What are the traditional Buddhist arguments for rebirth?

The Wikipedia page on Pramana says:

Buddhism accepts only two pranama (tshad ma) as valid means to knowledge: Pratyaksha (mngon sun tshad ma, perception) and Anumāṇa (rjes dpag tshad ma, inference). Rinbochay adds that Buddhism also considers scriptures as third valid pramana, such as from Buddha and other "valid minds" and "valid persons". This third source of valid knowledge is a form of perception and inference in Buddhist thought. Valid scriptures, valid minds and valid persons are considered in Buddhism as Avisamvadin (mi slu ba, incontrovertible, indisputable).

Perhaps, we can consider this third pramana the Śabda pramana, which according to this page is:

Sabda pramana is verbal testimony. It is also called ‘apta-vakyas’ (statement of a trust-worthy person’, and agama (authentic word).

In this essay entitled "Dhamma Without Rebirth?", Ven. Bhikkhu Bodhi wrote:

The aim of the Buddhist path is liberation from suffering, and the Buddha makes it abundantly clear that the suffering from which liberation is needed is the suffering of bondage to samsara, the round of repeated birth and death. To be sure, the Dhamma does have an aspect which is directly visible and personally verifiable. By direct inspection of our own experience we can see that sorrow, tension, fear and grief always arise from our greed, aversion and ignorance, and thus can be eliminated with the removal of those defilements. The importance of this directly visible side of Dhamma practice cannot be underestimated, as it serves to confirm our confidence in the liberating efficacy of the Buddhist path. However, to downplay the doctrine of rebirth and explain the entire import of the Dhamma as the amelioration of mental suffering through enhanced self-awareness is to deprive the Dhamma of those wider perspectives from which it derives its full breadth and profundity. By doing so one seriously risks reducing it in the end to little more than a sophisticated ancient system of humanistic psychotherapy.

Above, Ven. Bodhi states that part of the Buddha's teachings (Dhamma) is directly visible and personally verifiable, but he implies that the doctrine of rebirth is not. Therefore, this means that pratyaksha pramana is ruled out.

He also wrote in the same essay:

If we suspend our own predilections for the moment and instead go directly to our sources, we come upon the indisputable fact that the Buddha himself taught rebirth and taught it as a basic tenet of his teaching.

In the Maha-Saccaka Sutta (MN36), we have rebirth supported using the Śabda pramana, based on the Buddha's testimony to Saccaka, the Jain:

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of recollecting my past lives. I recollected my manifold past lives, i.e., one birth, two...five, ten...fifty, a hundred, a thousand, a hundred thousand, many eons of cosmic contraction, many eons of cosmic expansion, many eons of cosmic contraction & expansion: 'There I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose there. There too I had such a name, belonged to such a clan, had such an appearance. Such was my food, such my experience of pleasure & pain, such the end of my life. Passing away from that state, I re-arose here.' Thus I remembered my manifold past lives in their modes & details.

"This was the first knowledge I attained in the first watch of the night. Ignorance was destroyed; knowledge arose; darkness was destroyed; light arose — as happens in one who is heedful, ardent, & resolute. But the pleasant feeling that arose in this way did not invade my mind or remain.

"When the mind was thus concentrated, purified, bright, unblemished, rid of defilement, pliant, malleable, steady, & attained to imperturbability, I directed it to the knowledge of the passing away & reappearance of beings. I saw — by means of the divine eye, purified & surpassing the human — beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma: 'These beings — who were endowed with bad conduct of body, speech, & mind, who reviled the noble ones, held wrong views and undertook actions under the influence of wrong views — with the break-up of the body, after death, have re-appeared in the plane of deprivation, the bad destination, the lower realms, in hell. But these beings — who were endowed with good conduct of body, speech & mind, who did not revile the noble ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.' Thus — by means of the divine eye, purified & surpassing the human — I saw beings passing away & re-appearing, and I discerned how they are inferior & superior, beautiful & ugly, fortunate & unfortunate in accordance with their kamma.

