WARNER, C.J.

On November 27, 1988, the defendant's four month old daughter, Geneva, was found lying in her bedroom, motionless and gray. She had no pulse and was not breathing. She died that day. An autopsy revealed bruises and abrasions on various parts of her body, some of which were inflicted minutes to hours before her death. There were fractures in her arms and legs, consistent with her body having been shaken as she was held by the chest. Some of these fractures occurred from minutes to days before her death. She also had fractures of five ribs, one to six weeks old, in the process of healing. The infant's skull was fractured, and her brain was swollen and flattened. This injury resulted from a great force applied to the side of her head, as it hit a broad, flat surface. It was inflicted from minutes to hours before her death, and was consistent with the victim's having been held tightly by her ankles and forcefully swung so that the right side of her head struck a large flat object, such as a wall or the headboard of a bed. A jury found the defendant guilty of second degree murder in his daughter's death. We recite additional evidence in the course of our discussion of each of the defendant's claims of error.

1. The motion to dismiss the indictment. The defendant's motion for dismissal of the indictment, claiming that the evidence presented to the grand jury was insufficient for an indictment for murder, was denied. There was no error. In order to return a valid indictment a grand jury must hear "sufficient evidence to establish the identity of the accused . . . and probable cause to arrest him." Commonwealth v. McCarthy, 385 Mass. 160, 163 (1982). Probable cause is "reasonably trustworthy information . . . sufficient to warrant a prudent man in believing that the defendant had committed . . . an offense." Commonwealth v. O'Dell, 392 Mass. 445, 450 (1984).

The grand jury had the requisite probable cause to support an indictment for murder. There was the following testimony: the victim had no bruises when Suzanne Azar, the defendant's wife and mother of the victim, left home at around 8:15 A.M. on the day that she died; the victim suffered multiple injuries before she died; the defendant admitted that he was alone with the victim when she died; the defendant said to Judith Whyte, his wife's sister, when she arrived at the home, "Oh my god, I killed my baby, oh my God, I killed my baby." The defendant said to Suzanne Azar, "I'm sorry, it's my fault. I'm sorry."

In addition, an expert witness testified that the skull fracture was caused when the victim's head struck or was struck by a flat object with great force, and that death was caused by blunt force trauma to the head. The autopsy report also showed that the victim had five fractured ribs in various stages of healing and fractures to both arms and legs. The grand jury examined photos of the bruises and skull injury.

Although some of the grand jury testimony was hearsay, an indictment which is based in part or entirely on hearsay may stand as long as the integrity of the grand jury proceedings has not been impaired. Commonwealth v. McGahee, 393 Mass. 743, 746-747 (1985). In this case, the grand jury were clearly informed as to whether the evidence they heard was direct or hearsay testimony. There was no showing that the grand jury had been impaired in any way. Further, there was no showing that the testimony was false or deceptive, see Commonwealth v. Waters, 410 Mass. 224, 232 (1991), or vague.

The fact that the defendant was alone with the victim at the time that she sustained the injuries that caused her death, and his admissions that he killed her, were sufficient to show probable cause that the defendant murdered her. See Commonwealth v. Healy, 393 Mass. 367, 383 (1984). The severity of her injuries tended to show that the defendant acted with extreme atrocity or cruelty and with malice and premeditation. See Commonwealth v. Kane, 388 Mass. 128, 134 (1983); Commonwealth v. Hutchinson, 395 Mass. 568, 577-578 (1985); Commonwealth v. Glass, 401 Mass. 799, 803 (1988); Commonwealth v. Freiberg, 405 Mass. 282, 290 n. 3 (1989); Commonwealth v. Gallagher, 408 Mass. 510, 520-521 (1990).

