Fig 3. Bayesian repertoire overlap consistently estimates true overlap

Figure 3:

Repertoires with true overlaps ranging from 0 to 60 were subsampled in simulations. As sampling rates increase from $n_a = n_b = 30$ (left) to 40 (middle) and to 50 (right), the estimates of BRO (colored circles) approach the true values (dotted lines) symmetrically. Estimates from $\overset{\circ}{S}$ (crosses) approach the true values from below, systematically underestimating the true overlap. This bias is worse with lower sampling rates [7]. Similar results are found when $n_a \ne n_b$, and when the total repertoire sizes are different from each other (S1 Fig).

Link to the published figure:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
from scipy.stats import hypergeom
from scipy.stats import binned_statistic as binsta
from scipy.special import logsumexp
from util import *
import palettable as pal
clrx = pal.cartocolors.qualitative.Prism_10.mpl_colors
clr = tuple(x for n,x in enumerate(clrx) if n in [1,2,4,5,6])
clr2 = pal.cartocolors.sequential.agSunset_7.mpl_colors
import matplotlib.gridspec as gridspec
import matplotlib.patches as patches

import plotly.graph_objects as go
import as tls
from plotly.offline import plot, iplot, init_notebook_mode
from IPython.core.display import display, HTML
init_notebook_mode(connected = True)
config={'showLink': False, 'displayModeBar': False}

# CCP, the Coupon Collector's Problem
def ccp_sample(c,pool=60):
    return len(set(np.random.choice(pool,c)))

# Draw overlap
def nab_sample(s,na,nb,pool=60):
    sa = np.random.hypergeometric(s,pool-s,na)
    nab = np.random.hypergeometric(sa,pool-sa,nb)
    return nab

# Overlap between two PCRs of depth c and overlap s
def pcr_sample(c,s):
    na = ccp_sample(c)
    nb = ccp_sample(c)
    return nab_sample(s,na,nb),na,nb

# Draw na and nb samples from two populations of size pool_a and pool_b, with true overlap s
# and return empirical overlap between na and nb
# note that this is basically the same as nab_sample, but with two different size pools!
def nab_sample_unequal(s,na,nb,pool_a,pool_b):
    sa = np.random.hypergeometric(s,pool_a-s,na)
    nab = np.random.hypergeometric(sa,pool_b-sa,nb)
    return nab

def p_ccp(c, pool=60):
    p = np.zeros([c+1,pool+1])
    p[0,0] = 1;
    for row in range(1,c+1):
        for k in range(1,np.min([row+2,pool+1])):
            p[row,k] = p[row-1,k]*k/pool + p[row-1,k-1]*(1-(k-1)/pool)
    return p[-1,:]

def p_overlap(na,nb,nab,pool=60):
    p_s = np.zeros(pool+1)
    # reference: hypergeom.pmf(outcome, Total, hits, Draws, loc=0)
    for s in np.arange(pool+1):
        # p_sa is the probability that we'd get sa from the overlap (s), just in na draws of a
        p_sa = hypergeom.pmf(np.arange(pool+1),pool,s,na)
        # p_nab_given_sa is the probability of getting that nab, given sa
        p_nab_given_sa = hypergeom.pmf(nab,pool,np.arange(pool+1),nb)
        p_s[s] =,p_nab_given_sa)
    return p_s/np.sum(p_s)

def e_overlap(na,nb,nab,pool=60):
    p_s = p_overlap(na,nb,nab,pool=pool)

def credible_interval(na,nb,nab,pct=90,pool=60):
    p_s = p_overlap(na,nb,nab,pool=pool)
    cdf = np.cumsum(p_s)
    ccdf = np.flipud(np.cumsum(np.flipud(p_s)))
    # adjust for fractions vs percents; put everything as a fraction
    if pct > 1:
        pct = pct/100
    cutoff = (1-pct)/2
    # get the lower bound. 
    # it's the first index at which cdf ≥ cutoff
        lower = np.where(cdf >= cutoff)[0][0]
    except IndexError:
        lower = 0
    # get the upper bound
    # it's the first index at which ccdf ≥ 0.05
        upper = np.where(ccdf >= cutoff)[0][-1]
    except IndexError:
    expectation =,p_s)
    # Sanity and indexing check: uncomment this line to see true tail probability ≤ 0.05
    # print([cdf[lower-1],(1-ccdf[upper+1])])
    return lower,expectation,upper

def p_nab_given_c(s,c,pool=60):
    pna = p_ccp(c)
    pnb = p_ccp(c)
    nas = np.arange(1,len(pna))
    nbs = np.arange(1,len(pnb))
    p_gen = np.zeros([pool+1,pool+1,pool+1])
    for na in nas:
        p_sa = hypergeom.pmf(np.arange(pool+1),pool,s,na)
        for nb in nbs:
            pna_pnb = pna[na] * pnb[nb]
            for nab in range(0,np.minimum(na,nb)):
                p_nab_given_sa = hypergeom.pmf(nab,pool,np.arange(pool+1),nb)
                p_nab_given_s =,p_nab_given_sa)
                p_gen[na,nb,nab] = p_nab_given_s * pna_pnb
    return p_gen

