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PubMed1993

A cross-level units-of-analysis approach to individual differences in skill acquisition.

A recent multiple-stage model posits that the individual-difference factors influencing performance vary depending on skill acquisition stage (P. L. Ackerman, 1989, 1990). In the current study, the authors examine the effect of ability in early skill acquisition and extend earlier research by examining the roles of self-efficacy and task familiarity. Furthermore, learning-curve modeling with multilevel models is used to alleviate prior analytical problems. Subjects (N = 115) performed an air traffic control simulation task. Nonlinear learning-curve parameters were estimated for each subject using a negative exponential model (see D. R. Rogosa & J. B. Willett, 1985). Cognitive ability, self-efficacy, and task familiarity were then used to predict learning-curve parameters: learning-rate constant and asymptotic performance. Results revealed that ability, self-efficacy, and familiarity predicted the learning-rate constant, whereas self-efficacy predicted asymptotic performance.

Adult↗

PubMed1997

Two-portal endoscopic carpal tunnel release surgery: report of early experience.

Endoscopic carpal tunnel release is increasingly performed to treat median nerve entrapment neuropathy at the transverse carpal ligament. Proponents of these procedures claim that there are early postoperative advantages to be gained by the patient in the form of decreased pain and weakness, thus facilitating an earlier return to function. However, serious complications associated with the use of these techniques have been reported, especially during the surgeon's purported initial steep learning curve. A prospective analysis of the authors' first 51 cases using a two-portal endoscopic technique was conducted to determine whether these learning curve complications occurred. The authors did experience a learning curve; however, it was not significant. They encountered no serious complications and patient satisfaction was very high. It is concluded that the procedure is relatively easy to learn and safe to perform.

Journal Article↗

PubMed2002

A CUSUM analysis of ruptured abdominal aortic aneurysm repair.

Studies have investigated the role of hospital and surgeon case volume in outcome after ruptured abdominal aortic aneurysm repair (rAAA). Few have analyzed the learning curve of an individual surgeon. The purpose of this study was to analyze this learning curve in reducing morbidity and mortality after rAAA repair. Thirty-two consecutive patients who underwent rAAA repair during the initial 2.5 years of a vascular surgeon's career were reviewed retrospectively. They were divided temporally into two groups of 16 patients (groups 1 and 2). Outcome measures included mortality, postoperative myocardial infarction, stroke, and renal and respiratory failure. Perioperative variables previously associated with increased mortality were analyzed. The cumulative sum (CUSUM) method was used to analyze the learning curve with respect to published acceptable event rates and predetermined 80% alert and 95% alarm boundary lines. Groups 1 and 2 did not differ statistically in age, preoperative blood pressure, hemoglobin or creatinine. There was no difference in transfusion requirements (6.8 +/- 1.2 units vs. 6.4 +/- 1.0 units; p = 0.78), urine output (340 +/- 65 mL vs. 389 +/- 94 mL; p = 0.72) or clamp position. There was no difference in the incidence of postoperative myocardial infarction, stroke, or respiratory or renal failure. Thirty-day mortality in group 2 was 12% as compared to 50% in Group 1 (p = 0.03). On CUSUM analysis, the cumulative failure rate in group 2 progressed lower than the 80% reassurance line, indicating improved results with time. Mortality after rAAA repair decreased over time during an early period of an individual surgeon's career. The CUSUM method is a valuable tool in analyzing an individual surgeon's experience and shows promise in quality control in vascular surgery.

Aged↗

PubMed2001

Laparoscopic radical prostatectomy with the Heilbronn technique: an analysis of the first 180 cases.