Without citation to authority, the defendant also argues that he was entitled to a dismissal of the case because he did not have access to the instructions to the grand jury on the requirements of murder and manslaughter in order to ascertain the legal foundation of the grand jury indictment. There is no merit to the contention. A defendant is entitled to "the written or recorded statements of a person who has testified before a grand jury" (emphasis supplied). Mass.R.Crim.P. 14 (a)(1)(B), 378 Mass. 874 (1979). Commonwealth v. Stewart, 365 Mass. 99, 104-106 (1974). Commonwealth v. Liebman, 379 Mass. 671, 674 (1980). Similarly, G.L. c. 221, § 86, and Superior Court Rule 63 (1974) require only the transcription of testimony before the grand jury. Moreover, the defendant made no showing that the grand jury instructions were transcribed.

2. The motion to suppress the defendant's statements. The defendant argues that the judge improperly denied the motion to suppress his statements to the police, both because he was a "suspect" at the time he was questioned and because his statements were not voluntarily given.

Sergeant Bruce Noah, an emergency medical technician and veteran police officer of the Sudbury police department, testified that he did not suspect the defendant of committing any wrongdoing when he went to the Azar home on November 27, 1988, after receiving a call from the home requesting assistance with an infant who was not breathing. State police Trooper James Connolly testified that before going to the Azar home with two other officers to investigate a possible incident of sudden infant death syndrome on November 28, 1988, he and Trooper Gary Walsh had learned from Dr. Joann Richmond that the victim's skull had been fractured, that there were fractures of the extremities and healed rib fractures. At that time, the troopers were not told the cause or manner of the victim's death.

Trooper Connolly testified that he and the other officers had a discussion with the defendant in the living room of the Azar home but found that the conversation was frequently interrupted and, therefore, requested that the defendant come to the Sudbury police station, where they could talk without distraction. The defendant and his father, Mitchell Azar, drove to the station in a car separate from the police.

At the station, three officers and the defendant sat around a table in a conference room while the defendant's father waited outside the room. Trooper Walsh told the defendant that he was not under arrest or in custody and that he could leave at any time.

Troopers Connolly and Walsh testified that the defendant was cooperative, coherent, and sober during the interview, and he appeared to understand what was happening. After approximately twenty minutes, the defendant's father, who had entered the room upon the defendant's request, was becoming visibly upset at the questioning. Trooper Walsh asked to speak to the father outside, at which time he explained that they had questions about the injuries found on the victim. The father asked if he should call an attorney. Trooper Walsh told him that that was up to him and directed him to a telephone.

Troopers Walsh and Connolly testified that, upon returning to the conference room, Trooper Walsh read the defendant his Miranda warnings so that he would know that he was entitled to an attorney. The defendant indicated that he understood his rights. At the end of the recital of the Miranda warnings, the father returned to the room and announced that an attorney was coming. Troopers Walsh and Connolly testified that the defendant acknowledged this information.

After the attorney arrived, the police declined to question the defendant further and the discussion turned to where the defendant would stay for the night. The police said that they would have to make a report of abuse, and that until they completed an investigation, the State Department of Social Services (DSS) would probably separate the other two children of the defendant and his wife from them. Trooper Walsh told the defendant that if he found someplace else to stay, he could probably convince DSS to allow the children to stay at home. It was agreed that the defendant would stay with his father. Trooper Connolly testified that the defendant, his father, and the attorney then left the station, and that he did not arrest the defendant at that time because he did not have probable cause.

According to the defendant, the statements he made before he was read his Miranda warnings should not have been introduced at trial because of the custodial atmosphere of the questioning. He claims further that, even if the Commonwealth successfully established that the defendant was given his Miranda warnings, the statements should have been suppressed because he did not effectively and voluntarily waive his Miranda rights as required by Miranda v. Arizona, 384 U.S. 436, 444 (1966), and Commonwealth v. Day, 387 Mass. 915, 921 (1983), and because his statements were not freely and voluntarily given as required by Commonwealth v. Tavares, 385 Mass. 140, 152, cert. denied, 457 U.S. 1137 (1982).

The judge made the following findings. The police officers did not suspect the defendant of having caused the death when they went to the Azar home to investigate the death on November 28, 1988. When the defendant was requested to go to the station with the police, he reasonably should have been aware that he was not required to go. When the police asked the defendant to go to the station and allowed him to bring his father, they did not suspect that the defendant had killed his child.

