# Nonlinear model approximation

See the `method` vignette for details of the general `inlabru` method. Here, we’ll take a look at a small toy example to see the difference between a few posterior approximation methods.

Note: This vignette is not completed yet.

## A small toy problem

Hierarchical model: \[ \begin{aligned} \lambda &\sim \mathsf{Exp}(\gamma) \\ (y_i|\lambda) &\sim \mathsf{Po}(\lambda), \text{ independent across $i=1,\dots,n$} \end{aligned} \] With \(\overline{y}=\frac{1}{n}\sum_{i=1}^n y_i\), the posterior density is \[ \begin{aligned} p(\lambda | \{y_i\}) &\propto p(\lambda, y_1,\dots,y_n) \\ &\propto \exp(-\gamma\lambda) \exp(-n\lambda) \lambda^{n\overline{y}} \\ &= \exp\{-(\gamma+n)\lambda\} \lambda^{n\overline{y}}, \end{aligned} \] which is the density of a \(\mathsf{Ga}(\alpha = 1+n\overline{y}, \beta = \gamma+n)\) distribution.

## Latent Gaussian predictor version

Introducing a latent Gaussian variable \(u\sim\mathsf{N}(0,1)\), the model can be reformulated as \[ \begin{aligned} \lambda(u) &=-\ln\{1-\Phi(u)\}/\gamma \\ (y_i|u) &\sim \mathsf{Po}(\lambda(u)) \end{aligned} \] We will need some derivatives of \(\lambda\) with respect to \(u\): \[ \begin{aligned} \frac{\partial\lambda(u)}{\partial u} &= \frac{1}{\gamma}\frac{\phi(u)}{1-\Phi(u)} = \lambda'(u) \\ \frac{\partial^2\lambda(u)}{\partial u^2} &= - \frac{1}{\gamma}\frac{\phi(u)}{1-\Phi(u)} \left( u + \frac{\phi(u)}{1-\Phi(u)} \right) = -\lambda'(u)\left\{u-\gamma\lambda'(u)\right\} \\ \frac{\partial\ln\lambda(u)}{\partial u} &= \frac{1}{\lambda(u)} \frac{\partial\lambda(u)}{\partial u} =\frac{1}{-\ln\{1-\Phi(u)\}} \frac{\phi(u)}{1-\Phi(u)} = \frac{\lambda'(u)}{\lambda(u)} \\ \frac{\partial^2\ln\lambda(u)}{\partial u^2} &= \frac{1}{\lambda(u)} \frac{\partial^2\lambda(u)}{\partial u^2} -\frac{1}{\lambda(u)^2} \left(\frac{\partial\lambda(u)}{\partial u}\right)^2 = \frac{-\lambda'(u)\{u - \gamma\lambda'(u)\}}{\lambda(u)} - \frac{\lambda'(u)^2}{\lambda(u)^2}\\ &= -\frac{\lambda'(u)}{\lambda(u)}\left\{ u - \gamma\lambda'(u) +\frac{\lambda'(u)}{\lambda(u)} \right\} \end{aligned} \]

    lambda <- function(u, gamma) {
      -pnorm(u, lower.tail = FALSE, log.p = TRUE) / gamma
    }
    lambda_inv <- function(lambda, gamma) {
      qnorm(-lambda * gamma, lower.tail = FALSE, log.p = TRUE)
    }
    D1lambda <- function(u, gamma) {
      exp(dnorm(u, log = TRUE) - pnorm(u, lower.tail = FALSE, log.p = TRUE)) / gamma
    }
    D2lambda <- function(u, gamma) {
      D1L <- D1lambda(u, gamma)
      -D1L * (u - gamma * D1L)
    }
    D1log_lambda <- function(u, gamma) {
      D1lambda(u, gamma) / lambda(u, gamma)
    }
    D2log_lambda <- function(u, gamma) {
      D1logL <- D1log_lambda(u, gamma)
      -D1logL * (u - gamma * D1lambda(u, gamma = gamma) + D1logL)
    }

