2.2 MiB
Borrowing of strength in Bayesian hierarchical model¶
We consider the problem of estimating the average number of patients admitted to ER per day from various hospitals across a city. A Bayesian hierarchical model (BHM) allows us to partially pool information across hospitals, reducing variance in small-sample groups and leveraging strength across hospitals to estimate the population-level parameters. Here, we investigate a Poisson- lognormal model.
(a) Generate a dataset where the logarithm of the expected number of admissions per day for hospital $j$, $ln \lambda_j$, comes from a Normal distribution:
$$ ln \lambda_j \sim \mathcal{N}(\mu_0,\,\sigma_0^{2}), \text{ for } j = 1, ..., J $$
The admission counts data for each hospital for day $i$ follow a Poisson distribution:
$$ y_{ij} \mid \lambda_j \sim \text{Poisson}(\lambda_j), \text{ for } i = 1, ..., n_j $$
where each hospital $j$ has a different reporting frequency, which leads to different $n_j$ for each. To model this, you may draw $n_j$ randomly with equal probability from the set $\{52, 24, 12\}$, representing respectively weekly, twice a month and monthly reporting. Do this once, and keep $n_j$ fixed, treating it as known, throughout the exercise.
The DAG for this model is shown in the figure.
Since $\mu_0$ is the typical log-rate across hospitals, we may choose for its prior $$Pr(\mu_0) = \mathcal{N}(\log m, s^2)$$
where $m$ is a plausible choice for the median number of admissions/day, and s controls its spread. Since $\sigma_0$ controls the variability across hospitals, a prior choice that tends to reinforce shrinkage is $Pr(\sigma_0) = \exp(\tau)$.
Fix $m$, $s^2$, $\tau$ to sensible values, and use an MCMC algorithm of your choice to sample from the model and produce posterior distribution on $\{\lambda_j\}^J_{j=1}, \mu _0, \sigma _0$ (a sensible choice might be $J = 10$ hospitals).
from typing import List, Callable, Tuple import corner import emcee import matplotlib.pyplot as plt import numpy as np import pandas as pd from matplotlib.lines import Line2D from matplotlib.patches import Patch from scipy.special import gammaln from scipy.stats import pearsonr np.random.seed(42)
J = 10 reporting_frequency = np.random.choice( [54, 24, 12], J ) print(f"We have {J} hospitals reporting admission data.") for j in range(J): print( f"- Hospital number {j + 1:2} reports admission data " f"{reporting_frequency[j]} times a year." )
We have 10 hospitals reporting admission data. - Hospital number 1 reports admission data 12 times a year. - Hospital number 2 reports admission data 54 times a year. - Hospital number 3 reports admission data 12 times a year. - Hospital number 4 reports admission data 12 times a year. - Hospital number 5 reports admission data 54 times a year. - Hospital number 6 reports admission data 54 times a year. - Hospital number 7 reports admission data 12 times a year. - Hospital number 8 reports admission data 24 times a year. - Hospital number 9 reports admission data 12 times a year. - Hospital number 10 reports admission data 12 times a year.
def get_expected_admission_counts(mu: float, sigma: float, size: int): """Get expected number of admissions to a hospital at 'size' time points. The log of the admission rate comes from a normal distribution with mean 'mu' and standard deviation 'sigma'. The expected number of admissions comes from a Poisson distribution with parameter 'admission_rate'. From the same Poisson distribution, 'size' time points will be sampled. """ log_admission_rate = np.random.normal( mu, sigma ) admission_rate = np.exp(log_admission_rate) admission_counts = np.random.poisson( admission_rate, size=size ) return admission_counts, admission_rate
def get_admission_count_matrix( mu: float, sigma: float ) -> Tuple[pd.DataFrame, List[float]]: data = {} true_lambdas = [] total_array_length = max(reporting_frequency) for j in range(J): # For each hospital data_array_length = reporting_frequency[j] admission_records, admission_rate = get_expected_admission_counts( mu, sigma, size=data_array_length ) true_lambdas.append(admission_rate) padding = np.full(total_array_length - data_array_length, 0, dtype=int) admission_records = np.concatenate([ admission_records, padding ]) data[f"Hospital {j+1}"] = admission_records data = pd.DataFrame(data) data.index.name = "Time point" return data.transpose(), true_lambdas
- Prior for $\mu_0$: we choose $m$ (typical log-rate across hospitals) and $s$ (a measure of the spread of $m$); $Pr(\mu_0) = \mathcal{N}(\log m, s^2)$.