Once Mitchell Azar announced that an attorney was on his way, the defendant understood that he was not obligated to talk to the police, and that an attorney was coming and that he could remain silent; however, he continued talking for another twenty minutes. During that time, he chose to continue talking and "affirmative[ly] wished to answer the police questions." The defendant was not under arrest at any time. He fully and willingly waived his right to remain silent, and his cooperation was voluntary and not a result of threats or coercion. The judge found that the fact that the police consented to the defendant's staying at his father's home permitted the inference that he was free to leave at all times during his contact with the police.

While a motion judge's ultimate conclusions in ruling on a motion to suppress are subject to appellate review, his subsidiary findings of fact are accepted absent clear error. Commonwealth v. Robinson, 399 Mass. 209, 215 (1987).

The standards to be applied in determining whether a person is in custody are now familiar, see Commonwealth v. Bryant, 390 Mass. 729, 736 (1984), and were applied correctly in this case. The judge found that after answering questions in his home, the defendant agreed to go to the police station for further questioning. The police readily agreed to allow the defendant's father, on whom he relied heavily, to come, and he and his father drove to the station on their own. The defendant was not in custody "simply because he complie[d] with a request by the police that he accompany them to the station for questioning." Commonwealth v. Corriveau, 396 Mass. 319, 327 (1985). The questioning took place in a library-type conference room in the police station, and was conducted in a nonthreatening manner. It took the form of question and answer, and also narrative from the defendant. Although the investigation may have begun to focus on the defendant, "the fact that the police had focused their investigation on the defendant and desired to speak with him at the police station does not compel the conclusion that they took the defendant into custody." Commonwealth v. Corriveau, supra at 328. Finally, the defendant was explicitly told that he was not required to answer questions, and that he could leave the station at any time. He was in fact allowed to leave the station after the interview. The judge's conclusions that the defendant was not in custody during the questioning at his home and at the police station were fully supported. See Commonwealth v. Mayfield, 398 Mass. 615, 627 (1986); Commonwealth v. Buckley, 410 Mass. 209, 217 (1991); Commonwealth v. Barnes, 20 Mass. App. Ct. 748, 752 (1985). Since the defendant was not in custody, it was not necessary to read him his Miranda rights at all. Trooper Walsh did so out of "abundant caution." Commonwealth v. Bryant, supra at 737-738 n. 8. Commonwealth v. Barnes, supra at 752.

Similarly, there was ample basis for the judge's conclusion that the defendant's statements were freely and voluntarily given. To ensure that a defendant's statements were voluntary, the judge must examine the totality of the relevant circumstances, including the evidence of the defendant's conduct, his physical and mental condition, and the details of the interrogation. Commonwealth v. Parker, 402 Mass. 333, 340-341 (1988). The judge found that the defendant was sober, coherent, and cooperative during the interview at the police station. There was no indication that the defendant had trouble understanding what was going on. He felt safe in answering questions as long as his father was in the vicinity. He did so willingly and with knowledge of his right to remain silent and with a full and willing waiver of that right. See Commonwealth v. Brillante, 399 Mass. 152, 156-157 (1987); Commonwealth v. Callahan, 401 Mass. 627, 629-630 (1988); Commonwealth v. Parker, 402 Mass. at 340.

3. Jury sequestration. There was no abuse of discretion in the judge's denial of the defendant's motion to sequester the jury during trial (they were sequestered during deliberations). The judge conducted an extensive voir dire during jury selection, inquired of the trial jurors before each day's proceedings whether they had read or heard anything about the case and received negative answers, and at the end of each day reminded them not to give attention to out-of-court information. See Commonwealth v. Jackson, 376 Mass. 790, 800 (1978); Commonwealth v. Sperrazza, 379 Mass. 166, 170 (1979); Commonwealth v. Palmariello, 392 Mass. 126, 141-142 (1984); Commonwealth v. Cordle, 412 Mass. 172, 178-180 (1992). "The record in this case contains no indication that the jury members were less than fair and impartial." Commonwealth v. Colon-Cruz, 408 Mass. 533, 551 (1990). "The burden is on the defendant to demonstrate that the jurors were exposed to extraneous information and, in the absence of such evidence, the defendant's claim must fail." Commonwealth v. Cordle, supra at 180.