PURPOSE: In 1998 Guillonneau and Vallancien introduced laparoscopic radical prostatectomy with primary access to the seminal vesicle. In 1999 we developed a different laparoscopic technique similar to the classic retropubic radical prostatectomy. We focus on early results and the learning curve of the procedure in the first 180 patients. MATERIALS AND METHODS: A transperitoneal approach is used with immediate access to Retzius' space. After the dorsal vein complex is endoscopically sutured, the urethra is incised and distal pedicles of the prostate with or without the neurovascular bundle transected. The apex is then pulled ventrally followed with incision at the bladder neck, and transvesical access to vas deferens and seminal vesicle. After completing the posterior wall of the urethrovesical anastomosis with 5 interrupted endoscopic sutures, the Foley catheter is placed, bladder neck closed and specimen extracted via the umbilical incision. From March 1999 to December 2000 we have performed 180 procedures, including 3 for stage pT1 tumor, 88 pT2, 82 pT3 and 7 pT4. Mean preoperative PSA was 13.3 ng./ml. (range 1.4 to 148), mean specimen weight 37.4 gm. (10 to 125) and median Gleason score 6 (3 to 9). For evaluation of the learning curve a separate analysis of 3 groups with 60 patients in each was done. Differences between groups 1 (first 60 patients) and 3 (last 60) were analyzed for statistically significant differences. RESULTS: Mean operating time was 271 minutes (range 150 to 500) and transfusion rate 31%. The reintervention rate was 4.4% and complication rate 18.8%. Of the patients 92% did not require any analgesia on postoperative day 2. Positive margins were found in 16% of the patients. The rate of positive margins in pT2 tumors was 2.3%, pT3a 15% and pT3b 34%. After a median followup of 12 months (range 3 to 23) in 9 (5%) patients a prostate specific antigen relapse was observed. The anastomosis was tight after removal of the catheter in 83% of patients, with a median time of 7 days (range 5 to 30). An anastomotic stricture had to be treated with laser incision in 3.3% of patients. On discharge from the hospital 33% of patients were continent, after 6 months 74% and after 12 months 97%. Analysis of the learning curve revealed significant differences in operating time (324 versus 265 minutes), conversion rate to open surgery (8.1% versus 1.7%), complication rate (23.3% versus 11.7%) and rate of prolonged catheterization (31.6% versus 10%, respectively), whereas the percentage of positive margins and continence rates showed no influence. CONCLUSIONS: Laparoscopic radical prostatectomy requires significant laparoscopic expertise with an ongoing learning curve. Morbidity is low, oncological control similar to results of open surgery and functional results are promising. The procedure should be performed only at dedicated centers with adequate training and expertise.

Adult↗

PubMed2003

Laparoscopic gastric banding.

INTRODUCTION: Laparoscopic gastric banding (LGB) is currently the most popular purely restrictive bariatric operation in Europe and many other countries. It has a low operative morbidity, but is associated with a substantial late complication rate. Many late complications have been attributed to technical errors or to the learning curve. The aim of this paper is to present our results with gastric banding after the learning curve in order to disclose the true incidence of long-term complications. METHODS: LGB was introduced in our department in December 1995. Thirty patients were operated on until June 1997 using the early banding technique (band within the lesser sac), at which time the surgical technique was slightly modified in order to place the band above the lesser sac. Then another 300 patients underwent LGB using either the Lapband or the SAGB system. This report focuses on the latter patients. All the data were collected prospectively. RESULTS: The series includes 300 patients (257 women and 43 men) with a mean age of 38.3 years (19-64). The mean initial weight was 119.2 kg (57-179), initial body mass index (BMI) was 43.3 kg/m2 (21-64), and initial excess weight was 96.5% (0-191). The mean duration of surgery was 90 min, decreasing over time to a mean of 75 min for the last 50 cases. Early overall morbidity was 6.6%. Major complications occurred in 7 patients (2.3%). Excess weight loss (EWL) was at least 50% in 66% of the patients after 2 years, averaging 60%, with no substantial change until 4 years, and the BMI stabilized between 30 and 31 kg/m2. Forty-nine patients developed a total of 52 long-term complications, of which 23 (7.6%) were related only to the port or catheter. Band erosion occurred in 17 (5.6%), pouch dilatation with slippage in 8 (2.6%), and infection in 4 (1.3%) patients. Fifty-five reoperations were necessary. Twenty-five of these were related only to the port. The band was removed from 26 (9%) patients, of whom 17 were converted to Roux-en-Y gastric bypass. CONCLUSIONS: LGB gives satisfactory results in terms of weight loss in about two-thirds of the patients. Even beyond the learning curve, the long-term morbidity is not negligible, but is acceptable compared to other procedures such as vertical banded gastroplasty. Conversion to gastric bypass is possible when complications occur and can be performed when the band is removed in most cases.

Adult↗

PubMed1999

Effect of the diameter of the endoscope and of surgeon training on the duration and quality of laparoscopic surgery in a rabbit model.

