-flujo.png) + +Este flujo está dividido en tres partres: una pre-modelo y pre-datos, una pre-datos, +y uno cuando ya tenemos modelo y datos. Este enfoque es robusto desde el punto de vista +estadístico y computacional, aunque está escrito de una forma un poco diferente a como +lo planteamos en secciones anteriores. + +1. Pre-modelo y pre-datos: el análisis conceptual, definición de espacio de observaciones +y construcción de estadísticas resumen todos son basados en conocimiento de dominio y +deben tener una orientación causal. En esta parte es donde planteamos diagramas causales, +modelos generativos, y qué cantidades (con interpretación causal) que nos interesa estimar. + +2. En la segunda parte, construimos modelos y definimos estimadores en función de +estos modelos. Aquí es donde hacemos simulaciones para entender nuestra información +a priori o inicial, y las consecuencias que tienen nuestras primeras decisiones de +modelación. + +- (Supuestos de modelación) Podemos calcular qué tipo de valores obtenemos para las cantidades que queremos +estimar, y si son consistentes con el conocimiento de dominio (chequeos a priori, que incluye +también simular las posibles observaciones resultantes que implica el modelo) +- Podemos usar estas simulaciones para verificar que el algoritmo de ajuste que usamos está +bien calibrado (suponiendo que el modelo es correcto) +- Podemos también usar estas simulaciones para hacer calibración inferencial: ¿el modelo +recupera los valores que usamos para simular? ¿Existe evidencia de posible sobre ajuste o +información a priori incorrecta? + +3. En la tercera parte, ajustamos el modelo a los datos. Diagnosticamos el algoritmo +(diagnósticos de MCMC que vimos anterioremente), y podemos hacer también chequeos predictivos +posteriores para entender cómo se comporta el modelo con los datos reales. + + +Para introducir algunos análisis que consideraremos planteamos una situación simple, +donde estamos midiendo los resultados de 100 detectores de partículas en un experimento, +todos intentando medir la misma fuente. En este caso, todos los detectores están +midiendo la misma cantidad, pero hay cierto error en la medición, y podemos escribir el diagrama simple, +que supone que dada la fuente las observaciones son observaciones independientes. + +```{r} +#| out-width: 100% +#| code-fold: true +grViz(" +digraph { + graph [ranksep = 0.3, rankdir = LR] + node [shape=circle] + L + node [shape=plaintext] + y_1 + y_2 + y_3 + edge [minlen = 3] + L -> y_1 + L -> y_2 + L -> y_3 +} +")#, width = 200, height = 50) +``` + +Como las observaciones son enteros, supondremos que las observaciones son Poisson +con media $\lambda_F$. Nos interesa entonces estimar $\lambda_F$ que nos da la intensidad +de la fuente. + +Para poner una inicial necesitamos concocimiento de dominio. Supongamos que sabemos que +para este tipo de fuentes, detectores, y tiempo de detección es extremo obtener conteos mayores a 25 partículas: los pondremos en el 1% de la cola superior, por ejemplo. +Podemos experimentar con valores para $\sigma$ en una normal truncada en cero. + +```{r} +# valores de la inincial +lambda <- seq(3, 20, 2) +# simular observaciones +q_poisson <- lambda |> map_df(~ tibble(lambda = .x, q_99 = quantile(rpois(10000, .x), probs = 0.99))) +q_poisson +``` +Y podemos ver que requerimos aproximadamente $\lambda\leq 99$ con alta probabilidad. +Experimentando, podemos ver que si $\sigma=6$ es un valor razonable para la normal truncada: + +```{r} +quantile(abs(rnorm(10000, 0, 6)), probs = 0.99) +``` + +Ahora construimos nuestro modelo generativo y examinamos sus consecuencias. Simularemos +150 repeticiones de las 100 observaciones que esperamos: + +```{r} +library(cmdstanr) +N <- 100 +R <- 1500 +sim_datos <- list(N = N) + +mod_ensemble <- cmdstan_model("src/flujo-mb/1-simular-ensemble.stan") +print(mod_ensemble) +sims_priori <- mod_ensemble$sample(data = sim_datos, + iter_sampling = R, chains = 1, refresh = R, seed = 4838282, + fixed_param = TRUE) +``` + +Ahora podemos examinar algunas posibles configuraciones del modelo junto con las +observaciones que esperaríamos ver: + +```{r} +#| warning: false +#| message: false +sims_tbl <- sims_priori$draws(format = "df") +obs_priori_tbl <- sims_tbl |> + as_tibble() |> + pivot_longer(cols = starts_with("y"), names_to = "y", values_to = "valor") |> + separate(y, into = c("y", "n"), sep = "[\\[\\]]") |> select(-y) +ggplot(obs_priori_tbl |> filter(.draw < 5), aes(x = valor)) + + geom_histogram(bins = 30) + + facet_wrap(~lambda) + + labs(subtitle = "Simulaciones de observaciones a priori") +``` +Y podemos resumir estas simulaciones como + +```{r} +ggplot(obs_priori_tbl |> group_by(.draw, valor) |> count() |> + group_by(valor) |> + summarise(mediana = median(n), q_10 = quantile(n, 0.1), q90 = quantile(n, 0.9)) |> + pivot_longer(cols = c(mediana, q_10, q90), names_to = "tipo", values_to = "resumen"), + aes(x = valor)) + + geom_line(aes(y = resumen, group = tipo)) + + labs(subtitle = "Simulaciones de observaciones a priori") +``` +Y vemos que es muy poco probable observar cantidades medidas por arriba de 25: + +```{r} +obs_priori_tbl |> mutate(mayor_25 = valor > 25) |> + summarise(mayor_25 = mean(mayor_25)) +``` + +## Calibración algorítmica + +Bajo los supuestos del modelo, ahora podemos proponer nuestro algoritmo de estimación, +y checar en primer lugar que funciona apropiadamente. En este ejemplo, tomaremos +solamente 40 simulaciones y ajustaremos en cada caso el siguiente consecuencia de nuestros supuestos. Usualmente podemos usar 100 o más. + +En este paso podemos ajustar nuestro muestreador, número de cadenas y su longitud, etc. + +```{r} +#| message: false +mod_2 <- cmdstan_model("src/flujo-mb/2-modelo-poisson.stan") +print(mod_2) +ajustes_apriori <- purrr::map(1:40, function(rep){ + y <- obs_priori_tbl |> filter(.draw == rep) |> pull(valor) + datos_sim_lst <- list(y = y, N = length(y)) + ajuste <- mod_2$sample(data = datos_sim_lst, + iter_sampling = 1000, chains = 3, parallel_chains = 3, seed = 4838282, + refresh = 0, show_messages = FALSE) + list(ajuste = ajuste, lambda_sim = sims_tbl$lambda[rep]) +}) + +``` + +Podemos checar por ejemplo tamaño efectivo de muestra, rhat o divergencias: + +```{r} +ajustes_apriori |> map_df(~ .x$ajuste$summary("lambda")) |> + ggplot(aes(x = ess_bulk)) + geom_histogram(bins = 30) +``` + + + +```{r} +ajustes_apriori |> map_dbl( + ~ .x$ajuste$diagnostic_summary("divergences")$num_divergent |> sum()) |> + sum() +``` + + + + + + +Adicionalmente, podemos ver si recuperamos o no los parámetros de la simulación. +En primer lugar, calculamos el cuantil del valor verdadero para la posterior de la simulación: + +```{r} +resumen_cuantiles_tbl <- ajustes_apriori |> map_df(function(ajuste_lst){ + lambda_sim <- ajuste_lst$lambda_sim + cuantiles_tbl <- ajuste_lst$ajuste$draws("lambda", format = "df") |> + mutate(menor = lambda_sim < lambda) |> + summarise(q_menor = mean(menor)) + cuantiles_tbl |> mutate(lambda_sim = lambda_sim) +}) +``` + +Ahora podemos hacer una gráfica de cuantiles: si estamos recuperando correctamente +los parámetros, la distribución de los cuantiles de los valores verdaderos +en la posterior debe ser cercana a uniforme (pues si $y\sim F$, entonces +$P(F(y) map_df(function(ajuste_lst){ + lambda_sim <- ajuste_lst$lambda_sim + sd_previa <- 3.69 # ver inicial de lambda + post_media_sd_tbl <- ajuste_lst$ajuste$draws("lambda", format = "df") |> + summarise(media_post = mean(lambda), sd_post = sd(lambda)) + tibble(contraccion = 1 - post_media_sd_tbl$sd_post^2/sd_previa^2, + z = (lambda_sim - post_media_sd_tbl$media_post)/post_media_sd_tbl$sd_post) +}) +``` + +```{r} +ggplot(contraccion_z_tbl, aes(x = contraccion, y = z)) + + geom_point() + + xlab("Contracción") + ylab("Valor z") + + xlim(0, 1) +``` + +Y en este caso, obtenemos resultados informativos (alta contracción), que según +los valores $z$ capturan adecuadamente los valores verdaderos. + + +::: callout-note +# Sobreajuste + +Nótese que esta forma de ver el sobreajuste está más relacionada con la inferencia +acerca de parámetros de interés que al sobreajuste en modelos predictivos. + +Aunque también con modelos bayesianos podemos hacer validación cruzada para +predicciones (ya sea tradicional o con métodos computacionalmente más eficientes que +aproximan el desempeño predictivo), nuestro objetivo principal *no* es obtener buenas +predicciones, sino tener inferencia correcta e informativa acerca de las cantidades +de interés. + +::: + +El punto de vista predictivo también es importante, y puedes ver el texto +de McElreath para más detalles. + +Finalmente, vamos al ajuste de datos reales y diagnósticos asociados + +## Diagnóstico de ajuste + +Ya hemos visto antes cómo hacer diagnósticos del ajuste (checar MCMC, divergencias, etc.): + +```{r} +datos_obs <- read_csv("../datos/ejemplo-flujo.csv") +datos_lst <- list(y = datos_obs$y, N = nrow(datos_obs)) + ajuste <- mod_2$sample(data = datos_lst, + iter_sampling = 1000, refresh = 1000, chains = 3, parallel_chains = 3, seed = 282) +``` + +```{r} +ajuste$summary("lambda") |> +select(mean, sd, q5, q95, rhat, ess_bulk, ess_tail) +``` + +Los diagnósticos no muestran problemas. + +## Chequeos predictivos posteriores + +Ahora simulamos datos observados de la predictiva posterior ajustada, y comparamos con +los datos observados. + +```{r} +sims_post_pred_tbl <- ajuste$draws("y_sim", format = "df") |> + as_tibble() |> + pivot_longer(cols = starts_with("y_sim"), names_to = "y", values_to = "valor") |> + separate(y, into = c("y_sim", "n"), sep = "[\\[\\]]", extra = "drop") |> select(-y_sim) +``` + +```{r} +ggplot(sims_post_pred_tbl |> filter(.draw <=10) |> + bind_rows(datos_obs |> rename(valor = y) |> mutate(.draw = 11)), + aes(x = valor)) + + geom_histogram(bins = 30) + + facet_wrap(~.draw) + + labs(subtitle = "Chequeo predictivo posterior") +``` +Y vemos un desajuste claro en el modelo: los datos tienen exceso de ceros, y los +datos que nos son cero tienden a ser mayores que los simulados. En este punto, +es necesario regresar al análisis conceptual, pues hay algo fundamental en +el proceso generador de datos que no estamos considerando + +## Un segundo intento + +Supongamos que después de investigar, nos enteramos que es común que +algunos detectores fallen o estén defectuosos. En ese caso, marcan cero. Nótese +que la situación sería diferente, por ejemplo, si los detectores que se desbordan +marcan cero, etc. Es necesario regresar entonces al *análisis conceptual*, y repetir +todo el proceso. + +Nuestros siguientes pasos dependen de que podamos entender cuál es la razón +del exceso de ceros con respecto a nuestro modelo inicial. En este caso, +tenemos que considerar que hay cierta probabilidad de que los detectores fallen. + +Proponemos entonces un modelo como sigue: $y_n = 0$ con probabilidad $\pi$, y $y_n\sim \text{Poisson}(\lambda)$ con probabilidad $1-\pi$. El modelo de datos se puede escribir +como sigue: + +$$p(y|\lambda, \pi) = (1-\pi) W + \pi I(y=0)$$ +que es una Poisson reescalada con una masa $\pi$ en cero. + + +Recorremos los pasos con nuestro nuevo modelo, y consideraremos qué es lo que +sucede en la calibración algorítmica: + + +```{r} +N <- 100 +R <- 1500 +sim_datos <- list(N = N) + +mod_ensemble <- cmdstan_model("src/flujo-mb/2-simular-ensemble.stan") +print(mod_ensemble) +sims_priori <- mod_ensemble$sample(data = sim_datos, + iter_sampling = R, chains = 1, refresh = R, seed = 4838282, + fixed_param = TRUE) +``` + +Ahora podemos examinar algunas posibles configuraciones del modelo junto con las +observaciones que esperaríamos ver: + +```{r} +#| warning: false +#| message: false +sims_tbl <- sims_priori$draws(format = "df") +obs_priori_tbl <- sims_tbl |> + as_tibble() |> + pivot_longer(cols = starts_with("y"), names_to = "y", values_to = "valor") |> + separate(y, into = c("y", "n"), sep = "[\\[\\]]") |> select(-y) +ggplot(obs_priori_tbl |> filter(.draw < 5), aes(x = valor)) + + geom_histogram(bins = 30) + + facet_wrap(~lambda) + + labs(subtitle = "Simulaciones de observaciones a priori") +``` +Y podemos resumir estas simulaciones como + +```{r} +ggplot(obs_priori_tbl |> group_by(.draw, valor) |> count() |> + group_by(valor) |> + summarise(mediana = median(n), q_10 = quantile(n, 0.1), q90 = quantile(n, 0.9)) |> + pivot_longer(cols = c(mediana, q_10, q90), names_to = "tipo", values_to = "resumen"), + aes(x = valor)) + + geom_line(aes(y = resumen, group = tipo)) + + labs(subtitle = "Simulaciones de observaciones a priori") +``` +Y vemos que es muy poco probable observar cantidades medidas por arriba de 25: + +```{r} +obs_priori_tbl |> mutate(mayor_25 = valor > 25) |> + summarise(mayor_25 = mean(mayor_25)) +``` + +## Calibración algorítmica + +Bajo los supuestos del modelo, ahora podemos proponer nuestro algoritmo de estimación, +y checar en primer lugar que funciona apropiadamente. En este ejemplo, tomaremos +solamente 40 simulaciones y ajustaremos en cada caso el siguiente consecuencia de nuestros supuestos. Usualmente podemos usar 100 o más. + +En este paso podemos ajustar nuestro muestreador, número de cadenas y su longitud, etc. + +```{r} +#| message: false +mod_2 <- cmdstan_model("src/flujo-mb/2-modelo-poisson-cero-inflado.stan") +print(mod_2) +ajustes_apriori <- purrr::map(1:40, function(rep){ + y <- obs_priori_tbl |> filter(.draw == rep) |> pull(valor) + datos_sim_lst <- list(y = y, N = length(y)) + ajuste <- mod_2$sample(data = datos_sim_lst, + iter_sampling = 1000, chains = 3, parallel_chains = 3, seed = 4838282, + refresh = 0, show_messages = FALSE) + list(ajuste = ajuste, lambda_sim = sims_tbl$lambda[rep], p_sim = sims_tbl$p[rep]) +}) + +``` +Y vemos que nos encontramos con problemas. Examinamos los ajustes que producen divergencias: + + +```{r} +divergencias_lst <- ajustes_apriori |> map_dbl( + ~ .x$ajuste$diagnostic_summary("divergences")$num_divergent |> sum()) +div_sim <- which(divergencias_lst > 0) +div_sim +``` + +Veamos entonces que valores de $p$ y $lambda$ corresponden: + +```{r} +diag_tbl <- obs_priori_tbl |> as_tibble() |> select(lambda, p, .draw) |> unique() |> + filter(.draw <= 40) |> + mutate(problemas = .draw %in% div_sim) +ggplot(diag_tbl, aes(x = lambda, y = p, color = problemas, size = problemas)) + + geom_point() + + labs(subtitle = "Problemas de divergencia") +``` + +Y el problema aparece con valores extremos de $p$ y valores chicos de $\lambda$ +(prueba haciendo más ajustes). Cuando estos valores se presentan, +tenemos observaciones **que son principalmente ceros**, y es difícil distinguir +entre ceros que se deben a fallas en los detectores y ceros que se deben a una +tasa baja de Poisson. + +```{r} +filter(obs_priori_tbl |> filter(.draw == 17)) |> + select(valor) |> summarise(num_ceros = sum(valor == 0), num_no_ceros = sum(valor > 0)) +``` + + +Puedes ver más de esto en la discusión de M. Betancourt en +[aquí](https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow#42_Second_Iteration:_If_At_First_You_Don't_Succeed,_Try_Try_Again). + +## Tercer