MLE of Normal parameters

Dataset download

---

# Colab setup ------------------
import os, sys, subprocess
if "google.colab" in sys.modules:
    cmd = "pip install --upgrade watermark"
    process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)
    stdout, stderr = process.communicate()
    data_path = "https://s3.amazonaws.com/bebi103.caltech.edu/data/"
else:
    data_path = "../data/"
# ------------------------------

import numpy as np
import pandas as pd

---

Recall our first model for spindle length.

\begin{align} l_i \sim \text{Norm}(\phi, \sigma) \;\;\forall i. \end{align}

For this model, we have already worked out analytically that the maximum likelihood estimates for the parameters \(\phi\) and \(\sigma\) are given by their respective plug-in estimates.

\begin{align} &\phi^* = \hat{\phi}\\[1em] &\sigma^* = \hat{\sigma}. \end{align}

We can directly compute these and use parametric bootstrap to get their confidence intervals. We start by loading in the data and computing the MLEs for \(\phi\) and \(\sigma\).

df = pd.read_csv(os.path.join(data_path, 'good_invitro_droplet_data.csv'), comment='#')

spindle_length = df['Spindle Length (um)'].values

phi_mle = np.mean(spindle_length)
sigma_mle = np.std(spindle_length)

print('φ MLE:', phi_mle, 'µm')
print('σ MLE:', sigma_mle, 'µm')

φ MLE: 32.86402985074626 µm
σ MLE: 4.784665043782949 µm

Now we use parametric bootstrap to compute the confidence intervals. First, we’ll write functions to perform the bootstrapping.

rg = np.random.default_rng()

def draw_parametric_bs_reps(data, phi, sigma, size=1):
    """Parametric bootstrap replicates of parameters of
    Normal distribution."""
    bs_reps_phi = np.empty(size)
    bs_reps_sigma = np.empty(size)

    for i in range(size):
        bs_sample = np.random.normal(phi, sigma, size=len(data))
        bs_reps_phi[i] = np.mean(bs_sample)
        bs_reps_sigma[i] = np.std(bs_sample)

    return bs_reps_phi, bs_reps_sigma

Now we can get the bootstrap replicates.

bs_reps_phi, bs_reps_sigma = draw_parametric_bs_reps(
    spindle_length, phi_mle, sigma_mle, size=100000
)

With the bootstrap replicates in hand, we can compute the 95% confidence intervals using percentiles.

print('φ conf int:', np.percentile(bs_reps_phi, [2.5, 97.5]))
print('σ conf int:', np.percentile(bs_reps_sigma, [2.5, 97.5]))

φ conf int: [32.50257673 33.22919193]
σ conf int: [4.5262453  5.03866745]

So, we have estimated the parameters under this model.

Computing environment

%load_ext watermark
%watermark -v -p numpy,pandas,jupyterlab

CPython 3.8.5
IPython 7.19.0

numpy 1.19.2
pandas 1.1.3
jupyterlab 2.2.6