First commit of stuff
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Eric Ma
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web: streamlit run --server.port 5000 --server.baseUrlPath minimal-streamlit.herokuapp.com beta_distribution.py
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import streamlit as st
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from scipy.stats import beta, norm
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import numpy as np
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import matplotlib.pyplot as plt
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st.header("Beta Distribution Tutorial")
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st.write(
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"""
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The beta distribution describes a probability distribution
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over values from the range (0, 1).
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In our make believe protein engineering project,
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we use the beta distribution to help us
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with the estimation of a mutant's actual activity,
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defined as a fraction from 0 to 1,
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aggregated over multiple biological replicates (multiple colonies per mutant)
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and multiple technical replicates
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(replicate measurement of individual colonies).
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Before we go on to what Bayesian estimation is,
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please go ahead and change the values of the alpha and beta parameters
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using the sliders on the left sidebar
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(click the arrow on the left if it is closed).
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"""
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)
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# These are going to be "globally" defined, because I intend to use them
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# across multiple plots.
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st.sidebar.markdown(
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"""
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# Control Panel
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"""
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)
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alpha_slider = st.sidebar.slider(
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"Value of alpha parameter",
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min_value=0.1,
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max_value=100.0,
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step=1.0,
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value=2.0,
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)
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beta_slider = st.sidebar.slider(
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"Value of beta parameter",
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min_value=0.1,
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max_value=100.0,
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step=1.0,
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value=12.0,
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)
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def plot_dist(alpha_value: float, beta_value: float, data: np.ndarray = None):
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beta_dist = beta(alpha_value, beta_value)
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xs = np.linspace(0, 1, 1000)
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ys = beta_dist.pdf(xs)
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fig, ax = plt.subplots(figsize=(7, 3))
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ax.plot(xs, ys)
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ax.set_xlim(0, 1)
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ax.set_xlabel("x")
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ax.set_ylabel("P(x)")
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if data is not None:
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likelihoods = beta_dist.pdf(data)
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sum_log_likelihoods = np.sum(beta_dist.logpdf(data))
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ax.vlines(data, ymin=0, ymax=likelihoods)
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ax.scatter(data, likelihoods, color="black")
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st.write(
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f"""
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_Under your alpha={alpha_slider:.2f} and beta={beta_slider:.2f},
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the sum of log likelihoods is {sum_log_likelihoods:.2f}_
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"""
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)
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st.pyplot(fig)
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plot_dist(alpha_slider, beta_slider)
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st.subheader("What are the alpha and beta parameters?")
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st.write(
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"""
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The alpha and beta control the shape of the beta distribution.
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Their colloquial interpretation corresponds to
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the coin flip, roughly "number of successes" (alpha)
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and "number of failures" (beta).
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The only difference here is that alpha and beta can be non-integer values,
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i.e. having decimal places.
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"""
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)
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st.subheader("How do we use it?")
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st.write(
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"""
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The beta distribution is useful for modelling quantities
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that can only take on values between 0 and 1.
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Our conversion ratio is an example of this.
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"""
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)
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st.header("Bayesian Estimation of Fractions using Beta-Distribution")
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st.write(
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"""
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The core activity that we are trying to do
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is to obtain the parameters of the alpha and beta distribution
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that best explain the data that we observe.
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Let us assume that we made three observations of activity for a mutant:
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"""
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)
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with st.echo():
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data = [0.83, 0.86, 0.91]
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st.write(
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"""
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Can you find beta distribution parameters
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that best explain these three data points?
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The best ratio of alpha to beta is probably around 1:6.
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However, is it 1:6, or is it 6:36, or is it 15:90?
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Play around with different ratios to see which one maximizes the log likelihood.
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"""
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)
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plot_dist(alpha_slider, beta_slider, data)
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st.write(
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"""
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Let's try this out with another dataset.
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Which parameter values should best explain these values?
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(Hint, it's approximately a ratio of alpha:beta ~ 1:1)
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"""
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)
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with st.echo():
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data = [0.53, 0.56, 0.51]
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plot_dist(alpha_slider, beta_slider, data)
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st.write(
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"""
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As you might have noticed,
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the greater the magnitude of the value of your alpha and beta parameters,
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the tighter the distribution of the beta distribution.
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Let's take a look at it in the following chart.
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"""
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)
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alpha_value = st.radio(label="Select a value for alpha", options=(2, 5))
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beta_value = st.radio(label="Select a value for beta", options=(2, 5))
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beta_dist_1 = beta(alpha_value, beta_value)
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beta_dist_2 = beta(alpha_value * 5, beta_value * 5)
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beta_dist_3 = beta(alpha_value * 10, beta_value * 10)
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xs = np.linspace(0, 1, 1000)
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ys_1 = beta_dist_1.pdf(xs)
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ys_2 = beta_dist_2.pdf(xs)
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ys_3 = beta_dist_3.pdf(xs)
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fig4 = plt.figure(figsize=(7, 3))
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plt.plot(xs, ys_1, label=f"alpha={alpha_value}, beta={beta_value}")
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plt.plot(xs, ys_2, label=f"alpha={alpha_value * 5}, beta={beta_value * 5}")
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plt.plot(xs, ys_3, label=f"alpha={alpha_value * 10}, beta={beta_value * 10}")
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plt.ylabel("P(x)")
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plt.legend()
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st.pyplot(fig4)
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st.write(
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"""
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As you can see, the width of the curve decreases
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as the scale of the alpha and beta parameters increases.