4. Evidentiary issues. (a) Prior injuries. The defendant argues that the judge erroneously admitted evidence of injuries that the victim suffered before she died because there was no evidence linking him to these injuries and, even if the prosecution did establish such a link, the prejudicial impact of the evidence outweighed its probative value.

"Evidence of prior bad acts is not admissible to show that the defendant has a criminal propensity or is of bad character. Commonwealth v. Hoffer, 375 Mass. 369, 372 (1978). However, '[r]elevant evidence is not rendered inadmissible merely because it indicates that the defendant may have committed an offense other than that for which he is being tried.' Commonwealth v. Young, 382 Mass. 448, 462-463 (1981)." Commonwealth v. Robertson, 408 Mass. 747, 750 (1990). Evidence of prior bad acts may be used to show motive, state of mind, a pattern of conduct, the relationship between a defendant and a victim, or absence of accident. Commonwealth v. Calcagno, 31 Mass. App. Ct. 25, 26-27 (1991). Evidence of the injuries inflicted upon the victim was admissible to show that someone in a custodial relation to her bore her ill will. See Commonwealth v. Labbe, 6 Mass. App. Ct. 73, 76 (1978); Commonwealth v. Greene, 12 Mass. App. Ct. 982 (1981); Commonwealth v. Collins, 26 Mass. App. Ct. 1021, 1022 (1989). See also Estelle v. McGuire, 112 S.Ct. 475, 480 (1991).

In this case, the Commonwealth showed the prior injuries to be linked to the defendant. All of the bruises were connected to the defendant either by his own statements or through the observations of others. The testimony revealed that Suzanne Azar found bruises on the victim on several occasions when she came home from work after the victim had been in the defendant's sole custody. The victim had not been bruised while in Suzanne Azar's care, or in the care of a babysitter. In some instances, the defendant gave inconsistent explanations for the victim's bruises.

The rib fractures revealed during the autopsy were also linked to the defendant. Testimony revealed that the defendant told a police officer that he held the victim with his hands apart cupping her chest. He admitted that he might have held her too tightly and hurt her. Judith Whyte testified that she saw the defendant holding the victim too tightly on more than one occasion. She saw him hold the victim's face tightly against his chest with one arm under her bottom and one arm behind her head and neck, and she was screaming and crying. When she told him that he might be holding her too tightly, he left the room. Suzanne Azar saw the defendant squatting over the victim as she cried, holding her under her arms with his hands apart and around her ribs. Dr. Paul Kleinman, a pediatric radiologist, also testified about the victim's rib fractures. It was his opinion that the fractures were between one and six weeks old, and that they occurred as a result of severe compression of the chest.

The defendant argues that, even if the evidence of prior acts was admissible to show state of mind or absence of accident, the probative value of the evidence was outweighed by the prejudicial effect that it had on the jury. "Questions of relevancy and prejudicial effect are entrusted to the trial judge's discretion and will not be disturbed except for palpable error." Commonwealth v. LaSota, 29 Mass. App. Ct. 15, 24 (1990). There was no abuse of discretion.

The evidence of prior abuse had strong logical connection to the crime charged. It placed the defendant's conduct into a comprehensible context by providing the jury with a picture of the relationship between the defendant and the victim. See Commonwealth v. Calcagno, 31 Mass. App. Ct. 25, 28 (1991), and cases cited. The judge gave the jury careful limiting instructions regarding the rib fractures at the time the evidence was presented, and with respect to all prior injury evidence in his final instructions. He explained that the jury should not consider the acts at all if they did not think that the defendant committed them, and that they should only consider the acts as they related to the defendant's state of mind.

(b) Expert testimony. The defendant argues that the judge abused his discretion in allowing Dr. Joann Richmond to testify beyond the scope of her expertise; that her opinion was based on inadmissible evidence; and that she impermissibly testified as to an ultimate question in the case, the guilt of the defendant. The defendant also claims that Dr. Paul Kleinman's testimony was inadmissible because he did not connect the victim's injuries to the defendant.

