The defendant was convicted of murder in the first degree under theories of deliberate premeditation and extreme atrocity or cruelty. On appeal, he claims that (1) the affidavit offered in support of the application for the search warrant for his home and car failed to establish probable cause; and (2) pursuant to G. L. c. 278, § 33E, this court should reduce the verdict to murder in the second degree or order a new trial because the evidence of premeditation and extreme atrocity or cruelty was unfairly bolstered by the testimony of the psychiatrists who had examined him. We reject the defendant’s claims as to probable cause. We decline to exercise our power under G. L. c. 278, § 33E, to grant a new trial or reduce the verdict.
*711On October 17, 1997, the partially decomposed body of Elaine Donahue, a nurse and mother of four, was found by police in a fifty-gallon Rubbermaid container that had been stored in a rented storage locker. She had been missing since September 18, 1997, and at the time her body was discovered, she had been dead for approximately one month. Death resulted from at least two severe blows causing multiple skull fractures.
Before the victim’s body was discovered, extensive investigation had led police to suspect her husband, the defendant. On October 17, 1997, police executed a search warrant for the defendant’s home and car. A specially trained “cadaver-sniffing” dog signaled the presence of human remains, blood, body fluid, or decomposition in the closet of the master bedroom and in the basement of the home, as well as in the trunk of the defendant’s car and in a mattress stripped of its stuffing which the defendant had tried to hide in his backyard. Several other pieces of highly incriminating evidence were found, including (1) a large blood stain on the master bedroom window, (2) numerous blood stains on the wall, floor, and headboard in the master bedroom, (3) a bag of mattress stuffing which tested positive for blood, (4) the victim’s purse, (5) a receipt for the purchase of a fifty-gallon Rubbermaid container, and (6) two documents relating to the rental of a storage locker at an E-Z Mini Storage facility in Lynnfield. Based on this discovery, police obtained the search warrant for the storage locker. Inside the locker was the fifty-gallon Rubbermaid container containing the victim’s body. A shoe print in the storage locker matched that of the defendant. Along with the victim’s body, the defendant’s thumb print was found inside the fifty-gallon container.
On October 19, 1997, the defendant spoke from a jail cell at the Reading police department to his sister-in-law. During their conversation, the defendant said he had “just lost it,” and that there was “no excuse or explanation” for what he had done. From the sister-in-law’s testimony at trial, the jury could reasonably have found that the defendant admitted killing his wife while she slept.
1. Motion to suppress. The defendant argues that the affidavit supporting the application for the search warrant for his home and car failed to establish probable cause, and instead merely established that a person was missing. The argument has no merit.
To establish probable cause to search, “[a]n affidavit must *712contain enough information for an issuing .magistrate to determine that the items sought are related to the criminal activity under investigation, and that they reasonably may be expected to be located in the place to be searched at the time the search warrant issues.” Commonwealth v. Cinelli, 389 Mass. 197, 213, cert. denied, 464 U.S. 860 (1983). “[T]he nexus between the items to be seized and the place to be searched need not be based on direct observation . . . [but] . . . [t]he nexus may be found in ‘the type of crime, the nature of the missing items, the extent of the suspect’s opportunity for concealment, and normal inferences as to where a criminal would be likely to hide [evidence of the crime].’ ” Id., quoting United States v. Lucarz, 430 F.2d 1051, 1055 (9th Cir. 1970). See Commonwealth v. Wilson, 427 Mass. 336, 342 (1998). The affidavit need not convince the magistrate beyond a reasonable doubt, but must provide a substantial basis for concluding that evidence connected to the crime will be found on the specified premises. Commonwealth v. Stewart, 358 Mass. 747, 749 (1971). Moreover, affidavits for search warrants should be interpreted “in a commonsense and realistic fashion,” United States v. Ventresca, 380 U.S. 102, 108 (1965); see Commonwealth v. Burt, 393 Mass. 703, 714 (1985), and “read as a whole, not parsed, severed, and subjected to hypercritical analysis.” Commonwealth v. Blake, 413 Mass. 823, 827 (1992). Commonwealth v. Stewart, supra at 751. See United States v. Ventresca, supra. All reasonable inferences which may be drawn from the information in the affidavit may also be considered as to whether probable cause has been established. Commonwealth v. Smith, 370 Mass. 335, 342, cert. denied, 429 U.S. 944 (1976); Commonwealth v. Stewart, supra at 752.