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_We become less uncertain._
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"""
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)
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st.write(
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"""
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Now, if we focus our attention on
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the variance of the beta distribution when operating in the same scale
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(i.e. fix the sum of alpha and beta),
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let's see what happens to the variance of the likelihood distribution.
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Let's assume that alpha + beta = 50,
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and we will have you adjust the value of alpha.
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"""
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)
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max_val = 50
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alpha_value = st.slider("alpha", 1, max_val - 1)
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beta_value = max_val - alpha_value
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beta_dist = beta(alpha_value, beta_value)
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xs = np.linspace(0, 1, 1000)
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ys = beta_dist.pdf(xs)
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fig = plt.figure(figsize=(7, 3))
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plt.plot(xs, ys, label=f"alpha={alpha_value}, beta={beta_value}")
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plt.title(f"StDev: {beta_dist.std():.3f}")
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plt.ylabel("P(x)")
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plt.legend()
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st.pyplot(fig)
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st.write(
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"""
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As you can see, as we go more to the extremes, the variance decreases.
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A more comprehensive look at variance as a function of alpha and beta,
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keeping the "scale" (i.e. sum of alpha and beta) the same is below.
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"""
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)
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alpha_values = np.arange(1, 50)
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beta_dists = beta(alpha_values, 50 - alpha_values)
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ys = beta_dists.std()
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fig = plt.figure(figsize=(7, 4))
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plt.scatter(alpha_values, ys)
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plt.xlabel("alpha")
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plt.title("standard deviation of the beta distribution as a function of alpha")
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plt.ylabel("std")
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st.pyplot()
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st.header("Conclusions")
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st.write(
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"""
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What we have found thus far is that
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according to the math of the beta distribution,
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at a given scale, variance is always going to be highest in the middle,
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where the peak variance consistently is close to the middle
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of the beta distribution likelihood.
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The more noisy the calculated ratios are,
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the smaller the alpha + beta scale needed to maximize their likelihoods,
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and hence the greater their estimated likelihood distribution variance will be.
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When we do Bayesian estimation,
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we are estimating the parameters alpha and beta
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that explain both the central tendency measures
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of our measured ratios (i.e. mean)
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and the variance in the measured ratios.
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Ratios that are more variably distributed, such as `[0.81, 0.56, 0.93]`,
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will naturally get smaller estimated alphas and betas
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to explain the high variance in the ratios.
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(They will also not tend to be centered around the extremes,
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which also makes sense: to be generate data centered around the extremes,
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we need extremely tightly-distributed data.)
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Ratios that are more tightly distributed, such as `[0.51, 0.49, 0.48]`,
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will naturally get higher estimated alphas and betas.
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This is the spirit of what we are doing in the Bayesian estimation model.
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In short, there's both math + measurement contributing to the "wave" phenomena.
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"""
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)
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st.header("Try with your own data!")
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st.write(
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"""
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I'd like to invite you to upload your own data.
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Type in numbers into the text box below, separated by commas.
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"""
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)
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data = st.text_input("Your data")
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def process_data(data):
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# One thing nice about streamlit is it allows us to surface informative
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# errors to our end-users using built-in Python error handling.
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# This is done by capturing the output of stdout and surfacing it to HTML.
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try:
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data = np.array([float(i) for i in data.split(", ")])
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except ValueError as e:
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raise ValueError("The data that you input must be castable as floats!")
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if not (np.all(data > 0) and np.all(data < 1)):
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raise ValueError("Your input data must be 0 < x < 1.")
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return data
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plot_dist(alpha_slider, beta_slider, process_data(data))
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st.header("Congratulations!")
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st.write("Click the button below once you're done with this tutorial.")
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if st.button("CLICK ME!"):
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st.balloons()
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st.sidebar.markdown(
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"""
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# Thank you!
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Did you like this mini-tutorial?
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If you did, please give it a star on [GitHub](https://github.com/ericmjl/minimal-streamlit-example).
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This was hand-crafted using streamlit in under 3 hours.
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Created by [Eric J. Ma](https://ericmjl.github.io).
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"""
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)
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name: minimal-streamlit-example
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channels:
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- defaults
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- conda-forge
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- ericmjl
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dependencies:
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- python=3.8
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- conda
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- scikit-learn
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- scipy
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- pandas
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- numpy
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- matplotlib
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- pip
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- pylint
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- pydocstyle
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- flake8
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- black
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- pip:
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- streamlit
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@ -0,0 +1,4 @@
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numpy
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matplotlib
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streamlit
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scipy
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