The following arguments come from the book "Introduction to Abhidhamma" by Nina van Gorkom, from this chapter. This is likely to be the author's personal opinion that karma and rebirth can be inferred from the fact that sentient beings are born with very different tendencies and inclinations:

In order to understand what causes birth we should know what conditions the nāma and rūpa which arise at the first moment of a new lifespan. The citta which arises at that moment is called the rebirth-consciousness or paṭisandhi-citta. Paṭisandhi means relinking, it “ links” the previous life to the present life. It is usually translated as rebirth-consciousness, but, since there is no person who is reborn, birth-consciousness would be more correct. Since there isn’t any citta which arises without conditions, the paṭisandhi-citta must also have conditions. The paṭisandhi-citta is the first citta of a new life and thus its cause can only be in the past. One may have doubts about past lives, but how can people be so different if there were no past lives? We can see that people are born with different tendencies and talents. Cittas which arise and fall away succeed one another and thus each citta conditions the next one. The last citta of the previous life (dying-consciousness) is immediately succeeded by the first citta of this life, without there being any interval. That is why tendencies one had in the past can continue by way of accumulation from one citta to the next one and from past lives to the present life. Since people accumulated different tendencies in past lives, they are born with different tendencies and inclinations. Rebirth-consciousness is the result of kamma, it is vipākacitta. Our life starts at the moment the paṭisandhi-citta arises together with the rūpa which is at the same time produced by kamma. A lifespan ends when the last citta, the dying-consciousness (cuti-citta) falls away. Kamma produces rūpa not only at the first moment of life but throughout our life.

Apart from the Abhidhamma, it is likely that arguments could be found from Nagarjuna's Mūlamadhyamakakārikā or other Indian Mahayana writings, but I'm not sufficiently familiar with them.

You may also find Ven. Thanissaro's essay "The Truth of Rebirth And Why it Matters for Buddhist Practice" useful. He wrote:

To avoid wrong view — and the ridicule it deserves — the Buddha found it necessary to disclose his knowledge that there are lives after death. And he had to include the perspective not just of one lifetime after death, but of many. ....

This is why the Buddha never claimed to offer proof for either the efficacy of action or for rebirth, for he knew that the evidence for these teachings lay beyond the ken of most of his listeners. ....

These arguments don't prove the efficacy of action or the truth of rebirth, but they do show that it is a safer, more reasonable, and more honorable policy to assume the truth of these teachings than it would be to assume otherwise. The Buddha didn't press these arguments beyond that point.

Also comparing the Buddhist view to the Hindu and Jain views, he wrote:

In discussing rebirth, the Buddha differed from the other schools of the time in that he didn't base his position on a metaphysical view of personal identity — that is, on defining what it is that gets reborn. By placing rebirth in the context of dependent co-arising, he was presenting it in a phenomenological context — i.e., one that focused on phenomena as they can be directly experienced and that refused to take a stand on whether there is a reality of "things" underlying them.

So far as I know, the reasons or arguments include:

• It's Buddha-vacana, i.e. it's understood that the Buddha said so
• It's widespread traditional normal belief, which even pre-dates Buddhism (Buddhism isn't the only religion)

• There are historical events or witnesses (e.g. of young children who display knowledge which they could only have acquired in a previous life), e.g. associated with the recognition of Tulkus:

  In order to accept reincarnation or the reality of Tulkus, we need to accept the existence of past and future lives.

• And there are logical arguments such as (from the same reference):

  Generally, Buddhists believe that there is no beginning to birth and that once we achieve liberation from the cycle of existence by overcoming our karma and destructive emotions, we will not be reborn under the sway of these conditions. Therefore, Buddhists believe that there is an end to being reborn as a result of karma and destructive emotions, but most Buddhist philosophical schools do not accept that the mind-stream comes to an end. To reject past and future rebirth would contradict the Buddhist concept of the ground, path and result, which must be explained on the basis of the disciplined or undisciplined mind. If we accept this argument, logically, we would also have to accept that the world and its inhabitants come about without causes and conditions. Therefore, as long as you are a Buddhist, it is necessary to accept past and future rebirth.

  For those who remember their past lives, rebirth is a clear experience. However, most ordinary beings forget their past lives as they go through the process of death, intermediate state and rebirth. As past and future rebirths are slightly obscure to them, we need to use evidence-based logic to prove past and future rebirths to them.

  There are many different logical arguments given in the words of the Buddha and subsequent commentaries to prove the existence of past and future lives. In brief, they come down to four points: the logic that things are preceded by things of a similar type, the logic that things are preceded by a substantial cause, the logic that the mind has gained familiarity with things in the past, and the logic of having gained experience of things in the past.