intento + +En este punto, es necesario otra vez regresar al análisis conceptual: **los datos +no tienen información acerca de las causas de los ceros**. En primer +lugar, podríamos informarnos acerca de qué tan común es que los detectores fallen +(o si han existido chequeos recientes que nos den confianza que una proporción razonable +está funcionando), o adicionalmente cuál es el rango de tasas mínimas razonables +que se espera para el tipo de fuente con el que se está experimentando. En este caso, +podríamos evitar las zonas degeneradas y mal identificadas: + +- Poniendo una inicial en la tasa de Poisson que no sea muy cercana a cero (para un +detector que está funcionando), si tenemos información en este sentido +- Descartando probabilidades extremas de fallas de detección: puede ser muy factible +que algunos detectores fallen, pero no que la mayoría falle, y quizá tampoco esperamos +que todos estén en perfectas condiciones. + +La continuación de este caso (con otras dos interaciones) puedes seguirla +[en la misma referencia antes citada](https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow#42_Second_Iteration:_If_At_First_You_Don't_Succeed,_Try_Try_Again). + + + diff --git a/notas/_quarto.yml b/notas/_quarto.yml index a516dc1..c6cb40c 100644 --- a/notas/_quarto.yml +++ b/notas/_quarto.yml @@ -18,6 +18,7 @@ book: - 13-exp-naturales.qmd - 14-series-de-tiempo.qmd - 15-aplicaciones-series.qmd + - 16-mas-flujo-bayesiano.qmd bibliography: 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poisson_rng(lambda); + } +} diff --git a/notas/src/flujo-mb/2-ajustar-modelo.stan b/notas/src/flujo-mb/2-ajustar-modelo.stan new file mode 100644 index 0000000..db2adba --- /dev/null +++ b/notas/src/flujo-mb/2-ajustar-modelo.stan @@ -0,0 +1,13 @@ +data { + int N; + int y[N]; +} + +parameters { + real lambda; +} + +model { + lambda ~ normal(0, 6); + y ~ poisson(lambda); +} diff --git a/notas/src/flujo-mb/2-modelo-poisson-cero-inflado.stan b/notas/src/flujo-mb/2-modelo-poisson-cero-inflado.stan new file mode 100644 index 0000000..9f696f0 --- /dev/null +++ b/notas/src/flujo-mb/2-modelo-poisson-cero-inflado.stan @@ -0,0 +1,35 @@ +data { + int N; + array[N] int y; +} + +parameters { + real lambda; + real p; +} + +model { + lambda ~ normal(0, 6); + p ~ beta(1, 1); + for(n in 1:N){ + real lpdf = poisson_lpmf(y[n] | lambda); + if(y[n] == 0){ + target += log_mix(p, 0, lpdf); + } else { + target += log(1-p) + lpdf; + } + } +} + +generated quantities { + array[N] int y_sim; + + for (n in 1:N) { + real zero = bernoulli_rng(p); + if (zero == 1) { + y_sim[n] = 0; + } else { + y_sim[n] = poisson_rng(lambda); + } + } +} diff --git a/notas/src/flujo-mb/2-modelo-poisson.stan b/notas/src/flujo-mb/2-modelo-poisson.stan new file mode 100644 index 0000000..8835cf8 --- /dev/null +++ b/notas/src/flujo-mb/2-modelo-poisson.stan @@ -0,0 +1,21 @@ +data { + int N; + array[N] int y; +} + +parameters { + real lambda; +} + +model { + lambda ~ normal(0, 6); + y ~ poisson(lambda); +} + +generated quantities { + array[N] int y_sim; + + for (n in 1:N) { + y_sim[n] = poisson_rng(lambda); + } +} diff --git a/notas/src/flujo-mb/2-simular-ensemble.stan b/notas/src/flujo-mb/2-simular-ensemble.stan new file mode 100644 index 0000000..7175c38 --- /dev/null +++ b/notas/src/flujo-mb/2-simular-ensemble.stan @@ -0,0 +1,20 @@ +data { + int N; +} + +generated quantities { + real lambda; + real p; + array[N] int y; + + // Simular configuracion del modelo a partir de inicial + lambda = abs(normal_rng(0, 6)); + p = beta_rng(1, 1); + // Simular datos del modelo observacional + for (n in 1:N){ + y[n] = 0; + if(!bernoulli_rng(p)){ + y[n] = poisson_rng(lambda); + } + } +}

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