What are the traditional Buddhist arguments for rebirth? According to Buddhist philosophy, there are two Pramanas or means of valid knowledge: Pratyaksha or sensory perception and Anumana or inference.  (This is in contrast to most Hindus who believe in three or more Pramanas.) My question is, what are the traditional Buddhist arguments for the existence of rebirth?

Now at least ordinary humans don't observe rebirth directly, so I assume that these arguments will rely on Anumana.  I've seen Hindu arguments that use Anumana to prove the existence of rebirth, but those arguments are about establishing the existence of the Atma or soul, which Buddhists reject.  So I assume that traditional Buddhist works have a different Anumana-based argument than the one Hindus use.
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 &amp; now, timeless, inviting verification, pertinent, to be realized by the
  wise for themselves.' AN 4.92
  
  
  
  Monks, in this Teaching that is so well proclaimed by me and is plain, open, explicit and free of patchwork... MN 22
  
  
  
  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
  
  


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
  
  


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.