We will not reverse the admission of expert testimony absent an abuse of discretion. Commonwealth v. Garabedian, 399 Mass. 304, 310 (1987). Commonwealth v. Pikul, 400 Mass. 550, 553 (1987). Commonwealth v. Johnson, 410 Mass. 199, 202 (1991). The trial judge qualified Dr. Richmond, an associate chief medical examiner in Massachusetts, as an expert in forensic pathology without objection from the defendant. At the time of her testimony, she had performed twenty-seven hundred autopsies, with over two hundred and fifty of them involving the death of children aged from birth to two and one-half years. She had performed approximately sixteen to twenty autopsies involving homicides of small children under two and one-half years old, about half of which involved skull fractures, and had been qualified as an expert in such cases previously. This was the first time that she testified in Massachusetts in an infant skull fracture case.

The defendant argues that Dr. Richmond was unqualified to testify because she was not an expert in "pediatric pathology" (as opposed to forensic pathology). However, the professional specialty of a medical witness "need not be precisely and narrowly related to the medical issues of the case." Letch v. Daniels, 401 Mass. 65, 68 (1987), quoting from Kapp v. Ballantine, 380 Mass. 186, 192-193 n. 7 (1980). There was sufficient foundation to support the judge's finding that Dr. Richmond was qualified to give an opinion in the field of pediatric pathology. The fact that she had proportionately less experience with infant autopsies goes to the weight but not the admissibility of the testimony. Letch v. Daniels, 401 Mass. at 69. Commonwealth v. Siano, 4 Mass. App. Ct. 245, 249 (1976).

The defendant also contends that Dr. Richmond was unqualified to testify with respect to the victim's being a battered child. The defendant did not, however, object to this aspect of the doctor's testimony. "[Battered child] syndrome has come to be a well recognized medical diagnosis, dependent on inferences, not a matter of common knowledge, but within the area of expertise of physicians whose familiarity with numerous instances of injuries accidentally caused qualifies them to express with reasonable probability that a particular injury or group of injuries is not accidental or is not consistent with the explanation offered therefor but is instead the result of physical abuse by a person of mature strength." Commonwealth v. Day, 409 Mass. 719, 724 (1991), quoting from Commonwealth v. Labbe, 6 Mass. App. Ct. at 77. Dr. Richmond's broad experience in doing autopsies on children qualified her to testify on battered child syndrome. There was no abuse of discretion, much less a substantial risk of a miscarriage of justice.

The defendant next complains without specification that Dr. Richmond's testimony was based on inadmissible evidence. The doctor simply gave her opinion as to the manner in which the victim's injuries were inflicted within a reasonable degree of medical certainty. She said that "the child was held tightly around the legs above the ankles where the skin was squeezed by two fingers causing the star-shaped bruise, and then the child was swung with a force to strike an object on the right side of her head."

Lastly, the defendant claims that Dr. Richmond's testimony impermissibly touched on an ultimate issue in the case by offering an opinion as to whether the defendant was guilty. The doctor testified as to the autopsy that she performed, at which time she made the observation of the bruise on the victim's ankle, which in her opinion was inflicted within minutes to hours of death by squeezing or pinching the skin's top layers, and the head injury that caused the victim's death. She did not, however, express an opinion as to who inflicted the injuries on the victim. There was no error. See Commonwealth v. LaCorte, 373 Mass. 700, 705 (1977); Simon v. Solomon, 385 Mass. 91, 105 (1982); Commonwealth v. Pikul, 400 Mass. at 554.

There is no merit to the defendant's claims that Dr. Paul Kleinman made a speculative hypothesis that the victim had been shaken into unconsciousness on the day of her death and that his opinion was not based solidly on the facts in the record. The record shows that Dr. Kleinman gave his opinion to a reasonable degree of medical certainty that the fractures in the victim's arms, legs, and ribs, some of which were inflicted closer to the time of her death than others, were consistent with her chest having been compressed and her body having been shaken or severe force applied to her extremities. His testimony was based on the X-rays of the victim's body, on his experience as a pediatric radiologist, and his research in the area of child abuse.