STUDY OBJECTIVE: To evaluate the effect of diameter of the endoscope on the duration and quality of laparoscopic surgery. DESIGN: Prospective, randomized trial (Canadian Task Force classification I). SETTING: Centre for Surgical Technologies. SUBJECTS: Sixty adult, female, New Zealand, white rabbits. INTERVENTION: Two series of laparoscopic nephrectomies, one each performed by an experienced and an inexperienced surgeon comparing 10-, 5-, 4-, and 2-mm endoscopes. MEASUREMENTS AND MAIN RESULTS: Besides duration of surgery and occurrence of bleeding, the quality of dissection was scored by adding scores of dissection of renal vessels, ureters, and kidneys. During consecutive nephrectomies, the duration of surgery (p = 0. 001) and occurrence of bleeding (p = 0.02) decreased, whereas the quality of dissection increased (p = 0.002), demonstrating the learning curve, mainly for the less experienced surgeon. Duration of surgery (p = 0.04) decreased and quality of dissection increased (p = 0.05) when the larger endoscope was used. This was the case only for the less experienced surgeon, whereas for the experienced surgeon it had only a slight effect on learning curve. CONCLUSION: These results confirm a learning curve of nephrectomy consisting of some 20 animals. In addition, the endoscope diameter and thus quality of image affect both duration and quality of surgery, especially for less experienced surgeons.

Animals↗

PubMed2005

Laparoscopic Roux-en-Y gastric bypass: differences in outcome between attendings and assistants of different training backgrounds.

BACKGROUND: Laparoscopic Roux-en-Y gastric bypass (LRYGBP) is associated with a significant learning curve. We hypothesize that differences in surgeon and assistant training backgrounds may significantly impact outcomes during the learning curve. METHODS: Retrospective analysis was performed on patients undergoing LRYGBP at an academic medical center between January 1998 and August 2003. Operations were performed by surgeons with different training backgrounds: without formal laparoscopic fellowship (S1, n=95); immediately following laparoscopic fellowship (S2, n=100); and with extensive laparoscopic experience post fellowship (S3, n=88). First assistants were attendings, fellows, or residents. The variables analyzed included demographics, operative times, estimated blood loss (EBL), rate of conversion, length of stay (LOS), ICU stay, re-operation/re-admission rate, and complications. Results were analyzed by ANOVA and Fisher's exact test. RESULTS: There were significant differences among surgeons of different training backgrounds in EBL, LOS, rate of ICU admission, and intraoperative and late complications rates. Among assistants of different training levels, there were significant differences in operative time, EBL, intraoperative complication rates and re-admission rates. CONCLUSIONS: Differences in training background of the surgeons resulted in significant differences in outcome, including EBL, LOS, ICU admission and intraoperative and late complication rates. Lower assistant training levels significantly impacted efficiency through lengthened operative times and increased EBL, as well as increased intraoperative complication rates and re-admission rates. Our results suggested that participating in a laparoscopic fellowship and operating with a more experienced assistant may improve outcomes during the learning curve.

Adult↗

PubMed2003

Successful transfer of open surgical skills to a laparoscopic environment using a robotic interface: initial experience with laparoscopic radical prostatectomy.

PURPOSE: For a skilled laparoscopic surgeon the learning curve for achieving proficiency with laparoscopic radical prostatectomy (LRP) is estimated at 40 to 60 cases. For the laparoscopically naïve surgeon the curve is estimated at 80 to 100 cases. The development of a robotic interface might significantly shorten the LRP learning curve for an experienced open yet naïve laparoscopic surgeon. To our knowledge we report the initial experience with robot assisted LRP of a surgeon without laparoscopic experience. MATERIALS AND METHODS: Following a 1-day da Vinci (Intuitive Surgical, Mountain View, California) robotic laparoscopic training course and 2 cadaveric robotic LRPs an experienced oncologist (TEA) without laparoscopic experience performed 45 robotic LRPs. RESULTS: All procedures were successfully completed laparoscopically with no rectal injuries or transfusions. The learning curve to 4-hour proficiency was 12 patients and mean operating time subsequently was 3.45 hours (range 2.5 to 5.1). Mean blood loss was 145 cc (range 25 to 350), the mean postoperative day 1 decrease in hemoglobin was 2.6 mg/dl (range 1.9% to 5.1) and mean hospital stay was 36 hours (range 18 to 168). Mean Gleason score was 6.8, mean prostate volume was 50.5 gm (range 12.5 to 163) and the margin positive rate was 35.5%. Four patients (8.8%) had a total of 6 complications, which were managed conservatively. Catheterization time was 7 days (range 7 to 42). Continence (0 pads) was 33% at 1 week, 63% at 1 month and 81% at 3 months. CONCLUSIONS: A laparoscopically naïve yet experienced open surgeon successfully transferred open surgical skills to a laparoscopic environment in 8 to 12 cases using a robotic interface. This outcome is comparable to the reported experience of skilled laparoscopic surgeons after more than 100 LRPs.

