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The EcomPricer traffic and checkout monitoring dashboards are free to use. Use of the EcomPricer price optimization functionality is not free, but has been designed so that when the system is at least 90% confident it has generated a profit for the client over a billing period it will charge 30% of the increase in profit the system has generated. The code used to determine the increase in profit is detailed below:
# -*- coding: utf-8 -*- ''' Created on Fri Jul 29 13:32:02 2022 @author: Chris D'Arcy This is the production billing logic. Billing logic is subject to change but will always be open for viewing. While great pains have been taken to be as accurate as possible, we are always open to feedback, particularly with regards to prior-selection. The language used in the comment code has been intentionally chosen with a focus on readability. The price optimization code assigns customers randomly to two groups, an null group where no optimization is performed, and an experimental group where optimization is performed. In this way we can compare the profit per customer (where customer is defined as any user that visits the client site) between the two groups. Once the comparison has been made, we then seek to predict the probability the system is responsible for the increase in profit between the groups, and then return the difference in total profit the system is responsible for if it did so. At a very high level, this code performs two operations: -Estimates the probability that the experimental group generates more returns per customer than the control group -If the aforementioned probability is over 90%, returns the difference in profit per customer observed between the two groups Estimating the probability that the experimental group generates more returns per customer than the control group is undertaken using a bayesian approach, wherein estimates of the underlying probability of given profit-per-click values, P(ppc|data), are directly estimated using a Beta prior, with prior parameters set according to the variance and mean of a seperate data-set used to estimate the prior. ''' import numpy as np from scipy.special import gammaln def log_probability_density(theta,n,k,alpha,beta): ''' Non-normalized log probability density of given conversion rate(s), theta, having been the true conversion rate given n customers making k purchases with prior distribution parameters alpha and beta. This is the log of Binomial(k;theta,n)*Beta(alpha,beta) where each ditribution is defined from the following NIST sources: Beta(alpha,beta) = https://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm Binomial(k;theta,n) = https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm Parameters ---------- theta : FLOAT ARRAY Array of conversion rate values n : INTEGER number of customers in prior data k : INTEGER ARRAY Array of number of purchases for each product in prior data alpha : FLOAT value of alpha parameter for prior distribution beta : FLOAT value of beta parameter for prior distribution Returns ------- log_probabilities: FLOAT ARRAY Array of log probabilities of each conversion rate from theta ''' log_probabilities = (k+alpha-1)*np.log(theta)+(n-k+beta-1)*np.log(1-theta)-gammaln(k+alpha)+(k+alpha)*np.log(n-k+beta) return log_probabilities def p_dist(n,k,alpha,beta): ''' Returns a discrete approximation of the underlying probability density function of the probability of a particular true conversion rate, sampling 100 points along the interval, with ranges for the unbounded distribution heuristically capped by a CLT derived +/- 1/sqrt(n-k) bound. Parameters ---------- n : INTEGER number of customers in prior data k : INTEGER ARRAY Array of number of purchases for each product in prior data alpha : FLOAT value of alpha parameter for prior distribution beta : FLOAT value of beta parameter for prior distribution Returns ------- p : FLOAT ARRAY Array of associated conversion rate probabilities thetas : FLOAT ARRAY Array of associated conversion rate values ''' N = 100 s0 = max(0.00000001,(k/n)-1/np.sqrt(n-k)) s1 = min(0.99999999,(k/n)+1/np.sqrt(n-k)) thetas = np.arange(s0,s1,(s1-s0)/N) p = np.exp(log_probability_density(thetas,n,k,alpha,beta)); p = p/np.sum(p) return p,thetas def r_sample(p): ''' Randomly samples 10000 points from the PMF approximation and returns the number from each value. Parameters ---------- p : FLOAT ARRAY array of probabilities of returning a value equal to the asociated theta. Returns ------- X : FLOAT ARRAY array of number of random samples that fall on each theta. ''' u = np.random.random(size = 10000) u.sort() X =  cdf = np.cumsum(p) for c in cdf: d1 = len(u) u = u[u>c] if u is not None: X.append(d1-len(u)) else: X.append(d1) if len(X) != len(cdf): X.hcat(np.zeros(size=(len(cdf)-len(X)))) return X return X def convoluter(n,k,r,alpha,beta): ''' A basic monte-carlo method for convoluting each products probability distribution. Transforms a series of P_i(X)'s into a number of samples of a single probability distribution, P(sum X = Z). Parameters ---------- n : INTEGER number of customers in prior data k : INTEGER ARRAY Array of number of purchases for each product in prior data r : FLOAT ARRAY Array of profit values for each product sold alpha : FLOAT value of alpha parameter for prior distribution beta : FLOAT value of beta parameter for prior distribution