def p_shat_given_sc(s,c,shat,pool=60):
    masses = p_nab_given_c(s,c,pool=pool)
    if np.sum(masses)<0.99:
        print('Swapping to Monte Carlo')
        return p_shat_given_sc_montecarlo(s,c,shat,pool=pool)
    hist = binsta(np.ravel(shat),np.ravel(masses),statistic='sum',bins=(np.arange(pool+2)-0.5))
    return hist

def p_shat_given_sc_montecarlo(s,c,shat,pool=60,n_mc=int(1e5)):
    masses = np.zeros([pool+1,pool+1,pool+1])
    for ii in range(n_mc):
        nab,na,nb = pcr_sample(c,s)
        masses[na,nb,nab] += 1
    hist = binsta(np.ravel(shat),np.ravel(masses/n_mc),statistic='sum',bins=(np.arange(pool+2)-0.5))
    return hist

def compute_all_estimates(pool=60):
    shat = np.zeros([pool+1,pool+1,pool+1])
    for na in range(1,pool+1):
        for nb in range(1,pool+1):
            for nab in range(0,np.minimum(na+1,nb+1)):
                shat[na,nb,nab] = e_overlap(na,nb,nab,pool=pool)
    return shat

def p_overlap_unequal(na,nb,nab,pool_a,pool_b):
    # all loops are in terms of pool_a, which is assumed to be ≤ pool_b. 
    p_s = np.zeros(pool_a+1)
    # reference: hypergeom.pmf(outcome, Total, hits, Draws, loc=0)
    for s in np.arange(pool_a+1):
        # p_sa is the probability that we'd get sa from the overlap (s), just in na draws of a
        p_sa = hypergeom.pmf(np.arange(pool_a+1),pool_a,s,na)
        # p_nab_given_sa is the probability of getting that nab, given sa
        p_nab_given_sa = hypergeom.pmf(nab,pool_b,np.arange(pool_a+1),nb)
        p_s[s] =,p_nab_given_sa)
    return p_s/np.sum(p_s)

def e_overlap_unequal(na,nb,nab,pool_a,pool_b):
    # TODO. Code expects that pool_b > pool_a...
    p_s = p_overlap_unequal(na,nb,nab,pool_a,pool_b)

# shat = compute_all_estimates(pool=60)
shat = np.load('shat_60.npy')

shat = np.load('shat_60.npy')
recovered = np.zeros([6,reps,pool+1])
planted = np.zeros([6,reps,pool+1])
recovered_pts = np.zeros([6,reps,pool+1])
nas = [30,40,50]
for idxa,na in enumerate(nas):
    nb = na
    for s in np.arange(0,pool+1,1):
        for rep in range(3):
            nab = nab_sample(s,na,nb)
            planted[idxa,rep,s] = s
            recovered[idxa,rep,s] = e_overlap(na,nb,nab)
            recovered_pts[idxa,rep,s] = 2*nab/(na+nb)
z = np.zeros(60)
d = np.arange(1,60)
fig3 = go.Figure()

colors = ['rgba(129,105,177,0.85)', 'rgba(172,110,191,0.85)','rgba(215,128,193,0.85)']  

planted1 = np.zeros((4,244))
recovered1 = np.zeros((4,244))
recovered_pts1 = np.zeros((4,244))
for index in range(0, 3):
    tmp1 = planted[index,:,:]
    tmp2 = recovered[index,:,:]
    tmp3 = recovered_pts[index,:,:]
    planted1[index] = tmp1.flatten()
    recovered1[index] = tmp2.flatten()
    recovered_pts1[index] = tmp3.flatten()

fig3 = go.Figure()
fig3.add_trace(go.Scatter(x = d , y = d, line = dict(width=1, dash='dash', color = 'rgb(0,0,0)'), name = 'Reference', showlegend = False))

fig3.add_trace(go.Scatter(x=planted1[0], y=recovered1[0],
                    name = 'BRO',
fig3.add_trace(go.Scatter(x=planted1[0], y=pool*recovered_pts1[0],
                    name = 'S',
                    marker_symbol = "x",
for idxa in [1,2]:
    fig3.add_trace(go.Scatter(x=planted1[idxa], y=recovered1[idxa],
                    name = 'BRO',
                    visible = False,
    fig3.add_trace(go.Scatter(x=planted1[idxa], y=pool*recovered_pts1[idxa],
                    name = 'S',
                    visible = False,
                    marker_symbol = "x",

                dict(label="30 samples",
                     args=[{"visible": [True,True, True, False, False, False, False]},
                           {"title": ""}]),
                dict(label="40 samples",
                     args=[{"visible": [True,False, False, True, True, False, False]},
                           {"title": ""
                dict(label="50 samples",
                     args=[{"visible": [True,False, False,False,False, True, True]},
                           {"title": ""}]),
                        y = 1.2

fig3.update_layout(plot_bgcolor='rgb(255,255,255)',xaxis_title="True overlap s",
    yaxis_title='Estimated overlap \u015D', legend=dict(x=.85, y=.1))
fig3.update_xaxes(ticks = 'outside', showline=True, linecolor='black')
fig3.update_yaxes(ticks = 'outside', showline=True, linecolor='black')

# Plot figure
plot(fig3, filename = 'plotly_figures/fig3.html', config = config)