Aged↗

PubMed2006

Effects of antiepileptic drugs on learning as assessed by a repeated acquisition of response sequences task in rats.

Patients with epilepsy can have impaired cognitive abilities. Antiepileptic drugs (AEDs) may contribute to the cognitive deficits observed in patients with epilepsy, and have been shown to induce cognitive impairments in healthy individuals. However, there are few systematic data on the effects of AEDs on specific cognitive domains. We have previously demonstrated that a number of AEDs can impair working memory and attention. The purpose of the present study was to evaluate the effects of AEDs on learning as measured by a repeated acquisition of response sequences task in nonepileptic rats. The GABA-related AEDs phenobarbital and chlordiazepoxide significantly disrupted performance by shifting the learning curve to the right and increasing errors, whereas tiagabine and valproate did not. The sodium channel blockers carbamazepine and phenytoin suppressed responding at higher doses, whereas lamotrigine shifted the learning curve to the right and increased errors, and topiramate was without significant effect. Levetiracetam also shifted the learning curve to the right and increased errors. The disruptions produced by triazolam, chlordiazepoxide, lamotrigine, and levetiracetam were qualitatively similar to the effects of the muscarinic cholinergic receptor antagonist scopolamine. The present results indicate that AEDs can impair learning, but there are differences among AEDs in the magnitude of the disruption in nonepileptic rats, with drugs that enhance GABA receptor function and some that block sodium channels producing the most consistent impairment of learning.

Analysis of Variance↗

PubMed2002

Bilateral pneumothoraces, pneumomediastinum, pneumoperitoneum, pneumoretroperitoneum, and subcutaneous emphysema following intubation with a double-lumen endotracheal tube for thoracoscopic anterior spinal release and fusion in a patient with idiopathic scoliosis.

Recently, thoracoscopic approaches to the spine have taken on greater clinical applications in the treatment of spinal deformity with generally good results. However, the steep learning curve must be ascended by the surgeon and may lead to complications early in one's experience. There also exists a learning curve for the anesthesiologist to become adept at obtaining single lung ventilation and managing this throughout the operative procedure. We report a case of an 11-year-old patient with severe scoliosis who developed air in both chest cavities, mediastinum, peritoneum, retroperitoneum, and subcutaneous tissue after intubation with a double-lumen endotracheal tube. The patient remained hemodynamically stable throughout this period, and bilateral chest tubes were placed. The patient remained on the ventilator for 24 hours and was extubated without sequelae. Complications from a thoracoscopic approach to the spine for deformity are most often attributed to the learning curve of the surgeon; however, the entire operative team becomes exposed to the challenges of performing this procedure. This report documents a life-threatening complication of air throughout the chest, mediastinum, abdomen, and subcutaneous tissues in a patient with severe scoliosis.

Child↗

PubMed1999

Surgical experience and complications during endonasal sinus surgery.

OBJECTIVE/HYPOTHESIS: The introduction of optical aids for endonasal sinus surgery has not produced the expected drop in the rate of serious intraoperative complications. STUDY DESIGN/METHODS: 1. Retrospectively, consecutive procedures of different surgeons were analyzed in regard to major complications (periorbital injury, orbital lesion, dural injury, endocranial lesion, damage to the internal carotid artery). The chronological distribution was transformed into a personal learning curve. 2. From our own experience and as surveyors, we analyzed the experiences of surgeons having encountered severe complications and compared them with the above-mentioned learning curve. RESULT: In total, 1,500 operations carried out by five surgeons with 16 serious complications were assessed. For the learning curve, the following stages were defined. stage I: greatest risk of complication, with dural injury (1st to 30th operation); stage II, slighter risk of complication, with frequent periorbital injuries (31st to 180th operation); and stage III, least risk, corresponding to an experienced surgeon. Serious complications occur most frequently among experienced surgeons. CONCLUSION: The beginner enjoys the most effective type of assistance, in the form of personal guidance of an experienced surgeon who is constantly present during the first 30 operations, and who should then be readily available during the next 70 operative procedures. The use of multimedia software appears to be helpful, though its actual value still remains to be determined. The experienced surgeon in particular must be willing to exercise repeated self-criticism to keep his or her rate of complications to a minimum.

