In queezzz/qzmle: Maximum likelihood estimation

Project intro

Slide

What is this project?? Build R package for extensive MLE modelling with TMB/C++ backend

(briefly mentioned R, motivation to improve open-sourced community)

Intro of likelihood and modelling

Slide

What is MLE and why is it important?

• When we have real world data and we want to construct a model that explains the data...

• likelihood vs probability vs data (square diagram)

• example...

Slide 2

Given some sample observations $x_1, ..., x_n$ that follow a parametric distribution with probability density function $f(x_1,...,x_n | \theta)$ where $\theta$ is the model parameter, then the likelihood function, and log-likelihood function are denoted as: $$ L(\theta | x_i) = \prod_{i=1}^{n} f(x_i | \theta) \;\;\;\;\;\;\;\; l(\theta) = \sum_{i=1}^{n} ln(f(x_i | \theta))$$

Find the model parameter that maximize the the likelihood, or $ \hat \theta = \text{argmax}_\theta \textrm{L}(\theta)$ in which the value of $\hat \theta$ satisfies:

$$\frac{\partial l}{\partial \theta} = 0 $$

Slide 3

Basic: $ Y \sim f(\theta_i)$

e.g. $Y \sim \textrm{Binom}(p)$ or $Y \sim \textrm{Gamma}(\alpha, \beta)$

\bigskip \bigskip

Parameter $\theta$ depends on covariates $w_i$: $$Y \sim f(\theta) \;\;\;\;\; \theta = h(w_1,...,w_i)$$

\centerline{e.g. Linear model, $Y \sim Norm(\mu, \sigma), \;\;\;\; \mu = b_0+b_1 \cdot x$}

\bigskip \bigskip

Link functions $g$ on covariate $w_1$: $$Y \sim f(\theta) \;\;\;\;\; \theta = h(g(w_1),...,w_i)$$

\centerline{e.g. $g$ is logit function on $w_i$ where $0 < w_i < 1$ }

Slide AD

$$\frac{\partial l}{\partial b_i} = \frac{\partial l}{\partial \theta} \cdot \frac{\partial \theta}{\partial g(w_i)} \cdot \frac{\partial g(w_1)}{\partial w_1} \cdot \frac{\partial w_1}{\partial b_i} $$

RE

$$Y \sim f(\theta) $$ $$ \theta = \underbrace{h(w_1,..., w_i)}{\text{fixed effect}} + \underbrace{Z \cdot b}{\text{random effect}} \;\;\;\;\; b \sim \textrm{MVN}(0, G) $$

devtools::install_github("queezzz/qzmle")

qzmle::mle(form, start=list(logit_a=c(0), log_h=0),
parameters <- list(logit_a ~ 1 + (1|tank)), links <- list(a="logit", h="log"), data=rfpsim, method="TMB")

queezzz/qzmle documentation built on Nov. 24, 2021, 6:34 p.m.