## Latent Gaussian posterior approximations

A basic approximation of the posterior distribution for \(\lambda\) given \(y\) can be defined as a deterministic transformation of a Gaussian distribution obtained from a 2nd order Taylor approximation of \(\ln p(u|\{y_i\})\) at the posterior mode \(u_0\) of \(p(u|\{y_i\})\). The needed derivatives are \[ \begin{aligned} \frac{\partial\ln p(u|\{y_i\})}{\partial u} &= \frac{\partial\ln\phi(u)}{\partial u} - n\lambda'(u) + n\overline{y}\frac{\lambda'(u)}{\lambda(u)} = -u + n\frac{\lambda'(u)}{\lambda(u)}\left\{ \overline{y} - \lambda(u) \right\} \\ \frac{\partial^2\ln p(u|\{y_i\})}{\partial u^2} &= -1 - n \frac{\lambda'(u)}{\lambda(u)}\left\{ u - \gamma\lambda'(u) + \frac{\lambda'(u)}{\lambda(u)} \right\} \left\{ \overline{y} - \lambda(u) \right\} - n \frac{\lambda'(u)^2}{\lambda(u)} \end{aligned} \] At the mode \(u_0\), the first order derivative is zero, and \[ \begin{aligned} \left.\frac{\partial^2\ln p(u|\{y_i\})}{\partial u^2}\right|_{u=u_0} &= -1 - \left\{ u_0 - \gamma\lambda'(u_0) + \frac{\lambda'(u_0)}{\lambda(u_0)} \right\} u_0 - n \frac{\lambda'(u_0)^2}{\lambda(u_0)} . \end{aligned} \] The quadratic approximation of the log-posterior density at the mode \(u_0\) is then \[ \ln \breve{p}(u|\{y_i\}) = \text{const} - \frac{(u-u_0)^2}{2} \left[ - \left.\frac{\partial^2\ln p(u|\{y_i\})}{\partial u^2}\right|_{u=u_0} \right] \] In `inlabru`, the approximation first linearises \(\ln \lambda(u)\) at \(u_0\) before applying the Taylor approximation of \(\ln p(u|\{y_i\})\). The linearised log-predictor is \[ \ln \overline{\lambda}(u) = \ln \lambda(u_0) + \frac{\lambda'(u_0)}{\lambda(u_0)}(u - u_0) \] so that \[ \overline{\lambda}'(u) = \frac{\lambda'(u_0)}{\lambda(u_0)} \overline{\lambda}(u) \] and the second order derivative of the linearised log-posterior density is \[ \begin{aligned} \left.\frac{\partial^2\ln \overline{p}(u|\{y_i\})}{\partial u^2}\right|_{u=u_0} &= -1 - n \frac{\lambda'(u_0)^2}{\lambda(u_0)} . \end{aligned} \]

    log_p <- function(u, y, gamma) {
      L <- lambda(u, gamma)
      n <- length(y)
      dnorm(u, log = TRUE) - n * L + n * mean(y) * log(L) - sum(lgamma(y + 1))
    }
    D1log_p <- function(u, y, gamma) {
      n <- length(y)
      -u + n * D1log_lambda(u, gamma) * (mean(y) - lambda(u, gamma))
    }
    D2log_p <- function(u, y, gamma) {
      n <- length(y)
      -1 +
        n * D2log_lambda(u, gamma) * (mean(y) - lambda(u, gamma)) -
        n * D1log_lambda(u, gamma) * D1lambda(u, gamma)
    }

    g <- 1
    y <- c(0, 1, 2)
    y <- c(0, 0, 0, 0, 0)
    y <- rpois(5, 5)
    mu_quad <- uniroot(
      D1log_p,
      lambda_inv((1 + sum(y)) / (g + length(y)) * c(1 / 10, 10), gamma = g),
      y = y, gamma = g
    )$root
    sd_quad <- (-D2log_p(mu_quad, y = y, gamma = g))^-0.5
    sd_lin <- (1 + length(y) * D1log_lambda(mu_quad, gamma = g)^2 *
      lambda(mu_quad, gamma = g))^-0.5
    lambda0 <- lambda(mu_quad, gamma = g)

### Posterior densities

    ggplot() +
      xlim(
        lambda(mu_quad - 4 * sd_quad, gamma = g),
        lambda(mu_quad + 4 * sd_quad, gamma = g)
      ) +
      labs(x = "lambda", y = "Density") +
      geom_function(
        fun = function(x) {
          exp(log_p(lambda_inv(x, gamma = g), y = y, gamma = g)) /
            D1lambda(lambda_inv(x, gamma = g), gamma = g) / (
              exp(log_p(lambda_inv(lambda0, gamma = g), y = y, gamma = g)) /
                D1lambda(lambda_inv(lambda0, gamma = g), gamma = g)
            ) *
            dgamma(lambda0, shape = 1 + sum(y), rate = g + length(y))
        },
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = dgamma,
        args = list(shape = 1 + sum(y), rate = g + length(y)),
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = function(x) {
          dnorm(lambda_inv(x, gamma = g), mean = mu_quad, sd = sd_quad) /
            D1lambda(lambda_inv(x, gamma = g), gamma = g)
        },
        mapping = aes(col = "Quadratic"),
        n = 1000
      ) +
      geom_function(
        fun = function(x) {
          dnorm(lambda_inv(x, gamma = g), mean = mu_quad, sd = sd_lin) /
            D1lambda(lambda_inv(x, gamma = g), gamma = g)
        },
        mapping = aes(col = "Linearised"),
        n = 1000
      ) +
      geom_vline(mapping = aes(
        xintercept = (1 + sum(y)) / (g + length(y)),
        lty = "Bayes mean"
      )) +
      geom_vline(mapping = aes(xintercept = lambda0, lty = "Bayes mode")) +
      geom_vline(mapping = aes(xintercept = mean(y), lty = "Plain mean"))