- Prior for $\sigma_0$: $Pr(\sigma_0) = \text{Exponential}(\tau)$
m = 85 s = 0.4 tau = 0.1
true_mu_0 = np.random.normal(np.log(m), s) true_sigma_0 = np.random.exponential(tau) df, true_lambdas = get_admission_count_matrix(mu=true_mu_0, sigma=true_sigma_0) print(df)
Time point 0 1 2 3 4 5 6 7 8 9 ... 44 45 \ Hospital 1 149 147 161 152 144 155 171 161 146 173 ... 0 0 Hospital 2 180 166 160 178 182 147 153 151 154 161 ... 150 152 Hospital 3 174 147 163 149 146 156 144 142 177 166 ... 0 0 Hospital 4 154 178 170 138 178 165 145 167 148 149 ... 0 0 Hospital 5 163 155 156 165 160 160 169 155 183 175 ... 143 168 Hospital 6 163 162 165 161 156 180 136 167 153 151 ... 142 165 Hospital 7 167 165 179 178 164 147 154 161 173 146 ... 0 0 Hospital 8 145 155 174 159 151 176 162 150 164 166 ... 0 0 Hospital 9 183 150 153 150 143 145 165 154 159 146 ... 0 0 Hospital 10 154 167 171 138 181 151 169 148 159 154 ... 0 0 Time point 46 47 48 49 50 51 52 53 Hospital 1 0 0 0 0 0 0 0 0 Hospital 2 160 151 149 150 170 168 165 161 Hospital 3 0 0 0 0 0 0 0 0 Hospital 4 0 0 0 0 0 0 0 0 Hospital 5 141 159 158 164 130 165 176 146 Hospital 6 163 155 176 194 199 178 162 161 Hospital 7 0 0 0 0 0 0 0 0 Hospital 8 0 0 0 0 0 0 0 0 Hospital 9 0 0 0 0 0 0 0 0 Hospital 10 0 0 0 0 0 0 0 0 [10 rows x 54 columns]
We can use AIES (ensemble samplers with affine invariance), introduced by
Goodman and Weare in 2010 and implemented by the emcee python package, to
sample from the joint posterior without analytically deriving it.
We have to write a function that takes in all hyper-parameters (both fixed and
unknown) and returns the posterior value for those parameters; we will then
construct an emcee.EnsembleSampler (giving the number of unknown parameters
and the known ones) and we will finally run MCMC for a certain number of steps.
The sampler will return tuples of sampled parameters.
For analytical convenience, we will use $\ln(Posterior)$, $\ln(\text{Prior})$ and $\ln(\mathcal{L})$.
def log_posterior(parameters, m: float, s: float, tau: float, J: int, df: pd.DataFrame): # lambda_j is an array of length J with the admission rate for each hospital mu_0, sigma_0, *lambdas = parameters # Check parameters that have to be strictly positive if sigma_0 <= 0: # Standard deviation of a Normal distribution return -np.inf if any(l <= 0 for l in lambdas): # Frequency of a Poisson distribution return -np.inf log_prior = 0.0 # Prior for 'mu_0' (normally distributed, parameters 'm' and 's') log_prior += ( -0.5 * np.log(2 * np.pi * s**2) -0.5 * ((mu_0 - np.log(m))**2) / s**2 ) # Prior for 'sigma_0' (exponentially distributed, parameter 'tau') log_prior += ( np.log(tau) - tau * sigma_0 ) log_likelihood = 0.0 # Prior for 'lambdas' (each log-normally distributed, parameters 'mu_0', 'sigma_0') # and likelihood of 'observed_admissions' (Poisson distributed, parameter 'lambdas[j]') # We only iterate once over range(J) for the sake of efficiency for j in range(J): log_prior += ( -np.log(lambdas[j] * sigma_0 * np.sqrt(2 * np.pi)) - ((np.log(lambdas[j]) - mu_0)**2) / (2 * sigma_0**2) ) # Get the observed admission counts for hospital j, # and cast it to a numpy array for efficiency # This is equivalent to (but faster than) a for loop # for i in range(reporting_frequency[j]): # reporting_frequency = df.iloc[j, i] # log_likelihood += (...) observed_admissions = df.iloc[j,:reporting_frequency[j]].to_numpy() log_likelihood += np.sum( observed_admissions * np.log(lambdas[j]) - lambdas[j] - gammaln(observed_admissions+1) ) return log_prior + log_likelihood
def get_samples_from_full_bhm(initial_mu_0: float, initial_sigma_0: float, initial_lambdas: List[float], log_posterior: Callable, m: float, s: float, tau: float, J: int, df: pd.DataFrame): # Data: observed_admissions (from previous synthetic data generation) # Assuming observed_admissions is a pandas df (J * max(reporting_frequency)) # Number of dimensions and walkers ndim = 2 + J # [mu_0, sigma_0, lambda_1, ..., lambda_J] num_walkers = 32 # Initial guess for parameters initial_guess = [ initial_mu_0, initial_sigma_0, *initial_lambdas ] # Initialize walkers in a small ball around the initial guess pos = initial_guess + 1e-4 * np.random.randn(num_walkers, ndim) # Create the sampler sampler = emcee.EnsembleSampler( num_walkers, ndim, log_posterior, args=(m, s, tau, J, df) ) # Run MCMC nsteps = 5000 sampler.run_mcmc(pos, nsteps, progress=True) # Get one sample every 'thin' steps, discarding the burn-in (first 'discard' steps) return sampler.get_chain(discard=1000, thin=10, flat=True) samples = get_samples_from_full_bhm( initial_mu_0=true_mu_0 * 0.9, initial_sigma_0=true_sigma_0 * 1.1, initial_lambdas=[ lambda_j * (0.9 + 0.2 * np.random.random()) for lambda_j in true_lambdas ], log_posterior=log_posterior, m=m, s=s, tau=tau, J=J, df=df )
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# Labels and true values labels = [r'$\mu_0$', r'$\sigma_0$', *[rf'$\lambda_{{{j + 1}}}$' for j in range(J)]] truths = [true_mu_0, true_sigma_0, *true_lambdas] # Create a corner plot fig = corner.corner( samples, labels=labels, truths=truths, quantiles=[0.16, 0.5, 0.84], show_titles=True, title_kwargs={"fontsize": 12} ) plt.show()
(b) Determine the marginal posterior distribution for the admission rate for each hospital, and compare it with the posterior estimate in a model with no pooling (i.e., where each hospital’s rate is inferred exclusively from its observed counts, i.e. $\ln \lambda_j \sim \mathcal{N} (\mu_0, \sigma^2_0)$ and the same priors as above, and the posterior for hospital j comes exclusively from its own data). Check that the posterior means for hospitals with a smaller number of records (i.e., $n_j = 12$) exhibit stronger shrinkage towards the global mean, thus demonstrating borrowing of strength, and typically have 68% HPD credible intervals that are shorter than in the pooling model. You may use a violin plot to make this comparison.