(c) The defendant's silence. Contrary to the defendant's apparent argument on appeal, there was no evidence of the defendant's silence in the face of either pre- or postarrest police questioning. See Commonwealth v. Haas, 373 Mass. 545, 559-560 (1977), S.C., 398 Mass. 806 (1986); Commonwealth v. Nickerson, 386 Mass. 54, 57-62 (1982); Commonwealth v. Brown, 11 Mass. App. Ct. 288, 295-297 (1981). The only testimony as to the defendant's silence which remained in the case was that of Suzanne Azar. She testified that the defendant offered no explanation of the victim's death during discussions with Suzanne and others in the defendant and Suzanne's home on the evening of the death. The defendant did not object or move to strike the testimony or ask for limiting instructions. There was no substantial risk of a miscarriage of justice. The jury could have considered the situation to be one where the defendant would naturally be expected to offer an explanation of the circumstances of the death. See Commonwealth v. Nickerson, supra at 57, 61-62 n. 6. "Declarations or acts, or omissions to speak or to act when it would have been natural to do so if the fact were as testified to [here, the defendant's testimony that he dropped the victim on a kitchen counter], may be shown by way of contradiction or impeachment of the testimony of a witness, when they fairly tend to control or qualify his testimony." Langan v. Pianowski, 307 Mass. 149, 151-152 (1940), quoting from Foster v. Worthing, 146 Mass. 607, 608 (1888).

(d) Husband and wife conversations. The defendant argues that the judge erred in excluding certain conversations between the defendant and his wife, Suzanne, and in admitting others. General Laws c. 233, § 20, First (1988 ed.), provides, with exceptions not here material: "[N]either husband nor wife shall testify as to private conversations with the other" (emphasis supplied). The rule is one of disqualification and not of privilege. It depends upon the existence of the marital relationship at the time of the conversation and testimony of the contents of the conversation may not be received even if both spouses wish it to be; the spouses are incompetent to testify. See Gallagher v. Goldstein, 402 Mass. 457, 459-460 (1988); Liacos, Massachusetts Evidence 179-182 (5th ed. 1981). However, if no objection is made, testimony of a private conversation between spouses is admitted for its full probative value. See MacDonald's Case, 277 Mass. 418, 422 (1931); Commonwealth v. Stokes, 374 Mass. 583, 595 n. 8 (1978); Bak v. Bak, 24 Mass. App. Ct. 608, 611 n. 3 (1987).

In answer to the defendant's arguments, it is enough to say that (1) the conversations which he sought to admit were private and thus disqualified; (2) the testimony of Suzanne with respect to an incident in a beauty salon related to a conversation between an employee and the defendant; (3) there was a sufficient foundation laid to establish that the conversation between the defendant and Suzanne in their store following the victim's death was not private, see Linnell v. Linnell, 249 Mass. 51, 54 (1924); and (4) the defendant did not object to the last conversation, which appears to have been private.

5. The defendant's motions for a required finding of not guilty. The defendant's motions for a required finding of not guilty made at the close of the Commonwealth's case, at the close of all the evidence and after the jury verdict were denied. There was no error.

We assess the sufficiency of the evidence at the time of each motion. "In reviewing the denial of motions for [required findings of not guilty] in criminal cases, we have frequently said that 'we must consider and determine whether the evidence, in its light most favorable to the Commonwealth, notwithstanding the contrary evidence presented by the defendant [see Commonwealth v. Amazeen, 375 Mass. 73, 80 n. 5 (1978); Commonwealth v. Hastings, 22 Mass. App. Ct. 930, 931 (1986); Commonwealth v. Turner, 28 Mass. App. Ct. 909, 910 n. 3 (1989)] is sufficient . . . to permit the jury to infer the existence of the essential elements of the crime charged. . . .'" Brown v. Commonwealth, 407 Mass. 84, 85 (1990), quoting from Commonwealth v. Latimore, 378 Mass. 671, 676-677 (1979). "Further, '[c]ircumstantial evidence may be a thoroughly satisfactory basis for the conviction of the highest crimes.'" Id., quoting from Commonwealth v. Richmond, 207 Mass. 240, 246 (1911). "An inference drawn from circumstantial evidence 'need only be reasonable and possible; it need not be necessary or inescapable.'" Commonwealth v. Merola, 405 Mass. 529, 533 (1989), quoting from Commonwealth v. Beckett, 373 Mass. 329, 341 (1977).

