class: center, middle # Section I: Maximum Likelihood Estimation .course[450C] .institution[__Stanford University__ Department of Political Science --- Toby Nowacki] --- --- # Overview 1. Overview 2. Likelihood: An Intuition 3. Likelihood: A Recipe 4. Likelihood: An Example 5. MLE and Uncertainty --- # Likelihood: An Intuition * In previous classes, we asked: **"What is the effect of X on Y?"** **"Given the (known) DGP, what is the probability of observing our sample?"** * In a clean causal inference setting, we know what the data-generating process is because we are in charge of assigning treatment (in an experiment) or assume that it is as-if random. * But not all questions lend themselves to clean causal inference. * What if we *don't know* the DGP? **"How can we best describe the process that generated this data?"** **"Given our observed sample, what is the DGP?"** * When we don't know how X and Y are related and want to describe that relationship. --- # Likelihood: An Intuition (cont'd) * Previously: given our **model**, how likely are the **results** that we observe? * Now: given our **results**, how likely is the **model** that we assume? * This approach yields powerful solutions to some questions * Some methodological debates yielded dead ends (OLS v logit/probit) --- # Likelihood: A Recipe <ol> <li> Set up distribution that we assume has generated the data in question $$ Pr(y_i) = f(y_i) $$ <li> Write down likelihood function -- the joint probability of observing all events under the assumed distribution $$ L(\mathbf{\theta} | y_i) = f(y_i | \mathbf{\theta}) $$ $$ L(\theta | \mathbf{y}) = \prod f(\theta | y_i) $$ </ol> --- # Likelihood: A Recipe (cont'd) <ol start=3> <li> Refactor so that we can take the logs more easily; <li> Take the logs so we have the log-likelihood function: $$ \ell(\mathbf{\theta} | \mathbf{y}) = \log(L(\mathbf{\theta} | \mathbf{y})) $$ <li> Find parameters that maximise log-likelihood: $$ \frac{\partial \ell(\mathbf{\theta} | \mathbf{y})}{\partial \theta_1} = 0 \rightarrow \theta_1^* $$ $$ \frac{\partial \ell(\mathbf{\theta} | \mathbf{y})}{\partial \theta_2} = 0 \rightarrow \theta_2^* $$ <li> Derive Fisher information to calculate variance of MLE estimate: $$ I_n(\mathbf{\theta}) = - \mathbf{H}(L(\mathbf{\theta^*})) $$ $$ I_n(\theta_1) = - \frac{\partial^2 L(\mathbf{\theta})}{\partial \theta_1^2}(\theta^*) $$ </ol> --- # Likelihood: An Example * **Motivation:** Suppose that we have `\(N\)` elections with two parties `\((A, B)\)`. * A's vote share in the last five elections (`y_samp`) was: ``` ## [1] 51.88486 51.50774 44.50988 44.34797 36.01733 ``` * Can we describe the underlying data-generating process? * What's our best guess? --- # Breakout Activity I: Guessing How might we best model `\(A\)`'s vote share? What is your best guess for the vote share in the next election? --- # Normal MLE Estimation * **Let's assume:** A's vote share is drawn i.i.d. from a normal distribution with (unknown) mean `\(\mu\)` and (unknown) variance `\(\sigma^2\)`. * This gives us enough structure to proceed with MLE. * If we **knew** mean and variance, we could calculate the probability of observing any value: `$$f(x) = \textrm{pdf}(\mathcal{N}(\mu, \sigma^2))$$` * But we **don't**. So we have to make our best guess. --- # Normal MLE Estimation (cont'd) * (Step 2). We ask: what is the likelihood of observing any set combination of parameters ( `\(\mu, \sigma^2\)` ), **given the data that we observe?** `$$\begin{align} L(\mu, \sigma^2 | \mathbf{y}) & = \prod f(y_i | \mu, \sigma^2) \\ & = \prod \frac{\exp(-\frac{(y_i - \mu)^2}{2 \sigma^2})}{\sqrt{2 \pi \sigma^2}} \end{align}$$` * (Step 3). Refactoring. $$ L(\mu, \sigma^2 | \mathbf{y}) = \frac{\exp(-\sum \frac{(y_i - \mu)^2}{2 \sigma^2})}{(2 \pi)^{n/2} \sigma^{2n/2}} $$ --- # Normal MLE Estimation (cont'd) * (Step 4). Taking the logs. $$ \ell(\mu, \sigma^2 | \mathbf{y}) = - \sum \frac{(y_i - \mu)^2}{2\sigma^2} - \frac{n}{2} \log(\sigma^2) + C $$ * Now we can plug in any combination of candidate values for `\(\mu\)` and `\(\sigma^2\)` into this function and we get a score. * We have a nicely defined function `\(\rightarrow\)` time for some coding! --- # Normal MLE Estimation (cont'd) ```r likelihood_normal <- function(dvec, mu, sigma2){ - sum((dvec - mu)^2) / (2 * sigma2) - length(dvec) / 2 * log(sigma2) } # Quick example likelihood_normal(y_samp, 43, 2) ``` ``` ## [1] -52.77709 ``` --- # Normal MLE Estimation (cont'd) Let's set up some more code to plot the likelihood for every combination of `\(\mu\)` and `\(\sigma^2\)`. ```r mu_rg <- seq(30, 70, by = 0.5) sigma2_rg <- seq(3, 60, by = 0.5) viz_df <- expand.grid(mu = mu_rg, sigma2 = sigma2_rg) %>% as.data.frame %>% rowwise() %>% mutate(likelihood = likelihood_normal(y_samp, mu, sigma2)) max_row <- which.max(viz_df$likelihood) viz_df[max_row, ] ``` ``` ## Source: local data frame [1 x 3] ## Groups: <by row> ## ## # A tibble: 1 x 3 ## mu sigma2 likelihood ## <dbl> <dbl> <dbl> ## 1 45.5 34 -11.3 ``` --- ```r viz_mat <- viz_df %>% pivot_wider(names_from = mu, values_from = likelihood) %>% dplyr::select(-sigma2) %>% as.matrix plot_ly(x = mu_rg, y = sigma2_rg, z = viz_mat, type="surface") ``` <div id="htmlwidget-b06c7f37c729f14ff51c" style="width:504px;height:504px;" class="plotly html-widget"></div> <script type="application/json" data-for="htmlwidget-b06c7f37c729f14ff51c">{"x":{"visdat":{"14ca14df43b7d":["function () 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--- # Normal MLE Estimation (cont'd) * We can also find the parameter combination that optimises the likelihood algebraically. * Recall from Wednesday's lecture: `$$\begin{align} \mu^* &= \frac{\sigma^2 y_i}{n} = \bar{y} \\ \sigma^{2*} &= \frac{1}{n} \sum(y_i - \bar{y})^2 \end{align}$$` * Again, this is something that we can implement in code. --- # Breakout activity II Implement the two functions for the MLE of mean and variance in `R` and compute the estimated mean and variance of the MLE normal for `y_samp`. --- # MLE and uncertainty (cont'd) * What if I told you that the data were generated with: `$$\mu = 50, \sigma^2 = 25$$` * Our MLE estimate only takes the "sample" from the DGP. * We can't make any assumptions about the parameters in the DGP: that's the thing we're trying to estimate using MLE! * But because of the convergence in distribution, we can still infer how likely the observed MLE estimate is if we assume a true parameter `\(\theta_0\)`. --- # MLE and uncertainty (cont'd) * Let's create `\(M\)` samples with size `\(n\)` from our true DGP. * For each of these samples, we calculate the MLE estimate and its variance. ```r get_mean_variance <- function(n){ y_samp <- rnorm(n, 50, 5) y_mean <- mean(y_samp) y_sigma <- 1/n * sum((y_samp - y_mean)^2) return(c(y_mean, y_sigma)) } n <- 5 m <- 1000 rep_vec <- 1:m names(rep_vec) <- 1:m samp_df <- map_dfr(rep_vec, ~ get_mean_variance(n)) %>% t %>% as.data.frame ``` --- # MLE and uncertainty (cont'd) ```r ggplot(samp_df, aes(V1, V2)) + geom_point(alpha = .5) + theme_tn() + labs(x = "mu_hat", y = "sigma2_hat") ``` <!-- --> --- # MLE and uncertainty (cont'd) This is a two-dimensional distribution. We can characterise its (empirical) mean and variance. ```r map_dfr(samp_df, ~ tibble(sampling_mean = mean(.x), sampling_var = var(.x))) %>% mutate(dim = c("mle_mean", "mle_var")) ``` ``` ## # A tibble: 2 x 3 ## sampling_mean sampling_var dim ## <dbl> <dbl> <chr> ## 1 49.9 4.97 mle_mean ## 2 20.5 227. mle_var ``` What do these quantities correspond to? --- # MLE and uncertainty (cont'd) What happens if we increase the sample size? ```r n <- 100 samp_df <- map_dfr(rep_vec, ~ get_mean_variance(n)) %>% t %>% as.data.frame ``` --- # MLE and uncertainty (cont'd) ```r ggplot(samp_df, aes(V1, V2)) + geom_point(alpha = .5) + theme_tn() + labs(x = "mu_hat", y = "sigma2_hat") ``` <!-- --> --- # MLE and uncertainty (cont'd) ```r map_dfr(samp_df, ~ tibble(sampling_mean = mean(.x), sampling_var = var(.x))) %>% mutate(dim = c("mle_mean", "mle_var")) ``` ``` ## # A tibble: 2 x 3 ## sampling_mean sampling_var dim ## <dbl> <dbl> <chr> ## 1 50.0 0.239 mle_mean ## 2 24.6 12.4 mle_var ``` What do these quantities correspond to? --- # MLE and uncertainty (cont'd) Recall the asymptotic property of MLE estimators as `\(n \rightarrow \infty\)`: `$$p(\hat{\mu}, \hat{\sigma}^2) \xrightarrow{d} \text{MVN} \Big( (\bar{y}, \frac{1}{n} \sum (y_i - \bar{y})^2), \begin{bmatrix} \frac{\sigma^2}{n} & 0 \\ 0 & \frac{2(\sigma^2)^2}{n}\end{bmatrix} \Big)$$` Since we know the true parameters... - We have problems when `\(n = 5\)` - Mean ( `\(\mu = 50\)` ) and Variance ( `\(\sigma^2 = 25\)` ) parameters are correctly estimated with `\(n = 100\)` - "Sampling" uncertainty of these parameters falls with sample size - Variance of sampling distribution converges to `\(25/100 = 0.25\)` and `\((2 * 25^2) / 100 = 12.5\)`, respectively --- # Summary - Generic recipe for a how to think about likelihood. + Decide on model + Write down likelihood f'n: how likely is `\(\mathbf{\theta}\)` given the observed data? + Refactor and take the logs + Maximise w.r.t. `\(\mathbf{\theta}\)` (take first derivative) + Derive second derivative / Hessian for variance - Applied to normal distribution (both with algebra and with code) - Thinking about uncertainty and inference in the context of MLE