With pooling¶
rows = [] for j in range(J): mu_0_j = samples[:, 0] sigma_0_j = samples[:, 1] lambda_samples_j = np.log(samples[:, 2 + j]) lower, upper = np.percentile(lambda_samples_j, [16, 84]) rows.append({ "Hospital": j + 1, "Reporting frequency": reporting_frequency[j], "log lambda 16th percentile": lower, "log lambda 84th percentile": upper, "HPD width (68%)": upper - lower, "mu_0": np.mean(mu_0_j), "sigma_0": np.mean(sigma_0_j) }) temp = pd.DataFrame(rows) temp = pd.DataFrame(rows).set_index("Hospital") print(temp)
Reporting frequency log lambda 16th percentile \
Hospital
1 12 5.048979
2 54 5.060622
3 12 5.053617
4 12 5.063582
5 54 5.059486
6 54 5.076310
7 12 5.065790
8 24 5.068508
9 12 5.042735
10 12 5.059849
log lambda 84th percentile HPD width (68%) mu_0 sigma_0
Hospital
1 5.076839 0.027860 5.072097 0.015922
2 5.077944 0.017322 5.072097 0.015922
3 5.079737 0.026119 5.072097 0.015922
4 5.087553 0.023971 5.072097 0.015922
5 5.077057 0.017571 5.072097 0.015922
6 5.096485 0.020175 5.072097 0.015922
7 5.091870 0.026080 5.072097 0.015922
8 5.091746 0.023239 5.072097 0.015922
9 5.074290 0.031554 5.072097 0.015922
10 5.084888 0.025039 5.072097 0.015922
x = temp["Reporting frequency"].to_numpy() y = temp["HPD width (68%)"].to_numpy() r, p_value = pearsonr(x, y) r_squared = r**2 print(f"Pearson r = {r:.3f}") print(f"R^2 = {r_squared:.3f}") print(f"p-value = {p_value:.3e}")
Pearson r = -0.878 R^2 = 0.771 p-value = 8.381e-04
# Fit least-squares line slope, intercept = np.polyfit(x, y, 1) x_line = np.linspace(x.min(), x.max(), 100) y_line = slope * x_line + intercept plt.figure() plt.scatter(x, y) plt.plot(x_line, y_line) plt.xlabel("Reporting frequency (n_j)") plt.ylabel("HPD width (68%)") plt.title( f"HPD width vs reporting frequency\n" f"R² = {r_squared:.3f}, r = {r:.2f}, p = {p_value:.3f}" ) plt.show()
Without pooling¶
mu_fixed = np.log(m) sigma_fixed = s def log_posterior_without_pooling(parameters: np.ndarray, mu_0: float, sigma_0: float, j: int, df: pd.DataFrame): lambda_j, *_ = parameters if lambda_j <= 0: return -np.inf # Prior on 'lambda_j' (log-normally distributed, parameters 'mu_0' and 'sigma_0') log_prior = ( -np.log(lambda_j * sigma_0 * np.sqrt(2 * np.pi)) - (np.log(lambda_j) - mu_0)**2 / (2 * sigma_0**2) ) observed_admissions = df.iloc[j,:reporting_frequency[j]].to_numpy() # Likelihood: Poisson log_likelihood = np.sum( observed_admissions * np.log(lambda_j) - lambda_j - gammaln(observed_admissions + 1) ) return log_prior + log_likelihood def get_samples_without_pooling(initial_lambdas: List[float], log_posterior_without_pooling: Callable, mu_fixed: float, sigma_fixed: float, J: int, df: pd.DataFrame): samples_without_pooling = {} for j in range(J): # Number of dimensions and walkers ndim = 1 # Only lambda_j num_walkers = 32 # Initial guess for parameters initial_guess = [ initial_lambdas[j] ] # Initialize walkers in a small ball around the initial guess pos = initial_guess + 1e-4 * np.random.randn(num_walkers, ndim) # Create the sampler sampler = emcee.EnsembleSampler( num_walkers, ndim, log_posterior_without_pooling, args=(mu_fixed, sigma_fixed, j, df) ) # Run MCMC nsteps = 3000 sampler.run_mcmc(pos, nsteps, progress=True) # Get one sample every 'thin' steps, discarding the burn-in (first 'discard' steps) samples_without_pooling[j] = sampler.get_chain(discard=1000, thin=10, flat=True) return samples_without_pooling samples_without_pooling = get_samples_without_pooling( initial_lambdas=[ lambda_j * (0.9 + 0.2 * np.random.random()) for lambda_j in true_lambdas ], log_posterior_without_pooling=log_posterior_without_pooling, mu_fixed=mu_fixed, sigma_fixed=sigma_fixed, J=J, df=df )