Under these principles, the evidence was sufficient to sustain the jury's verdict of second degree murder. The defendant was the victim's sole caretaker on Wednesdays and Sundays from late September, 1988, until her death on November 27, 1988. Bruises were discovered on her body at least five times after she had been under his care. The defendant admitted having inflicted at least one of those injuries. During this period he stated that he had a "personality problem" with the infant.

The day of the victim's death, the defendant was alone with his children from 8:15 A.M. until noon. At noon, the victim was found lying unresponsive in her bedroom. She died at 1:30 that afternoon. An expert testified that a fatal blow to her head had been inflicted minutes to hours before she died. She had a star-shaped bruise on her ankle, fractures at the ends of her leg bones, massive bruises and hemorrhaging on the right side of her head, brain swelling, and a fractured skull. These injuries were consistent with someone having held her tightly by the ankles and swinging her forcefully so that her head struck a broad, flat surface. She also had fractures of five ribs which were from one to six weeks old, and multiple bruises and bone fractures. The victim's injuries, which were in various stages of healing, were characteristic of battered child syndrome. The defendant admitted that he had killed the victim. The day after her death he told his wife the victim had been "her usual shitty self" on the day she died.

The evidence supports an inference that the head injury which caused the infant's death resulted from the defendant's swinging her by the ankles so that her head hit a hard object with extreme force. "This conclusion in turn supports an inference of malice which in turn supports the verdict of the jury." Commonwealth v. Kane, 388 Mass. 128, 134 (1983). "The jury could find that according to common experience there was a plain and strong likelihood that death would follow the defendant's blow [to the infant] and could infer that he foresaw serious injury." Commonwealth v. Starling, 382 Mass. 423, 426 (1981).

6. The prosecutor's closing argument. At trial, the defendant testified that the skull fracture the victim suffered occurred when he attempted to use the telephone while he was holding her in his arms. As he reached for the phone, he dropped her and she hit her head on the kitchen counter. During closing argument, over the defendant's objection, the prosecutor attempted to impeach the defendant's credibility by referring to his failure to give this explanation of the death to his family and to the police.

The defendant said nothing to his family soon after the death, the prosecutor argued, because at the time he believed that everyone thought the baby had died of sudden infant death syndrome. The prosecutor thus implied that the defendant later fabricated the story about accidentally dropping the infant. The defendant's failure to tell his family how the victim died was properly in evidence. See part 4(c), supra. The prosecutor's comments contrasting the defendant's response to his family with his subsequent testimony were permissible attacks on his credibility based on evidence presented at trial. See Commonwealth v. Lamrini, 392 Mass. 427, 431 (1984); Commonwealth v. Andrews, 403 Mass. 441, 457-458 (1988); Commonwealth v. Moore, 408 Mass. 117, 127-128 (1990).

It was improper, however, for the prosecutor to suggest that the defendant had been silent when he was questioned by the police about the cause of the victim's death. To the contrary, the defendant had responded to police questioning and his responses were introduced in evidence at trial. See part 4(c), supra. A prosecutor may not misstate the evidence or refer to facts not in evidence. See Commonwealth v. Kozec, 399 Mass. 514, 516 (1987). Nevertheless, the prosecutor's remark could not have harmed the defendant. Immediately after referring to his "silence," she proceeded to recount his answers to the police. Based on those answers, she argued that the defendant's subsequent testimony at trial was not credible. Taken in context, the references to the defendant's "silence" were hyperbolic descriptions of his responses, which might more appropriately be characterized as evasive. The jury would not have been misled by the mischaracterization because they immediately heard the statements themselves. "We can fairly assume . . . some measure of jury sophistication with regard to sorting out hyperbole and speculation." Commonwealth v. Good, 409 Mass. 612, 626 (1991), and cases cited. See Commonwealth v. Kozec, supra at 517.

