In SAS® Enterprise Miner™, the HPBNET procedure uses the Additive Smoothing technique to smooth categorical data. The formula is shown below and you can get details from the following Wikipedia page: https://en.wikipedia.org/wiki/Additive_smoothing.  

Given an observation x = (x₁, …, x_d) from a multinomial distribution with N trials and parameter vector Ө = (θ₁, …, θ_d), a "smoothed" version of the data gives this estimator:

 

Pseudo count  α  ( > 0) is the smoothing parameter. 

 

PROC HPBNET performs Add-one Smoothing (α=1) for the calculation of the conditional probability.  This calculation addresses the zero-frequency problem. Without an adjustment, the conditional probability is zero when there is a zero entry in the contingency table.

Click the Full Code tab to see an example that generates output to demonstrates how Add-one Smoothing is performed in PROC HPBNET.  The conditional probability table network_A is created from a Naive Bayes network using PROC HPBNET.  Also, a PROC FREQ statement is used to create a contingency table between the target variable Pain and the input variable Age for the treatment group A.

 

 

As shown in the table created by the FREQ procedure, there is an entry with a zero-frequency count.  A table with this entry is called a zero-frequency problem, which means that the conditional probability P( Group=1 / Pain = Yes)  is 0.  Add-one Smoothing adjusts the probability so that the estimated probability is not zero.  By setting the pseudo count smoothing parameter to α=1, you can calculate the conditional probabilities as follows:

          P( Group=1 / Pain = Yes) = (0+1)/(5+1*3) = 0.12500

          P( Group=2 / Pain = Yes) = (2+1)/(5+1*3) = 0.37500

          P( Group=3 / Pain = Yes) = (3+1)/(5+1*3) = 0.50000

          P( Group=1 / Pain = No) = (7+1)/(15+1*3) = 0.44444

          P( Group=2 / Pain = No) = (7+1)/(15+1*3) = 0.44444

          P( Group=3 / Pain = No) = (1+1)/(15+1*3) = 0.11111

          P( Pain = Yes) = (5+1) /  (20+1*2) = 0.27273

          P( Pain = No) = (15+1) / (20+1*2) = 0.72727

 

These results are matched to those in the network_A table as shown below:

 

 

 

Operating System and Release Information

Product Family	Product	System	Product Release		SAS Release
			Reported	Fixed*	Reported	Fixed*
SAS System	SAS Enterprise Miner	Microsoft® Windows® for x64	14.2		9.4 TS1M4
		Microsoft Windows 8 Enterprise 32-bit	14.2		9.4 TS1M4
		Microsoft Windows 8 Enterprise x64	14.2		9.4 TS1M4
		Microsoft Windows 8 Pro 32-bit	14.2		9.4 TS1M4
		Microsoft Windows 8 Pro x64	14.2		9.4 TS1M4
		Microsoft Windows 8.1 Enterprise 32-bit	14.2		9.4 TS1M4
		Microsoft Windows 8.1 Enterprise x64	14.2		9.4 TS1M4
		Microsoft Windows 8.1 Pro 32-bit	14.2		9.4 TS1M4
		Microsoft Windows 8.1 Pro x64	14.2		9.4 TS1M4
		Microsoft Windows 10	14.2		9.4 TS1M4
		Microsoft Windows Server 2008	14.2		9.4 TS1M4
		Microsoft Windows Server 2008 R2	14.2		9.4 TS1M4
		Microsoft Windows Server 2008 for x64	14.2		9.4 TS1M4
		Microsoft Windows Server 2012 Datacenter	14.2		9.4 TS1M4
		Microsoft Windows Server 2012 R2 Datacenter	14.2		9.4 TS1M4
		Microsoft Windows Server 2012 R2 Std	14.2		9.4 TS1M4
		Microsoft Windows Server 2012 Std	14.2		9.4 TS1M4
		Windows 7 Enterprise 32 bit	14.2		9.4 TS1M4
		Windows 7 Enterprise x64	14.2		9.4 TS1M4
		Windows 7 Home Premium 32 bit	14.2		9.4 TS1M4
		Windows 7 Home Premium x64	14.2		9.4 TS1M4
		Windows 7 Professional 32 bit	14.2		9.4 TS1M4
		Windows 7 Professional x64	14.2		9.4 TS1M4
		Windows 7 Ultimate 32 bit	14.2		9.4 TS1M4
		Windows 7 Ultimate x64	14.2		9.4 TS1M4
		64-bit Enabled AIX	14.2		9.4 TS1M4
		64-bit Enabled Solaris	14.2		9.4 TS1M4
		HP-UX IPF	14.2		9.4 TS1M4
		Linux for x64	14.2		9.4 TS1M4
		Solaris for x64	14.2		9.4 TS1M4

* For software releases that are not yet generally available, the Fixed Release is the software release in which the problem is planned to be fixed.

The Neuralgia data is available via this page: (http://support.sas.com/documentation/onlinedoc/miner/index.html). Select SAS Enterprise Miner 14.2: High-Performance Procedures ► The HPBNET Procedure ► Getting Started: HPBNET Procedure. The code is used to demonstrate how PROC HPBNET adjusts the conditional probability when a zero-frequency is observed.

---

Data Neuralgia;
input Treatment $ Sex $ Age Duration Pain $ @@;
datalines;
P F 68 1 No B M 74 16 No P F 67 30 No
P M 66 26 Yes B F 67 28 No B F 77 16 No
A F 71 12 No B F 72 50 No B F 76 9 Yes
A M 71 17 Yes A F 63 27 No A F 69 18 Yes
B F 66 12 No A M 62 42 No P F 64 1 Yes
A F 64 17 No P M 74 4 No A F 72 25 No
P M 70 1 Yes B M 66 19 No B M 59 29 No
A F 64 30 No A M 70 28 No A M 69 1 No
B F 78 1 No P M 83 1 Yes B F 69 42 No
B M 75 30 Yes P M 77 29 Yes P F 79 20 Yes
A M 70 12 No A F 69 12 No B F 65 14 No
B M 70 1 No B M 67 23 No A M 76 25 Yes
P M 78 12 Yes B M 77 1 Yes B F 69 24 No
P M 66 4 Yes P F 65 29 No P M 60 26 Yes
A M 78 15 Yes B M 75 21 Yes A F 67 11 No
P F 72 27 No P F 70 13 Yes A M 75 6 Yes
B F 65 7 No P F 68 27 Yes P M 68 11 Yes
P M 67 17 Yes B M 70 22 No A M 65 15 No
P F 67 1 Yes A M 67 10 No P F 72 11 Yes
A F 74 1 No B M 80 21 Yes A F 69 3 No
;

data Trt_A;
  set Neuralgia;
  if Treatment= 'A' ;
run;

ods listing close;
proc hpbnet data=Trt_A numbin=3 structure=Naive maxparents=1 prescreening=0 varselect=0;
   target Pain;
   input Age/level=INT;
   output network=network_A parameter=parameter_A pred=predictions_A;
run;
ods listing;

title "data=Trt_A";
data Trt_A;
  set Trt_A;
  if age < 67.333333 then group= 1;
  else if age <72.666667 then group =2;
  else group= 3;
run;

proc freq data=Trt_A;
   table group*pain / norow nocol nopercent;
run;


title "data=network_A"; 
proc print data=network_A(where=( _TYPE_ eq 'PROBABILITY'))  noobs;
  var  _CHILDCOND_  _PARENTCOND_  _CHILDCONDID_ _VALUE_;
run;

Type:	Usage Note
Priority:

Date Modified:	2017-08-17 09:38:54
Date Created:	2017-08-14 14:52:54