We summarize the information in the affidavit. Elaine Donahue was reported missing by the defendant on September 18, 1997. The police interviewed several of her family members, friends, and coworkers, and learned that she was a very stable, responsible person of the highest moral character who would never have simply abandoned her children, friends, or job. She had no prior history of leaving home without notice, and no history of emotional problems. There was no indication that she was romantically involved with another person. Although the defendant told the police that his wife had gone shopping the last time he had seen her, no corresponding activity was reported by her bank or credit card issuer. When police found her car in *713the parking lot of a local shopping center, they noted that the driver’s seat was pushed to the “far back position,” an unusual position for a woman of Elaine Donahue’s diminutive physical size.
Between September 18, the date of her disappearance, and October 17, the day the search warrant was executed, no one had heard from Elaine Donahue. Police conducted an aerial search for her and found nothing. With cadaver-sniffing dogs, they searched the wooded areas near her home and those next to the parking lot where her car was discovered, but found nothing. Police also requested permission from the defendant on September 20 to conduct a canine search of the interior of his home, but the defendant refused. The media broadcast a picture of Elaine Donahue with a story about her disappearance; flyers were posted and distributed; hotels, hospitals, shelters, transportation service providers, and law enforcement agencies were contacted, all without any response.
Elaine Donahue and the defendant had serious marital problems stemming from the defendant’s addiction to gambling. At one point, the defendant threatened to kill his wife if she ever tried to take their children away from him. The defendant admitted to police that he and his wife were rarely intimate, but had decided to stay together for the sake of their children. Elaine Donahue agreed to stay with the defendant on the condition that she maintain complete control over the family finances. The defendant, an accountant by occupation, found his wife’s total control over the finances to be “frustrating.” He was required to go to her each time he needed money, even for mundane items such as gas, bowling fees, and fast food. The defendant was aware that he was the beneficiary of his wife’s life insurance policies, which had an aggregate face value of $350,000. Years before she disappeared, Elaine Donahue told her best friend, “If anything happens to me, don’t let Ed get away with it.”
Over the' course of their investigation, the police noted behavior by the defendant which, viewed as a whole, caused them to focus on him as a suspect. He failed to answer his own telephone when it rang, notwithstanding the fact that his wife was inexplicably missing and could have been trying to reach him. He cashed checks totaling $900 on the same day that his wife disappeared, despite the fact that his wife did not ordinarily permit him to handle money. On the same day Elaine Donahue *714disappeared, the defendant was seen riding a bicycle near where her car was found. When asked the last time he had been on a bicycle, the defendant replied, “Who can put me on a bicycle?” Unprompted, the defendant called the police in an “anxious” voice to ask when he would be eliminated as a suspect. He later asked police another unprompted question about how suspects in the murder of ten year old Jeffrey Curley recently were apprehended. On October 8, 1997, the police asked the defendant to tell them in what area his wife’s body could be found. The defendant responded that “she’s not in an area.” On October 9, 1997, the defendant again declined consent to a canine search of his home. On October 14, 1997, the police renewed their request to conduct a canine search of the defendant’s home and property. The defendant at first resisted this request, but then consented. Police agreed to conduct the search between 8:15 a.m. and 8:30 a.m. on October 17, 1997, after his children had gone to school. On the day before the scheduled search, the defendant purchased a fifty-gallon Rubbermaid container. Before making his purchase, the defendant asked the clerk whether the store had a larger container he could purchase. At approximately 2 a.m. on October 17, 1997, six hours before the scheduled search, the defendant walked out the rear entry of his house, visually scanned his backyard, then dragged a large mattress stripped of its cloth covering from inside his home to a wooded area at the rear of his property.1
The magistrate could have inferred from the averments in the *715affidavit that (1) Elaine Donahue was the victim of foul play, (2) the defendant had a strong financial incentive to murder his wife and was capable of carrying it out, (3) the defendant knew that his wife was no longer alive and thus was unable to telephone him or control his access to money, (4) the defendant purchased a fifty-gallon Rubbermaid container in order to dispose of his wife’s body, (5) he feared being caught and experienced consciousness of guilt, and (6) the defendant’s mattress contained incriminating evidence which he did not want the police to discover. Such inferences would have been reasonable, and therefore permissible. See Commonwealth v. Taglieri, 378 Mass. 196, 199, cert. denied, 444 U.S. 973 (1979). Taken as a whole and interpreted in a commonsense fashion, the affidavit provides a substantial basis for a magistrate to conclude that Elaine Donahue was the victim of a crime, and that evidence related to that crime would be found in the defendant’s home and car. We conclude that the affidavit established probable cause on which the search warrant properly issued.
2. Psychiatrists’ testimony. The defendant next argues that we should either reduce his sentence or grant a new trial pursuant to G. L. c. 278, § 33E, because the expert testimony by psychiatrists who examined the defendant unfairly bolstered the evidence concerning premeditation and extreme atrocity or cruelty. This argument is also without merit.