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rows = [] for j in range(J): swp = samples_without_pooling[j] lambda_samples_j = np.log(swp[:, 0]) lower, upper = np.percentile(lambda_samples_j, [16, 84]) rows.append({ "Hospital": j + 1, "Reporting frequency": reporting_frequency[j], "log lambda 16th percentile": lower, "log lambda 84th percentile": upper, "HPD width (68%)": upper - lower, "mu_0": np.mean(mu_fixed), "sigma_0": np.mean(sigma_fixed) }) temp = pd.DataFrame(rows) temp = pd.DataFrame(rows).set_index("Hospital") print(temp)
Reporting frequency log lambda 16th percentile \
Hospital
1 12 5.017071
2 54 5.057248
3 12 5.027429
4 12 5.057141
5 54 5.054694
6 54 5.085284
7 12 5.070768
8 24 5.072616
9 12 4.999103
10 12 5.044466
log lambda 84th percentile HPD width (68%) mu_0 sigma_0
Hospital
1 5.063500 0.046429 4.442651 0.4
2 5.079072 0.021824 4.442651 0.4
3 5.072223 0.044794 4.442651 0.4
4 5.102917 0.045777 4.442651 0.4
5 5.076253 0.021559 4.442651 0.4
6 5.106803 0.021519 4.442651 0.4
7 5.116844 0.046076 4.442651 0.4
8 5.104699 0.032083 4.442651 0.4
9 5.046609 0.047507 4.442651 0.4
10 5.089842 0.045376 4.442651 0.4
data = [] positions = [] groups = [] pos, offset, step = 1.0, 0.35, 1.2 for j in range(J): # With pooling data.append(np.log(samples[:, 2 + j])) positions.append(pos) groups.append("bhm") # Without pooling data.append(np.log(samples_without_pooling[j][:, 0])) positions.append(pos + offset) groups.append("nopool") pos += step plt.figure(figsize=(14, 6)) vp = plt.violinplot( data, positions=positions, widths=0.3, showmeans=False, showmedians=True, showextrema=False ) # Color each violin plot according to group (with or without pooling) for body, group in zip(vp["bodies"], groups): body.set_facecolor( { "bhm": "orange", "nopool": "darkblue" }[group] ) body.set_alpha(0.7) vp["cmedians"].set_color("white") vp["cmedians"].set_alpha(1.0) vp["cmedians"].set_linewidth(2.0) # Add ground truth x_true_lambdas = [1 + step * j + offset / 2 for j in range(J)] y_true_lambdas = np.log(true_lambdas) plt.scatter( x_true_lambdas, y_true_lambdas, color="darkgreen", marker="x", s=80, linewidths=2, zorder=100000, label="Ground truth" ) # Add 86% HPD intervals cap_width = 0.06 for i, x in enumerate(positions): lower = np.quantile(data[i], 0.16) higher = np.quantile(data[i], 0.84) plt.hlines( y=[lower, higher], xmin=x - cap_width, xmax=x + cap_width, color="black", linewidth=2, zorder=10 ) plt.vlines( x=x, ymin=lower, ymax=higher, color="black", linewidth=2, zorder=10 ) # Set axes labels and plot title plt.xticks( [1 + step * j + offset/2 for j in range(J)], [f"Hospital {j+1}\n(n={reporting_frequency[j]})" for j in range(J)] ) plt.ylabel(r"$\ln \lambda_j$") plt.title(r"Distribution of $\ln \lambda_y$""\n"r"Full BHM vs w/out pooling") plt.legend( handles=[ Patch(color="orange", label="With pooling"), Patch(color="darkblue", label="No pooling"), Line2D( [0], [0], marker="x", color="darkgreen", linestyle="None", markersize=10, markeredgewidth=2, label="Ground truth" ), Line2D( [0], [0], color="black", markersize=10, markeredgewidth=2, label="68% HPD intervals" ) ], loc="upper right" ) plt.tight_layout() plt.show()
(c) Consider the prior predictive distribution for possible data within the BHM:
$$ Pr( y_{ij} \mid priors) = \int Pr ( y_{ij} \mid \lambda_j) Pr(\lambda_j \mid \mu_0, \sigma_0) Pr (\mu_0,\sigma_0) d \mu_0 d \sigma_0$$
and simulate from it predictions for the possible counts $y_{ij}$ from the model (before you see any data). Evaluate the degree of shrinkage by using as metric the average shrinkage in the standard deviation of the posterior with respect to the no-pooling scenario, i.e.
$$ S = \frac {1} {J} \sum_{j=1}^{J}{1 - \frac{\text{std}_\text{BHM}(\ln \lambda_j \mid \bold y)}{\text{std}_\text{no pool}(\ln \lambda_j \mid \bold y)}}$$
where a smaller S corresponds to higher degree of shrinkage.