Returns ------- X : FLOAT ARRAY Array of samples of the convoluted probability distribution. ''' X = None for h in range(len(r)): p,thetas = p_dist(n,k[h],alpha,beta) if X is None: rs = r_sample(p) X =  for h2 in range(len(rs)): X += ([thetas[h2]*r[h]]*rs[h2]) X = np.array(X) else: rs = r_sample(p) X2 =  for h2 in range(len(rs)): X2 += ([thetas[h2]*r[h]]*rs[h2]) np.random.shuffle(X2) X = X+X2 return X def prior_parameters(n,k): ''' These parameters set our prior distribution for the returns presented. We use a beta distribution as our prior. The parameters that define the beta distribution, alpha and beta, can be defined via the expectation and variance of the distribution. This code sets the values of alpha and beta based on the desired expectation and variance of the prior distribution. As the expectation of a beta distribution is alpha/(alpha+beta), we can use the expectation of prior data to set E(x) = alpha/(alpha+beta) and thus alpha = beta*E(X)/(1-E(X)). When K is 0, we use the relaxation K_0 = K_0 + 1, meaning E(x) = 1-(1/(G*N)), where G is the number of products in the prior data. When K is equal to N for all K, we use the relaxation K_0 = K_0 + 1, meaning E(x) = 1/(G*N). The variance of a beta distribution is given by: var(x) = (alpha*beta)/((alpha+beta+1)*(alpha+beta)^2) Using this variance formula, and by defining gamma = E(X)/(1-E(X)), we can insert our formula for alpha, alpha = beta*gamma to show: var(x) = (gamma*beta^2)/((beta*(gamma+1)+1)*(beta*(gamma+1))^2) and by factoring out terms: var(x) = gamma/((beta*(gamma+1)+1)*(gamma+1)^2) Reformulating the above we then find: beta = ((gamma/(var(x)*(gamma+1)^2))-1)/(gamma+1) In cases where variance equals 0, we instead use var(x) = (1/sqrt(N)) as would be the case via the application of the CLT to the variance of the mean of a binomial sample. Parameters ---------- n : INTEGER number of customers in prior data k : INTEGER ARRAY Array of number of purchases for each product in prior data Returns ------- alpha : FLOAT value of alpha parameter for prior distribution beta : FLOAT value of beta parameter for prior distribution ''' if np.sum(k)==0: k += 1 E_X = np.mean(k/n) Var_X = np.var(k/n) elif np.mean(k)==n: k -= 1 E_X = np.mean(k/n) Var_X = np.var(k/n) else: E_X = np.mean(k/n) Var_X = np.var(k/n) if Var_X == 0: Var_X = (1/np.sqrt(n)) gamma = E_X/(1-E_X) beta = ((gamma/(Var_X*((gamma+1)**2)))-1)/(gamma+1) alpha = beta*E_X/(1-E_X) return alpha,beta def billing_output(alpha,beta,n_null,r_null,k_null,n_experimental,r_experimental,k_experimental): ''' This function outputs two values. The confidence, which is the probability that the experimental group has an increased profit-per-click (PPC) value, and the expectation, which is the change in profit in the experimental groups PPC value that is due to the experimental-group treatment. If confidence is below the 90% threshold a value of 0 is returned. Parameters ---------- alpha : FLOAT value of alpha parameter for prior distribution beta : FLOAT value of beta parameter for prior distribution n_null : INTEGER number of customers in control data r_null : FLOAT ARRAY Array of profit values for each product sold in control data k_null : INTEGER ARRAY Array of number of purchases for each product in control data n_experimental : INTEGER number of customers in experimental data r_experimental : FLOAT ARRAY Array of profit values for each product sold in experimental data k_experimental : INTEGER ARRAY Array of number of purchases for each product in experimental data Returns ------- confidence : FLOAT Calculated probability profit has been increased. expectation : FLOAT Total increase in PPC attributable to system. ''' X_null = convoluter(n_null,k_null,r_null,alpha,beta) X_experimental = convoluter(n_experimental,k_experimental,r_experimental,alpha,beta) #random sampling of differences in PPC between 10000 simulated customers np.random.shuffle(X_experimental) DX = X_experimental-X_null #proportion of sample which presents a positive change in PPC #i.e. approximate probability a profit has been induced confidence = len(DX[DX>=0].flatten())/len(DX.flatten()) #Expectation of increase in profit associated with the system if confidence>0.9: real_PPC_experimental = np.sum(np.array(r_experimental)*k_experimental)/n_experimental real_PPC_null = np.sum(np.array(r_null)*k_null)/n_null expectation = n_experimental*(real_PPC_experimental-real_PPC_null) else: expectation = 0 return confidence, expectation def demo(): n = 500000 k = np.array([55,140,182,306,210]) alpha,beta = prior_parameters(n,k) n_null = 41000 k_null = np.array([5,10,0,40,0]) r_null = np.array([20,30,40,10,20]) n_experimental = 39000 k_experimental = np.array([2,12,0,60,4]) r_experimental = np.array([20,30,40,11,20]) confidence, expectation = billing_output(alpha,beta,n_null,r_null,k_null,n_experimental,r_experimental,k_experimental) print('Total sales: $'+str(np.sum(k_experimental*r_experimental)+np.sum(k_null*r_null))) print('Total profit increase: $'+str(expectation)) print('Confidence: '+str(confidence)+'%') print('Bill total: $'+str(expectation*0.3)) print('------------') print('Overall earnings without system: $'+str(np.sum(k_experimental*r_experimental)+np.sum(k_null*r_null)-expectation)) print('Overall earnings with system: $'+str(np.sum(k_experimental*r_experimental)+np.sum(k_null*r_null)-expectation*0.3))
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