Dura Mater↗

PubMed2003

Influence of identification and preservation of pelvic autonomic nerves in rectal cancer surgery on bladder dysfunction after total mesorectal excision.

PURPOSE: Given the improvement in oncologic outcome after the introduction of total mesorectal excision for the treatment of rectal cancer, the objective of the present study was to determine the frequency of identification and preservation of the pelvic autonomic nerves and to identify a possible link between postoperative micturition disturbances and the extent of the radical resection. METHODS: Between March 1997 and December 2001, 150 patients with adenocarcinoma of the rectum (<or=16 cm from the anal verge) underwent surgery, with sphincter preservation in 112 cases (74.7 percent). Sixty-three patients (42 percent) were classified as American Society of Anesthesiologists Stage III and two (1.3 percent) as Stage IV. The number of cases with complete identification, partial identification, or nonidentification of the autonomic nerves (superior hypogastric plexus, hypogastric nerve, and inferior hypogastric plexus) was documented and correlated with micturition disturbances (need for a long-term urinary catheter). Urine volumes were measured by ultrasound before and after surgery. RESULTS: The pelvic autonomic nerves were identified completely in 108 patients (72 percent), partially in 16 (10.7 percent), and not at all in 26 (17.3 percent). After the initial phase of the study (n = 50 patients), complete identification was realized in 78 percent of cases. Multivariate analysis showed that of the predetermined parameters (learning curve for Group I vs. Groups II or III, gender, T stage, blood loss, curative surgery, and previous surgery), gender (P = 0.006), learning curve (P = 0.019), and depth of penetration of the rectal wall (T1/T2 vs. T3/T4; P = 0.028) exerted an independent influence on achievement of complete pelvic nerve identification. Sixteen patients (10.7 percent) were discharged from the hospital with a urinary catheter. Identification and preservation of the pelvic autonomic nerves was associated with low bladder dysfunction rates (4.5 vs. 38.5 percent; P < 0.001). In the evaluation of preoperative and postoperative bladder function, a urologic history and residual urine volume measurements by ultrasound were essential. The information obtained from urodynamic studies was of no relevance. CONCLUSIONS: Identification and preservation of the pelvic autonomic nerves was achieved in the majority of patients and led to the prevention of urinary dysfunction. Gender (P = 0.006), learning curve (P = 0.019), and T stage are independent parameters that influence outcome.

Adenocarcinoma↗

PubMed2006

Evaluation of the new Viewmax laryngoscope in a simulated difficult airway.

BACKGROUND: In this study, we evaluated the learning curve and performance of the Viewmax laryngoscope during simulated difficult laryngoscopy in an intubation manikin (Laerdal Airway Management Trainer). METHODS: To determine the learning curve, 25 anaesthesiologists without previous experience with the Viewmax laryngoscope performed 10 successive intubations in an intubation manikin with a normal airway. Time to intubation and failed intubation attempts were recorded. Another manikin was modified to enable comparison of the Viewmax laryngoscope with Macintosh and McCoy laryngoscopes. The time to intubation, number of failed intubation attempts, modified Cormack and Lehane (MCL) laryngeal view grading, percentage of glottic opening (POGO score), use of gum elastic bougie and subjective rating of degree of difficulty were recorded. RESULTS: The learning curve for the Viewmax laryngoscope showed a progressive decrease in time to successful intubation and reached a plateau at the sixth attempt. In simulated difficult laryngoscopy, the Viewmax laryngoscope demonstrated significantly better laryngeal view than the Macintosh and McCoy laryngoscopes in terms of MCL grading (Macintosh, P = 0.01; McCoy, P < 0.01) and POGO score (Macintosh, P < 0.01; McCoy, P < 0.01). The time required for intubation in simulated difficult laryngoscopy for the Viewmax laryngoscope was significantly longer than that for the Macintosh (P = 0.02) and McCoy (P < 0.01) laryngoscopes. There was no significant difference in the degree of difficulty, number of failed intubations and use of gum elastic bougie. CONCLUSION: When compared with the Macintosh and McCoy laryngoscopes in a manikin, the Viewmax laryngoscope appears to improve the view of the larynx but requires a longer time for tracheal intubation.

Clinical Competence↗

PubMed2005

Consideration of ergonomic aspects in the development of a new endoscopic navigation system.