Determine which hyperparameter among $(m, s, \tau)$ has the most effect on the degree of shrinkage, and tune each to avoid predictions that are too diffuse (i.e., the predictive spread is unreasonably wide) or too narrow (i.e., the predictive spread is over-constraining).
S = np.mean([ 1 - ( np.std(np.log(samples[:, 2 + j])) / np.std(np.log(samples_without_pooling[j][:, 0])) ) for j in range(J) ]) print( f"The degree of shrinkage S for m = {m}, s = {s}, tau = {tau} is: {S:.3f}" )
The degree of shrinkage S for m = 85, s = 0.4, tau = 0.1 is: 0.291
m = 85 s = 0.4 tau = 0.1 def evaluate_degree_of_shrinkage(m, s, tau): print( f"Evaluating degree of shrinkage S for m = {m}, s = {s}, tau = {tau}..." ) true_mu_0 = np.random.normal(np.log(m), s) true_sigma_0 = np.random.exponential(tau) df, true_lambdas = get_admission_count_matrix( mu=true_mu_0, sigma=true_sigma_0 ) initial_mu_0 = true_mu_0 * 0.9 initial_sigma_0 = true_sigma_0 * 1.1 initial_lambdas = [ lambda_j * (0.9 + 0.2 * np.random.random()) for lambda_j in true_lambdas ] # Samples from joint posterior (full BHM) samples_bhm = get_samples_from_full_bhm( initial_mu_0=initial_mu_0, initial_sigma_0=initial_sigma_0, initial_lambdas=initial_lambdas, log_posterior=log_posterior, m=m, s=s, tau=tau, J=J, df=df ) # Samples from non-pooling model samples_without_pooling = get_samples_without_pooling( initial_lambdas=initial_lambdas, log_posterior_without_pooling=log_posterior_without_pooling, mu_fixed=initial_mu_0, sigma_fixed=initial_sigma_0, J=J, df=df ) S = np.mean([ 1 - ( np.std(np.log(samples_bhm[:, 2 + j])) / np.std(np.log(samples_without_pooling[j][:, 0])) ) for j in range(J) ]) print( f"The degree of shrinkage S for m = {m}, s = {s}, tau = {tau} is: {S:.3f}" ) evaluate_degree_of_shrinkage(m=(1.1 * m), s=s, tau=tau) evaluate_degree_of_shrinkage(m=m, s=(1.1 * s), tau=tau) evaluate_degree_of_shrinkage(m=m, s=s, tau=(1.1 * tau))
Evaluating degree of shrinkage S for m = 93.50000000000001, s = 0.4, tau = 0.1...
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The degree of shrinkage S for m = 93.50000000000001, s = 0.4, tau = 0.1 is: 0.015 Evaluating degree of shrinkage S for m = 85, s = 0.44000000000000006, tau = 0.1...
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The degree of shrinkage S for m = 85, s = 0.44000000000000006, tau = 0.1 is: 0.021 Evaluating degree of shrinkage S for m = 85, s = 0.4, tau = 0.11000000000000001...
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The degree of shrinkage S for m = 85, s = 0.4, tau = 0.11000000000000001 is: -0.003
Tentative interpretation of these results¶
The parameter that affects $S$ more is $\tau$, which controls the variability of lambdas across hospitals. Therefore, the BHM is most useful when the sources being pooled have similar values for the latent variables (in this case, smaller $\tau$), which makes sense.
Also $s$ (the uncertainty about the population mean admission rate) affects $S$ (lower $s$ means more shrinkage).
$m$, the population mean admission rate, affects $S$ indirectly through $\sigma_0$.
To support this conclusion, the following table reports the increase of each parameter and the decrease in shrinkage it produced when comparing the full hierarchical model with the model without pooling across hospitals.
The smallest change in parameters ($\tau + 0.01$) also produced the biggest deterioration in BHM performance, supporting the hypothesis that $\tau$ is the most influent parameter in determining the efficacy of borrowing of strength.
Parameter | 10% increase in parameter | $\Delta S$ | :-----: | :-----------------------: | :--------: | m | 8.5 | 27.6% s | 0.04 | 27.0% $\tau$ | 0.01 | 29.4%