  Ultimately all these arguments are based on the idea that the nature of the mind, its clarity and awareness, must have clarity and awareness as its substantial cause. It cannot have any other entity such as an inanimate object as its substantial cause. This is self-evident. Through logical analysis we infer that a new stream of clarity and awareness cannot come about without causes or from unrelated causes. While we observe that mind cannot be produced in a laboratory, we also infer that nothing can eliminate the continuity of subtle clarity and awareness.

I don't understand these arguments well enough to argue them myself: I could only quote them. If you want to question one of these arguments, try posting that as a new question tagged tibetan-buddhism.

I have been unable to find the word anumāna in the Pali suttas but found it in the later-day Milindapañha, as follows:

He who, himself set free in that bless’d state In which the Upadhis have ceased to be, —Lusts, sin, and Karma—has brought safe ashore, Saved from the sea of woe, great multitudes— Only by inference can it be known That he, the best of men, existed once. Mil 6.4 1

In AN 2.25 there is the word nītattha (inferred meaning), which I have been unable to find elsewhere in the Pali suttas (although the prefix 'nita' is often used unfavourably, such as 'led' by ignorance or 'led' to hell).

The Dhamma of the Buddha is explicitly defined in the Pali suttas as follows:

The Dhamma is well-expounded by the Blessed One, to be seen here & now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.' AN 4.92

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Monks, in this Teaching that is so well proclaimed by me and is plain, open, explicit and free of patchwork... MN 22

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I have set forth the Dhamma without making any distinction of esoteric and exoteric doctrine; there is nothing, Ananda, with regard to the teachings that the Tathagata holds to the last with the closed fist of a teacher who keeps some things back. DN 16

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The metaphysical ideas about reincarnation or rebirth, such as Jataka Stories, relinking consciousness (paṭisandhi-viññāṇa), 3-lifetime dependent origination, etc, were obviously brought into Buddhism at a time after the Buddha's passing, probably during to era of King Ashoka, who was zealous in growing the religion socially, Since the core essence of Buddhism does not appeal to common people, a religion would have to be created to make Buddhism appealing, which ultimately resulted in the demise of Buddhism in India (per common scholarly opinion) because Buddhism became virtually indistinguishable from Hinduism.

Buddhism's distinctiveness diminished with the rise of Hindu sects. Though Mahayana writers were quite critical of Hinduism, the devotional cults of Mahayana Buddhism and Hinduism likely seemed quite similar to laity, and the developing Tantrism of both religions were also similar. Buddhist ideas, and even the Buddha himself, were absorbed and adapted into orthodox Hindu thought,while the differences between the two systems of thought were emphasized. Wikipedia

If you are Indian, you must know the word 'jati', which is the central term used in Pali Buddhism, does not mean physical birth or childbirth. It means self or social identity or caste:

Jāti (in Devanagari: जाति, Bengali: জাতি, Telugu:జాతి, Kannada:ಜಾತಿ, Malayalam: ജാതി, Tamil:ஜாதி, literally "birth") is a group of clans, tribes, communities and sub-communities, and religions in India. Each jāti typically has an association with a traditional job function or tribe. Religious beliefs (e.g. Sri Vaishnavism or Veera Shaivism) or linguistic groupings may define some jatis. A person's surname typically reflects a community (jati) association: thus Gandhi = perfume seller, Dhobi = washerman, Srivastava = military scribe, etc. Wikipedia

In summary, the numerous words in the Pali scriptures, commonly universally translated as "rebirth", simply refer to the re-arising of 'self-view' (see discussion here); just as the words 'pubbe nivasa', which is wrongly translated as 'past lives', really mean 'past adherences', namely, when in the past the mind ignorantly clung to the five aggregates as 'self'.

Thus, when the Buddha recollected his "past lives" or "past adherences" (properly translated as "past abodes" or "past dwellings", literally "former homes"), he recollected each time the mind in the past ignorantly gave "birth" ('jati') to the idea of 'self', as explicitly explained in the Khajjaniya Sutta, Haliddakani Sutta and elsewhere.

153. Through many a birth in samsara have I wandered in vain, seeking the builder of this house. Repeated birth is indeed suffering!

154. O house-builder, you are seen! You will not build this house again. For your rafters are broken and your ridgepole shattered. My mind has reached the Unconditioned; I have attained the destruction of craving.

First Words of the Buddha: Dhammapada 153

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In his discussions with Brahmans (MN 95; DN 13), monks (MN 38) and even lay people (AN 3.65), the Buddha often condemned the idea of blind faith or unsubstantiated beliefs.