    ggplot() +
      xlim(
        lambda_inv(lambda0, gamma = g) - 4 * sd_quad,
        lambda_inv(lambda0, gamma = g) + 4 * sd_quad
      ) +
      xlab("u") +
      ylab("Density") +
      geom_function(
        fun = function(x) {
          exp(log_p(x, y = y, gamma = g) -
            log_p(lambda_inv(lambda0, gamma = g), y = y, gamma = g)) *
            (dgamma(lambda0, shape = 1 + sum(y), rate = g + length(y)) *
              D1lambda(lambda_inv(lambda0, gamma = g), gamma = g))
        },
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = function(x) {
          dgamma(lambda(x, gamma = g), shape = 1 + sum(y), rate = g + length(y)) *
            D1lambda(x, gamma = g)
        },
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = dnorm,
        args = list(mean = mu_quad, sd = sd_quad),
        mapping = aes(col = "Quadratic"),
        n = 1000
      ) +
      geom_function(
        fun = dnorm,
        args = list(mean = mu_quad, sd = sd_lin),
        mapping = aes(col = "Linearised"),
        n = 1000
      ) +
      geom_vline(
        mapping = aes(
          xintercept = lambda_inv((1 + sum(y)) / (g + length(y)),
            gamma = g
          ),
          lty = "Bayes mean"
        )
      ) +
      geom_vline(
        mapping = aes(
          xintercept = lambda_inv(lambda0, gamma = g),
          lty = "Bayes mode"
        )
      ) +
      geom_vline(
        mapping = aes(
          xintercept = lambda_inv(mean(y), gamma = g),
          lty = "Plain mean"
        )
      )

### Posterior CDFs

    ggplot() +
      xlim(
        lambda(mu_quad - 4 * sd_quad, gamma = g),
        lambda(mu_quad + 4 * sd_quad, gamma = g)
      ) +
      xlab("lambda") +
      ylab("CDF") +
      geom_function(
        fun = pgamma,
        args = list(shape = 1 + sum(y), rate = g + length(y)),
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = function(x) {
          pnorm(lambda_inv(x, gamma = g), mean = mu_quad, sd = sd_quad)
        },
        mapping = aes(col = "Quadratic"),
        n = 1000
      ) +
      geom_function(
        fun = function(x) {
          pnorm(lambda_inv(x, gamma = g), mean = mu_quad, sd = sd_lin)
        },
        mapping = aes(col = "Linearised"),
        n = 1000
      ) +
      geom_vline(mapping = aes(
        xintercept = (1 + sum(y)) / (g + length(y)),
        lty = "Bayes mean"
      )) +
      geom_vline(mapping = aes(xintercept = lambda0, lty = "Bayes mode")) +
      geom_vline(mapping = aes(xintercept = mean(y), lty = "Plain mean"))

    ggplot() +
      xlim(
        lambda_inv(lambda0, gamma = g) - 4 * sd_quad,
        lambda_inv(lambda0, gamma = g) + 4 * sd_quad
      ) +
      xlab("u") +
      ylab("CDF") +
      geom_function(
        fun = function(x) {
          pgamma(lambda(x, gamma = g), shape = 1 + sum(y), rate = g + length(y))
        },
        mapping = aes(col = "Theory"),
        n = 1000
      ) +
      geom_function(
        fun = pnorm,
        args = list(mean = mu_quad, sd = sd_quad),
        mapping = aes(col = "Quadratic"),
        n = 1000
      ) +
      geom_function(
        fun = pnorm,
        args = list(mean = mu_quad, sd = sd_lin),
        mapping = aes(col = "Linearised"),
        n = 1000
      ) +
      geom_vline(mapping = aes(
        xintercept = lambda_inv((1 + sum(y)) / (g + length(y)),
          gamma = g
        ),
        lty = "Bayes mean"
      )) +
      geom_vline(mapping = aes(
        xintercept = lambda_inv(lambda0, gamma = g),
        lty = "Bayes mode"
      )) +
      geom_vline(mapping = aes(
        xintercept = lambda_inv(mean(y), gamma = g),
        lty = "Plain mean"
      ))