7. The jury instructions. The defendant claims several errors in the jury instructions. He preserved none of these claims for appeal. We therefore consider whether the instructions created a substantial risk of a miscarriage of justice. Commonwealth v. Freeman, 352 Mass. 556, 563-564 (1967). Commonwealth v. Keevan, 400 Mass. 557, 564-565 (1987).

(a) Instructions on voluntary manslaughter. The defendant asserts that he was entitled to an instruction on voluntary manslaughter because the evidence permitted a finding that the infant victim's crying "caused the defendant to lose control of himself and take his daughter's life." There was no evidence presented of provocation adequate to mitigate the crime of murder to manslaughter. See Commonwealth v. Young, 326 Mass. 597, 601 (1950); Commonwealth v. Walden, 380 Mass. 724, 728 (1980); Commonwealth v. Amaral, 389 Mass. 184, 188-189 (1983); Commonwealth v. Garabedian, 399 Mass. at 313-315.

(b) Instructions on accident. The defendant argues that the judge gave an insufficient instruction on accident. When the issue is fairly raised, a judge is required to instruct the jury that the Commonwealth must prove beyond a reasonable doubt that the killing was not accidental. Commonwealth v. Palmariello, 392 Mass. 126, 145 (1984). Commonwealth v. Ferguson, 30 Mass. App. Ct. 580, 583 (1991). The defendant requested an accident instruction. The judge instructed the jury that "the Government must prove beyond a reasonable doubt that the death didn't result from some kind of accident as opposed to intentional act, and the consequences that a reasonable person would understand to come from the intentional act." The instruction was adequate. "The judge . . . was not required to instruct the jury in the terms urged by the defendant, so long as he covered adequately the substance of the requested instructions." Commonwealth v. Lowe, 391 Mass. 97, 109 (1984).

(c) Prior bad acts instructions. The defendant contends that the judge erred in failing to instruct the jury that prior bad acts of the defendant, which could be considered as evidence of the defendant's state of mind, had to be proved beyond a reasonable doubt. Likewise, he claims that the judge erroneously failed to instruct the jury that inferential or circumstantial evidence had to be proved beyond a reasonable doubt. As a result, claims the defendant, the jury were permitted to convict on the basis of a preponderance of the evidence. There was no error.

The judge repeatedly instructed the jury that the Commonwealth had to prove every element of the crime beyond a reasonable doubt. He correctly defined reasonable doubt using the language of Commonwealth v. Webster, 5 Cush. 295, 320 (1850). See Commonwealth v. Pires, 389 Mass. 657, 664 (1983). There is no requirement that evidence of prior bad acts or that each inference made must be proved beyond a reasonable doubt. "Every necessary element of the crime must be proved beyond a reasonable doubt, but it does not follow that every piece of evidence must be admissible beyond a reasonable doubt." Commonwealth v. Polian, 288 Mass. 494, 498 (1934). See Commonwealth v. Helfant, 398 Mass. 214, 226 n. 9 (1986). Inferences drawn from circumstantial evidence need not be necessary, but only "reasonable and possible." Brown v. Commonwealth, 407 Mass. 84, 89 (1990). Commonwealth v. Arias, 29 Mass. App. Ct. 613, 618 (1990), S.C., 410 Mass. 1005 (1991).

There is no merit to the defendant's further claim that the judge's instructions concerning prior acts either shifted the burden of proof to the defendant or did not clarify who had that burden. The instruction the defendant refers to concerned the limited use to which the jury could put evidence of the defendant's prior acts, not to the burden of proof. The judge did instruct the jury that they could draw an inference or conclusion favoring the government from a set of facts only if that inference was stronger than other inferences which did not favor the government.

The instructions of which the defendant now complains created no substantial risk of a miscarriage of justice.

Judgment affirmed.