The evidence of the defendant’s guilt in this case was overwhelming. The defendant concedes that there was enough evidence to convict him of murder in the second degree with malice aforethought. We also conclude that, even without the psychiatrists’ testimony, the evidence was sufficient to convict the defendant of murder in the first degree under a theory of deliberate premeditation and extreme atrocity or cruelty.2 There was evidence to permit the jury to find that the victim was killed in her bed while she was asleep, and that while she slept the defendant had ample time to reflect before he raised a blunt *716object and struck her head, and again before he struck her at least a second time. Such evidence may support a conviction of murder in the first degree on a theory of deliberate premeditation.3 See Commonwealth v. Skinner, 408 Mass. 88, 92-93 (1990), quoting Commonwealth v. Tucker, 189 Mass. 457, 494-495 (1905) (deliberate premeditation may be “a matter of days, hours, or seconds”).
In the face of such strong evidence, defense counsel had little with which to work. His strategy to pursue an insanity defense based on statements the defendant made to Dr. Keith Ablow, a psychiatrist and expert witness who had interviewed the defendant and reviewed the circumstances of the homicide, was not unreasonable. Although the defendant had no history of mental illness, Dr. Ablow testified that the defendant suffered from a “dissociative disorder” and was radically disconnected from his emotions. He claimed that, as a result of this disorder, the defendant lacked substantial capacity to conform his behavior to the requirements of law, and did not have the ability to form a specific intent to kill. To support his contention, Dr. Ablow recounted statements that the defendant had made to him regarding the circumstances of his wife’s death.4
In rebuttal, the Commonwealth presented expert testimony *717from Dr. Alison Fife, another psychiatrist who had interviewed the defendant and reviewed the record. Based on a series of hypothetical questions, Dr. Fife testified that she did not believe that the defendant’s account of the killing of his wife was consistent with a dissociative disorder.5 She opined that the sole mental disease or defect suffered by the defendant was pathological gambling, and that the defendant’s pathological gambling did not cause him to lack the substantial capacity to conform his behavior to the requirements of the law on September 18, 1997.
The statements offered by both psychiatrists were indeed damaging to the defendant. Before Dr. Ablow testified, however, the judge instructed the jury not to consider the defendant’s statements to the psychiatrist as “estabhshing the truth of any facts contained therein,” and that they were admissible “only as they relate to the basis of the doctor’s opinion of the defendant’s mental condition.” The judge repeated the instruction during Dr. Ablow’s cross-examination,6 and then reminded the jury that the defendant’s statements to any psychiatrist, even his own, *718could not be considered as evidence of premeditation, extreme atrocity or cruelty, or specific intent to kill. The judge gave three similar instructions during the testimony of Dr. Fife. Furthermore, the judge reiterated the limiting instructions in his final instructions to the jury. The jury therefore heard seven limiting instructions regarding the psychiatrists’ testimony. We presume that a jury understand and follow limiting instructions, Commonwealth v. Jackson, 384 Mass. 572, 579 (1981); see Commonwealth v. Kappler, 416 Mass. 574, 576 n.1 (1993), and that the application of such instructions ordinarily renders any potentially prejudicial evidence harmless. Id. In this case we see no reason to conclude otherwise, especially where the experienced trial judge reiterated the limited use of the statements at every conceivable opportunity, to the point of inculcation. Moreover, “[i]n a case raising a defense of insanity, defense counsel is likely to be confronted with difficult decisions concerning trial tactics. Review under G. L. c. 278, § 33E, is not intended to relieve the defendant of the consequences of those decisions absent a substantial risk of a miscarriage of justice.” Commonwealth v. Louraine, 390 Mass. 28, 40 (1983). See Commonwealth v. Wright, 411 Mass. 678, 681-682 (1992) (in considering issues argued on direct appeal as to which the defendant’s appellate rights were not preserved by an objection or in some other manner, this court must apply the standard of G. L. c. 278, § 33E, whether any error created a substantial likelihood of a miscarriage of justice). Here, in the face of overwhelming evidence of his client’s guilt, defense counsel’s pursuit of the insanity defense was not unreasonable, even though the strategy failed. In the absence of any error in admitting the statements, and without the psychiatrists’ testimony, the jury heard abundant evidence to convict the defendant of murder in the first degree on theories of both premeditation and extreme atrocity or cruelty. We see no reason to exercise our power under G. L. c. 278, § 33E, to reduce the verdict to a lesser degree of guilt or to order a new trial.