During the development of new navigation systems, the ergonomic aspects of the accompanying software are seldom taken into account. The VN system is a navigation system working with real and previously stored neuroendoscopic images. The latter enable the module to execute virtual back movement, e.g. in case of bleeding. Several other modules are offered. The aim of the study was the ergonomic analysis of different modules and their learning curves in cadaveric heads. The endoscope was navigated by the neurosurgeon while another person operated the computer. A total of 128 experiments were performed with two software versions. When the landmark tracking module was used, a real learning curve could be observed. By contrast, testing the measurement module did not produce a learning curve. A significant reduction of the time required by the three modules investigated could be observed with software version 2. The module for virtual back movement works best with a minimum time of 20 s for image storage. During machine-human interactions ergonomic software use is important, especially if operative procedures are performed.

Clinical Competence↗

PubMed1999

[On the validity of applying associative learning model to the acquisition process of human contingency judgment].

The assessment process of contingency between two binary events was examined in the present experiments using university students. Some researchers have obtained a learning curve in judging contingency and have thereby applied an associative model to an explanation of human contingency judgment. Other researchers, however, claimed that the task structure did not adequately reflect the structure of 2 x 2 contingency tables and failed to obtain learning curves. After having resolved methodological problems of task structure and procedure (Experiments 1 & 2), we demonstrated little evidence of learning curve in judging contingency (Experiment 3). These results were discussed in terms of associative viewpoints and rule-based models.

Adolescent↗

PubMed1997

Positive effect of surgical experience with implants on second-stage implant survival.

This Dental Implant Clinical Research Group study defined a learning curve for dental implant placement. Implants placed by inexperienced surgeons (< 50 implants) failed twice as often as those placed by experienced surgeons (> or = 50 implants). Implants placed during the first 6, 8, 10, 12, and 16 cases were compared with all others. The greatest difference was seen between the first nine cases and all others (P = .001), with later cases failing significantly less often. Inexperienced surgeons had more failures in the first nine cases (5.9%) than more experienced surgeons (2.4%). Surgeons with little or no previous experience must expect a definite learning curve. Previous experience may transfer and result in a shallower learning curve for subsequent systems.

Clinical Competence↗

PubMed2003

When can I be proficient in laparoscopic surgery? A systematic review of the evidence.

PURPOSE: The aim of this study was to quantify the learning curve in laparoscopic surgery. METHODS: A systematic review of the evidence using a defined search strategy (PubMed, Medline, OVID, Embase, ERIC, Cochrane databases) was performed. Studies without statistical evaluation of the learning curve and opinion articles were excluded. The authors analysed 7 common laparoscopic procedures: cholecystectomy, fundoplication, colectomy, herniorrhaphy, splenectomy, appendicectomy, and pyloromyotomy. The "initial" and "late" stages of experience were compared with regards to the following outcome measures: operating time, conversion rate, complication rate, and length of stay in hospital. RESULTS: A total of 3,641 articles were reviewed, of which, 37 (25,777 patients) fulfilled the entry criteria (5 in children). In all articles, the definition of proficiency was subjective, and the number of operations required to reach it was highly variable. There were improvements in all 4 outcome measures for cholecystectomy, fundoplication, colectomy, herniorrhaphy, and splenectomy between the "initial" and "late" experience. No data were available for the learning curves in appendicectomy or pyloromyotomy. CONCLUSIONS: The number of procedures required to reach proficiency in laparoscopic surgery has not been defined clearly. These findings are important for training, ethical and medico-legal issues.

Appendectomy↗

PubMed2001

Algebraic analysis for nonidentifiable learning machines.

This article clarifies the relation between the learning curve and the algebraic geometrical structure of an unidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously prove that the Bayesian stochastic complexity or the free energy is asymptotically equal to lambda(1) log n - (m(1) - 1) log log n + constant, where n is the number of training samples and lambda(1) and m(1) are the rational number and the natural number, which are determined as the birational invariant values of the singularities in the parameter space. Also we show an algorithm to calculate lambda(1) and m(1) based on the resolution of singularities in algebraic geometry. In regular statistical models, 2lambda(1) is equal to the number of parameters and m(1) = 1, whereas in nonregular models, such as multilayer networks, 2lambda(1) is not larger than the number of parameters and m(1) > or = 1. Since the increase of the stochastic complexity is equal to the learning curve or the generalization error, the nonidentifiable learning machines are better models than the regular ones if Bayesian ensemble learning is applied.

Algorithms↗