Judgment affirmed.
The defendant also took a polygraph test, the results of which indicated that he had been extremely deceptive in answering questions as to whether he injured his wife, removed her body, or left her vehicle in the shopping center parking lot. The defendant’s argument that the results of his failed polygraph examination should not have been included in the affidavit consists of two sentences in his brief. We do not consider that to be adequate for appellate consideration. See Mass. R. A. R 16 (a) (4), as amended, 367 Mass. 921 (1975). See Commonwealth v. Angiulo, 415 Mass. 502, 523-524 n.17 (1993); Snow v. E.L. Dauphinais, Inc., 13 Mass. App. Ct. 330, 338 n.8 (1982). We will consider the issue pursuant to G. L. c. 278, § 33E. We have previously stated that evidence which is not admissible for trial purposes may in some instances be permitted in an affidavit. See Commonwealth v. Von Utter, 355 Mass. 597, 599-600 (1969) (hearsay may be used to establish probable cause). While we recognize that polygraph evidence is not ordinarily admissible in criminal trials, Commonwealth v. Stewart, 422 Mass. 385, 389 (1996); Commonwealth v. Mendes, 406 Mass. 201, 208-212 (1989), we conclude without deciding that the polygraph results did not render the affidavit invalid. There is no claim that the polygraph test results were intentionally falsified or stated with reckless disregard as to their accuracy. Compare Commonwealth v. Am*715ral, 407 Mass. 511, 519 (1990); Commonwealth v. Singer, 29 Mass. App. Ct. 708, 711 (1991). There was, in any event, sufficient information to establish probable cause without resort to the polygraph test results.
We have affirmed convictions of murder in the first degree based solely on circumstantial evidence. See, e.g., Commonwealth v. Marquetty, 416 Mass. 445, 452-453 (1993); Commonwealth v. Salim, 399 Mass. 227, 232-233 (1987); Commonwealth v. Smith, 350 Mass. 600 (1966); Commonwealth v. Tucker, 189 Mass. 457 (1905); Commonwealth v. Webster, 5 Cush. 295, 310-320 (1850).
Because the jury were warranted in finding the defendant guilty of murder in the first degree on a theory of deliberate premeditation, we need not address the sufficiency of the evidence on which the defendant also was convicted on the alternate theory of extreme atrocity or cruelty. See Commonwealth v. Chipman, 418 Mass. 262, 270 n.5 (1994).
Dr. Ablow testified to the following: “[Wjhat happens is on that morning he tells me his wife had worked until about [7 p.m.] the night before, had gone to bed at eight. She was sleeping a little bit late. He usually rises early. He was getting the kids ready for school and did so.
“They left. He walked past his office. In the office he noticed a bat. He went in, he picked up the bat. He walked upstairs, standing outside the bedroom. His wife was inside sleeping, in the bedroom.
“He says that for thirty minutes to an hour he stood outside the room with the impulse, you have to do it, the strong thought, the impulse, you have to do it, seizing him at the same time as .he responded, you can’t do it. You have to do it, you can’t do it.
“This builds, and he stands there for thirty minutes. He then goes numb. Hits the wife with the bat, is surprised not only by the event but how much damage he has caused, how bloody it is, says, 7 could not have done that,’ to himself, 7 could not have done that’ and then in a sequence separated perhaps by ten or fifteen minutes, cleans the bat and noticing that his wife is suffering, he delivers an additional or additional blows in order, he says, for her suffer*717ing to end. And he prays to God to take his wife. And that’s what he says happened.”
Dr. Fife’s testimony included the following exchanges:
A.: “[If] there is clear memory [by the defendant of killing his wife], then, there is clear memory, and there’s no amnesia, and there’s no fugue, and there’s no multiple personality. There’s no dissociation. It’s not there. There’s no evidence.”
Q.: “[Assuming again that certain facts are in evidence before you, that the defendant struck his wife in the head with a baseball bat, stood and looked at her body, left the room, went to the basement, cleaned the bat, put the bat away, went back upstairs and discovered that she was alive, went back downstairs and got the baseball bat, went back up to the room, stood over her body and struck her again.
“Could you tell us, with those facts before you in evidence, how that would aid and assist your opinion as to whether the defendant has dissociative disorder or, if not, something else?”
A.: “In my opinion that’s inconsistent with dissociative disorder.”
Q.: “And how so?”
A.: “The behavior is well organized. There is memory for the behavior. The actions are organized, purposeful, toward an end. There’s no disorganization, there’s no ■ — • there’s no out of touch with reality.”
The judge on that occasion added the admonition that the jury pay attention to the instruction because it was “very important.”